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

Percentage Accurate: 100.0% → 100.0%

Time: 11.5s

Alternatives: 19

Speedup: 1.0×

• 58.6% 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: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+125}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+67}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -5e+125) (exp x) (if (<= t_0 2e+67) (exp (- z)) (pow y y)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y * log(y));
	double tmp;
	if (t_0 <= -5e+125) {
		tmp = exp(x);
	} else if (t_0 <= 2e+67) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y * log(y))
    if (t_0 <= (-5d+125)) then
        tmp = exp(x)
    else if (t_0 <= 2d+67) then
        tmp = exp(-z)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y * Math.log(y));
	double tmp;
	if (t_0 <= -5e+125) {
		tmp = Math.exp(x);
	} else if (t_0 <= 2e+67) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y * math.log(y))
	tmp = 0
	if t_0 <= -5e+125:
		tmp = math.exp(x)
	elif t_0 <= 2e+67:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y * log(y)))
	tmp = 0.0
	if (t_0 <= -5e+125)
		tmp = exp(x);
	elseif (t_0 <= 2e+67)
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y * log(y));
	tmp = 0.0;
	if (t_0 <= -5e+125)
		tmp = exp(x);
	elseif (t_0 <= 2e+67)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+125], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 2e+67], N[Exp[(-z)], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -4.99999999999999962e125

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.2%

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

   if -4.99999999999999962e125 < (+.f64 x (*.f64 y (log.f64 y))) < 1.99999999999999997e67

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 84.4

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

   5. Simplified 84.4%

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

   if 1.99999999999999997e67 < (+.f64 x (*.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

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}} \]

      3. log-rec N/A

         \[\leadsto e^{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right)} \]

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 76.0

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

   5. Simplified 76.0%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. pow-lowering-pow.f64 76.0

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

   7. Applied egg-rr 76.0%

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

Alternative 3: 32.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := \left(x \cdot x\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+20}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (* (* x x) 0.5)))
   (if (<= t_0 -2e+26) t_1 (if (<= t_0 2e+20) (+ x 1.0) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -2e+26) {
		tmp = t_1;
	} else if (t_0 <= 2e+20) {
		tmp = x + 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + (y * log(y))) - z
    t_1 = (x * x) * 0.5d0
    if (t_0 <= (-2d+26)) then
        tmp = t_1
    else if (t_0 <= 2d+20) then
        tmp = x + 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + (y * Math.log(y))) - z;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -2e+26) {
		tmp = t_1;
	} else if (t_0 <= 2e+20) {
		tmp = x + 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + (y * math.log(y))) - z
	t_1 = (x * x) * 0.5
	tmp = 0
	if t_0 <= -2e+26:
		tmp = t_1
	elif t_0 <= 2e+20:
		tmp = x + 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	t_1 = Float64(Float64(x * x) * 0.5)
	tmp = 0.0
	if (t_0 <= -2e+26)
		tmp = t_1;
	elseif (t_0 <= 2e+20)
		tmp = Float64(x + 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + (y * log(y))) - z;
	t_1 = (x * x) * 0.5;
	tmp = 0.0;
	if (t_0 <= -2e+26)
		tmp = t_1;
	elseif (t_0 <= 2e+20)
		tmp = x + 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+26], t$95$1, If[LessEqual[t$95$0, 2e+20], N[(x + 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -2.0000000000000001e26 or 2e20 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) 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

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

   5. Simplified 45.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 22.5

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 22.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 28.7

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

   11. Simplified 28.7%

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

   if -2.0000000000000001e26 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 2e20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 88.8%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 75.5

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

   8. Simplified 75.5%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq -2 \cdot 10^{+26}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(x + y \cdot \log y\right) - z \leq 2 \cdot 10^{+20}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{t\_0}{0.25}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-178}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{t\_0}{z \cdot \left(z \cdot 0.027777777777777776\right)}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* z (* z z)) -0.004629629629629629 0.125)))
   (if (<= z -1e+62)
     (fma z (fma z (/ t_0 0.25) -1.0) 1.0)
     (if (<= z -2.35e-178)
       (exp x)
       (if (<= z -1.25e-304)
         (fma z (fma z (/ t_0 (* z (* z 0.027777777777777776))) -1.0) 1.0)
         (if (<= z 5e+80) (exp x) (* (* x x) (* x 0.16666666666666666))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z * (z * z)), -0.004629629629629629, 0.125);
	double tmp;
	if (z <= -1e+62) {
		tmp = fma(z, fma(z, (t_0 / 0.25), -1.0), 1.0);
	} else if (z <= -2.35e-178) {
		tmp = exp(x);
	} else if (z <= -1.25e-304) {
		tmp = fma(z, fma(z, (t_0 / (z * (z * 0.027777777777777776))), -1.0), 1.0);
	} else if (z <= 5e+80) {
		tmp = exp(x);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125)
	tmp = 0.0
	if (z <= -1e+62)
		tmp = fma(z, fma(z, Float64(t_0 / 0.25), -1.0), 1.0);
	elseif (z <= -2.35e-178)
		tmp = exp(x);
	elseif (z <= -1.25e-304)
		tmp = fma(z, fma(z, Float64(t_0 / Float64(z * Float64(z * 0.027777777777777776))), -1.0), 1.0);
	elseif (z <= 5e+80)
		tmp = exp(x);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision]}, If[LessEqual[z, -1e+62], N[(z * N[(z * N[(t$95$0 / 0.25), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, -2.35e-178], N[Exp[x], $MachinePrecision], If[LessEqual[z, -1.25e-304], N[(z * N[(z * N[(t$95$0 / N[(z * N[(z * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 5e+80], N[Exp[x], $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.00000000000000004e62

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 91.4

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

   5. Simplified 91.4%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 83.2

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 83.2%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

      5. unpow-prod-down N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \color{blue}{\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}}\right)}, -1\right), 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}}, -1\right), 1\right) \]

      16. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}, -1\right), 1\right) \]

      17. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{z \cdot \left(\frac{-1}{6} \cdot \frac{1}{2}\right)}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{1}{2}\right)\right)}\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 14.6%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25 + \mathsf{fma}\left(z \cdot z, 0.027777777777777776, z \cdot 0.08333333333333333\right)}}, -1\right), 1\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

   13. Simplified 91.4%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, -1\right), 1\right) \]

   if -1.00000000000000004e62 < z < -2.35e-178 or -1.24999999999999991e-304 < z < 4.99999999999999961e80

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.9%

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

   if -2.35e-178 < z < -1.24999999999999991e-304

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 8.2

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

   5. Simplified 8.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 8.2

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 8.2%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

      5. unpow-prod-down N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \color{blue}{\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}}\right)}, -1\right), 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}}, -1\right), 1\right) \]

      16. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}, -1\right), 1\right) \]

      17. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{z \cdot \left(\frac{-1}{6} \cdot \frac{1}{2}\right)}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{1}{2}\right)\right)}\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 8.2%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25 + \mathsf{fma}\left(z \cdot z, 0.027777777777777776, z \cdot 0.08333333333333333\right)}}, -1\right), 1\right) \]

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{36} \cdot {z}^{2}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{36} \cdot \color{blue}{\left(z \cdot z\right)}}, -1\right), 1\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\frac{1}{36} \cdot z\right) \cdot z}}, -1\right), 1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{z \cdot \left(\frac{1}{36} \cdot z\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{z \cdot \left(\frac{1}{36} \cdot z\right)}}, -1\right), 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{z \cdot \color{blue}{\left(z \cdot \frac{1}{36}\right)}}, -1\right), 1\right) \]

       6. *-lowering-*.f64 94.9

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{z \cdot \color{blue}{\left(z \cdot 0.027777777777777776\right)}}, -1\right), 1\right) \]

   13. Simplified 94.9%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{\color{blue}{z \cdot \left(z \cdot 0.027777777777777776\right)}}, -1\right), 1\right) \]

   if 4.99999999999999961e80 < 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

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

   5. Simplified 32.1%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 11.8

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 11.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 45.6

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

   11. Simplified 45.6%

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

Alternative 5: 74.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 140:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 140.0) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 140.0) {
		tmp = exp(x);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 140.0d0) then
        tmp = exp(x)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 140.0) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 140.0:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 140.0)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 140.0)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 140.0], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 140

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 63.9%

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

   if 140 < 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

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

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

         \[\leadsto e^{\color{blue}{y \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}} \]

      3. log-rec N/A

         \[\leadsto e^{y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right)} \]

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 82.0

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

   5. Simplified 82.0%

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. pow-lowering-pow.f64 82.0

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

   7. Applied egg-rr 82.0%

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

Alternative 6: 52.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{t\_0}{0.25}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{t\_0}{z \cdot \left(z \cdot 0.027777777777777776\right)}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 - \frac{4 + \frac{\frac{16}{z} + -8}{z}}{z}}{z}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* z (* z z)) -0.004629629629629629 0.125)))
   (if (<= z -1.32e+23)
     (fma z (fma z (/ t_0 0.25) -1.0) 1.0)
     (if (<= z -1.75e-164)
       (fma x (fma x 0.5 1.0) 1.0)
       (if (<= z 1.7e-153)
         (fma z (fma z (/ t_0 (* z (* z 0.027777777777777776))) -1.0) 1.0)
         (if (<= z 2.3e-68)
           (fma
            (* z (fma 0.25 (* z z) -1.0))
            (/ (- 2.0 (/ (+ 4.0 (/ (+ (/ 16.0 z) -8.0) z)) z)) z)
            1.0)
           (* (* x x) (* x 0.16666666666666666))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z * (z * z)), -0.004629629629629629, 0.125);
	double tmp;
	if (z <= -1.32e+23) {
		tmp = fma(z, fma(z, (t_0 / 0.25), -1.0), 1.0);
	} else if (z <= -1.75e-164) {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	} else if (z <= 1.7e-153) {
		tmp = fma(z, fma(z, (t_0 / (z * (z * 0.027777777777777776))), -1.0), 1.0);
	} else if (z <= 2.3e-68) {
		tmp = fma((z * fma(0.25, (z * z), -1.0)), ((2.0 - ((4.0 + (((16.0 / z) + -8.0) / z)) / z)) / z), 1.0);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125)
	tmp = 0.0
	if (z <= -1.32e+23)
		tmp = fma(z, fma(z, Float64(t_0 / 0.25), -1.0), 1.0);
	elseif (z <= -1.75e-164)
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	elseif (z <= 1.7e-153)
		tmp = fma(z, fma(z, Float64(t_0 / Float64(z * Float64(z * 0.027777777777777776))), -1.0), 1.0);
	elseif (z <= 2.3e-68)
		tmp = fma(Float64(z * fma(0.25, Float64(z * z), -1.0)), Float64(Float64(2.0 - Float64(Float64(4.0 + Float64(Float64(Float64(16.0 / z) + -8.0) / z)) / z)) / z), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision]}, If[LessEqual[z, -1.32e+23], N[(z * N[(z * N[(t$95$0 / 0.25), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, -1.75e-164], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1.7e-153], N[(z * N[(z * N[(t$95$0 / N[(z * N[(z * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 2.3e-68], N[(N[(z * N[(0.25 * N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 - N[(N[(4.0 + N[(N[(N[(16.0 / z), $MachinePrecision] + -8.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.3199999999999999e23

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 89.2

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

   5. Simplified 89.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 74.5%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

      5. unpow-prod-down N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \color{blue}{\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}}\right)}, -1\right), 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}}, -1\right), 1\right) \]

      16. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}, -1\right), 1\right) \]

      17. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{z \cdot \left(\frac{-1}{6} \cdot \frac{1}{2}\right)}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{1}{2}\right)\right)}\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 13.4%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25 + \mathsf{fma}\left(z \cdot z, 0.027777777777777776, z \cdot 0.08333333333333333\right)}}, -1\right), 1\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

   13. Simplified 81.8%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, -1\right), 1\right) \]

   if -1.3199999999999999e23 < z < -1.75e-164

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 55.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 43.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   if -1.75e-164 < z < 1.6999999999999999e-153

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 18.7

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

   5. Simplified 18.7%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 18.7

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 18.7%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

      5. unpow-prod-down N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \color{blue}{\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}}\right)}, -1\right), 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}}, -1\right), 1\right) \]

      16. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}, -1\right), 1\right) \]

      17. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{z \cdot \left(\frac{-1}{6} \cdot \frac{1}{2}\right)}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{1}{2}\right)\right)}\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 18.7%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25 + \mathsf{fma}\left(z \cdot z, 0.027777777777777776, z \cdot 0.08333333333333333\right)}}, -1\right), 1\right) \]

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{36} \cdot {z}^{2}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{36} \cdot \color{blue}{\left(z \cdot z\right)}}, -1\right), 1\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\frac{1}{36} \cdot z\right) \cdot z}}, -1\right), 1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{z \cdot \left(\frac{1}{36} \cdot z\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{z \cdot \left(\frac{1}{36} \cdot z\right)}}, -1\right), 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{z \cdot \color{blue}{\left(z \cdot \frac{1}{36}\right)}}, -1\right), 1\right) \]

       6. *-lowering-*.f64 68.9

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{z \cdot \color{blue}{\left(z \cdot 0.027777777777777776\right)}}, -1\right), 1\right) \]

   13. Simplified 68.9%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{\color{blue}{z \cdot \left(z \cdot 0.027777777777777776\right)}}, -1\right), 1\right) \]

   if 1.6999999999999999e-153 < z < 2.29999999999999997e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 14.5

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

   5. Simplified 14.5%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 14.5

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 14.5%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4}}, z \cdot z, -1\right) \cdot z, \frac{1}{\frac{1}{2} \cdot z - -1}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, \color{blue}{z \cdot z}, -1\right) \cdot z, \frac{1}{\frac{1}{2} \cdot z - -1}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \color{blue}{\frac{1}{\frac{1}{2} \cdot z - -1}}, 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{1}{\color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(-1\right)\right)}}, 1\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{1}{\color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(-1\right)\right)}, 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{1}{z \cdot \frac{1}{2} + \color{blue}{1}}, 1\right) \]

      18. accelerator-lowering-fma.f64 14.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \frac{1}{\color{blue}{\mathsf{fma}\left(z, 0.5, 1\right)}}, 1\right) \]

   10. Applied egg-rr 14.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)} \]

   11. Taylor expanded in z around -inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{16 \cdot \frac{1}{z} - 8}{z} - 4}{z} - 2}{z}}, 1\right) \]

   13. Simplified 59.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \color{blue}{\frac{2 - \frac{4 + \frac{\frac{16}{z} + -8}{z}}{z}}{z}}, 1\right) \]

   if 2.29999999999999997e-68 < 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

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

   5. Simplified 41.1%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 18.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 18.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 39.0

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

   11. Simplified 39.0%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{z \cdot \left(z \cdot 0.027777777777777776\right)}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 - \frac{4 + \frac{\frac{16}{z} + -8}{z}}{z}}{z}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{t\_0}{0.25}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{t\_0}{z \cdot \left(z \cdot 0.027777777777777776\right)}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 660:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* z (* z z)) -0.004629629629629629 0.125)))
   (if (<= z -3.5e+22)
     (fma z (fma z (/ t_0 0.25) -1.0) 1.0)
     (if (<= z -4.5e-164)
       (fma x (fma x 0.5 1.0) 1.0)
       (if (<= z 1.45e-154)
         (fma z (fma z (/ t_0 (* z (* z 0.027777777777777776))) -1.0) 1.0)
         (if (<= z 660.0)
           (fma x (* x (fma x 0.16666666666666666 0.5)) 1.0)
           (* (* x x) (* x 0.16666666666666666))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z * (z * z)), -0.004629629629629629, 0.125);
	double tmp;
	if (z <= -3.5e+22) {
		tmp = fma(z, fma(z, (t_0 / 0.25), -1.0), 1.0);
	} else if (z <= -4.5e-164) {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	} else if (z <= 1.45e-154) {
		tmp = fma(z, fma(z, (t_0 / (z * (z * 0.027777777777777776))), -1.0), 1.0);
	} else if (z <= 660.0) {
		tmp = fma(x, (x * fma(x, 0.16666666666666666, 0.5)), 1.0);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125)
	tmp = 0.0
	if (z <= -3.5e+22)
		tmp = fma(z, fma(z, Float64(t_0 / 0.25), -1.0), 1.0);
	elseif (z <= -4.5e-164)
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	elseif (z <= 1.45e-154)
		tmp = fma(z, fma(z, Float64(t_0 / Float64(z * Float64(z * 0.027777777777777776))), -1.0), 1.0);
	elseif (z <= 660.0)
		tmp = fma(x, Float64(x * fma(x, 0.16666666666666666, 0.5)), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision]}, If[LessEqual[z, -3.5e+22], N[(z * N[(z * N[(t$95$0 / 0.25), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, -4.5e-164], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1.45e-154], N[(z * N[(z * N[(t$95$0 / N[(z * N[(z * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 660.0], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -3.5e22

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 89.2

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

   5. Simplified 89.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 74.5%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

      5. unpow-prod-down N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \color{blue}{\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}}\right)}, -1\right), 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}}, -1\right), 1\right) \]

      16. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}, -1\right), 1\right) \]

      17. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{z \cdot \left(\frac{-1}{6} \cdot \frac{1}{2}\right)}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{1}{2}\right)\right)}\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 13.4%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25 + \mathsf{fma}\left(z \cdot z, 0.027777777777777776, z \cdot 0.08333333333333333\right)}}, -1\right), 1\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

   13. Simplified 81.8%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, -1\right), 1\right) \]

   if -3.5e22 < z < -4.4999999999999997e-164

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 55.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 43.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   if -4.4999999999999997e-164 < z < 1.45e-154

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 17.2

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

   5. Simplified 17.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 17.2

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 17.2%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

      5. unpow-prod-down N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \color{blue}{\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}}\right)}, -1\right), 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}}, -1\right), 1\right) \]

      16. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}, -1\right), 1\right) \]

      17. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{z \cdot \left(\frac{-1}{6} \cdot \frac{1}{2}\right)}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{1}{2}\right)\right)}\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 17.2%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25 + \mathsf{fma}\left(z \cdot z, 0.027777777777777776, z \cdot 0.08333333333333333\right)}}, -1\right), 1\right) \]

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{36} \cdot {z}^{2}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{36} \cdot \color{blue}{\left(z \cdot z\right)}}, -1\right), 1\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\left(\frac{1}{36} \cdot z\right) \cdot z}}, -1\right), 1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{z \cdot \left(\frac{1}{36} \cdot z\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{z \cdot \left(\frac{1}{36} \cdot z\right)}}, -1\right), 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{z \cdot \color{blue}{\left(z \cdot \frac{1}{36}\right)}}, -1\right), 1\right) \]

       6. *-lowering-*.f64 69.8

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{z \cdot \color{blue}{\left(z \cdot 0.027777777777777776\right)}}, -1\right), 1\right) \]

   13. Simplified 69.8%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{\color{blue}{z \cdot \left(z \cdot 0.027777777777777776\right)}}, -1\right), 1\right) \]

   if 1.45e-154 < z < 660

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 69.6%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 42.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 42.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. associate-*l* N/A

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

       6. lft-mult-inverse N/A

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

       7. metadata-eval N/A

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

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

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. accelerator-lowering-fma.f64 42.1

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

   11. Simplified 42.1%

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

   if 660 < 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

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

   5. Simplified 34.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 12.7

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 12.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 42.3

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

   11. Simplified 42.3%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 50.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-228}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4}{z}}{z}, 1\right)\\ \mathbf{elif}\;z \leq 580:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.32e+23)
   (fma
    z
    (fma z (/ (fma (* z (* z z)) -0.004629629629629629 0.125) 0.25) -1.0)
    1.0)
   (if (<= z 2.9e-228)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (if (<= z 1.45e-154)
       (fma (* z (fma 0.25 (* z z) -1.0)) (/ (+ 2.0 (/ -4.0 z)) z) 1.0)
       (if (<= z 580.0)
         (fma x (* x (fma x 0.16666666666666666 0.5)) 1.0)
         (* (* x x) (* x 0.16666666666666666)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.32e+23) {
		tmp = fma(z, fma(z, (fma((z * (z * z)), -0.004629629629629629, 0.125) / 0.25), -1.0), 1.0);
	} else if (z <= 2.9e-228) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else if (z <= 1.45e-154) {
		tmp = fma((z * fma(0.25, (z * z), -1.0)), ((2.0 + (-4.0 / z)) / z), 1.0);
	} else if (z <= 580.0) {
		tmp = fma(x, (x * fma(x, 0.16666666666666666, 0.5)), 1.0);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.32e+23)
		tmp = fma(z, fma(z, Float64(fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125) / 0.25), -1.0), 1.0);
	elseif (z <= 2.9e-228)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	elseif (z <= 1.45e-154)
		tmp = fma(Float64(z * fma(0.25, Float64(z * z), -1.0)), Float64(Float64(2.0 + Float64(-4.0 / z)) / z), 1.0);
	elseif (z <= 580.0)
		tmp = fma(x, Float64(x * fma(x, 0.16666666666666666, 0.5)), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.32e+23], N[(z * N[(z * N[(N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 2.9e-228], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1.45e-154], N[(N[(z * N[(0.25 * N[(z * z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(-4.0 / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 580.0], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.3199999999999999e23

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 89.2

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

   5. Simplified 89.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 74.5%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

      5. unpow-prod-down N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \color{blue}{\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}}\right)}, -1\right), 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}}, -1\right), 1\right) \]

      16. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}, -1\right), 1\right) \]

      17. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{z \cdot \left(\frac{-1}{6} \cdot \frac{1}{2}\right)}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{1}{2}\right)\right)}\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 13.4%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25 + \mathsf{fma}\left(z \cdot z, 0.027777777777777776, z \cdot 0.08333333333333333\right)}}, -1\right), 1\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

   13. Simplified 81.8%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, -1\right), 1\right) \]

   if -1.3199999999999999e23 < z < 2.9000000000000001e-228

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 59.0%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 42.7

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   if 2.9000000000000001e-228 < z < 1.45e-154

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 12.4

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

   5. Simplified 12.4%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 12.4

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 12.4%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4}}, z \cdot z, -1\right) \cdot z, \frac{1}{\frac{1}{2} \cdot z - -1}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, \color{blue}{z \cdot z}, -1\right) \cdot z, \frac{1}{\frac{1}{2} \cdot z - -1}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \color{blue}{\frac{1}{\frac{1}{2} \cdot z - -1}}, 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{1}{\color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(-1\right)\right)}}, 1\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{1}{\color{blue}{z \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(-1\right)\right)}, 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{1}{z \cdot \frac{1}{2} + \color{blue}{1}}, 1\right) \]

      18. accelerator-lowering-fma.f64 12.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \frac{1}{\color{blue}{\mathsf{fma}\left(z, 0.5, 1\right)}}, 1\right) \]

   10. Applied egg-rr 12.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \frac{1}{\mathsf{fma}\left(z, 0.5, 1\right)}, 1\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \color{blue}{\frac{2 - 4 \cdot \frac{1}{z}}{z}}, 1\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \color{blue}{\frac{2 - 4 \cdot \frac{1}{z}}{z}}, 1\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{\color{blue}{2 + \left(\mathsf{neg}\left(4 \cdot \frac{1}{z}\right)\right)}}{z}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{\color{blue}{2 + \left(\mathsf{neg}\left(4 \cdot \frac{1}{z}\right)\right)}}{z}, 1\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{2 + \left(\mathsf{neg}\left(\color{blue}{\frac{4 \cdot 1}{z}}\right)\right)}{z}, 1\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{2 + \left(\mathsf{neg}\left(\frac{\color{blue}{4}}{z}\right)\right)}{z}, 1\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{2 + \color{blue}{\frac{\mathsf{neg}\left(4\right)}{z}}}{z}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, z \cdot z, -1\right) \cdot z, \frac{2 + \color{blue}{\frac{\mathsf{neg}\left(4\right)}{z}}}{z}, 1\right) \]

       8. metadata-eval 72.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \frac{2 + \frac{\color{blue}{-4}}{z}}{z}, 1\right) \]

   13. Simplified 72.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25, z \cdot z, -1\right) \cdot z, \color{blue}{\frac{2 + \frac{-4}{z}}{z}}, 1\right) \]

   if 1.45e-154 < z < 580

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 69.6%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 42.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 42.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. associate-*l* N/A

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

       6. lft-mult-inverse N/A

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

       7. metadata-eval N/A

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

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

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. accelerator-lowering-fma.f64 42.1

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

   11. Simplified 42.1%

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

   if 580 < 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

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

   5. Simplified 34.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 12.7

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 12.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 42.3

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

   11. Simplified 42.3%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-228}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(0.25, z \cdot z, -1\right), \frac{2 + \frac{-4}{z}}{z}, 1\right)\\ \mathbf{elif}\;z \leq 580:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 1020:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.5e+20)
   (fma
    z
    (fma z (/ (fma (* z (* z z)) -0.004629629629629629 0.125) 0.25) -1.0)
    1.0)
   (if (<= z 1020.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (* (* x x) (* x 0.16666666666666666)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.5e+20) {
		tmp = fma(z, fma(z, (fma((z * (z * z)), -0.004629629629629629, 0.125) / 0.25), -1.0), 1.0);
	} else if (z <= 1020.0) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.5e+20)
		tmp = fma(z, fma(z, Float64(fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125) / 0.25), -1.0), 1.0);
	elseif (z <= 1020.0)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.5e+20], N[(z * N[(z * N[(N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision] / 0.25), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1020.0], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5e20

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 89.2

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

   5. Simplified 89.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 74.5%

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

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. +-commutative N/A

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

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

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

      5. unpow-prod-down N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \color{blue}{\frac{-1}{216}}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}}, -1\right), 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \frac{1}{2} \cdot \left(z \cdot \frac{-1}{6}\right)\right)}, -1\right), 1\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) - \color{blue}{\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}}\right)}, -1\right), 1\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\left(\left(z \cdot \frac{-1}{6}\right) \cdot \left(z \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}}, -1\right), 1\right) \]

      16. swap-sqr N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)\right)}, -1\right), 1\right) \]

      17. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \color{blue}{\mathsf{fma}\left(z \cdot z, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(\color{blue}{z \cdot z}, \frac{-1}{6} \cdot \frac{-1}{6}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \color{blue}{\frac{1}{36}}, \mathsf{neg}\left(\left(z \cdot \frac{-1}{6}\right) \cdot \frac{1}{2}\right)\right)}, -1\right), 1\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \mathsf{neg}\left(\color{blue}{z \cdot \left(\frac{-1}{6} \cdot \frac{1}{2}\right)}\right)\right)}, -1\right), 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\frac{1}{4} + \mathsf{fma}\left(z \cdot z, \frac{1}{36}, \color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{1}{2}\right)\right)}\right)}, -1\right), 1\right) \]

   10. Applied egg-rr 13.4%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{0.25 + \mathsf{fma}\left(z \cdot z, 0.027777777777777776, z \cdot 0.08333333333333333\right)}}, -1\right), 1\right) \]

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

   13. Simplified 81.8%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)}{\color{blue}{0.25}}, -1\right), 1\right) \]

   if -6.5e20 < z < 1020

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 63.1%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 40.6

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 40.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   if 1020 < 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

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

   5. Simplified 34.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 12.7

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 12.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 42.3

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

   11. Simplified 42.3%

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

Alternative 10: 47.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 520:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+88)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= z 520.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (* (* x x) (* x 0.16666666666666666)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+88) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (z <= 520.0) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+88)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (z <= 520.0)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+88], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[z, 520.0], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.50000000000000003e88

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 90.9

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

   5. Simplified 90.9%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 87.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

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

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 87.5

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

   11. Simplified 87.5%

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

   if -1.50000000000000003e88 < z < 520

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 39.2

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   if 520 < 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

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

   5. Simplified 34.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 12.7

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 12.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 42.3

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

   11. Simplified 42.3%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 520:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 580:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+88)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= z 580.0)
     (fma x (* x (fma x 0.16666666666666666 0.5)) 1.0)
     (* (* x x) (* x 0.16666666666666666)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+88) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (z <= 580.0) {
		tmp = fma(x, (x * fma(x, 0.16666666666666666, 0.5)), 1.0);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+88)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (z <= 580.0)
		tmp = fma(x, Float64(x * fma(x, 0.16666666666666666, 0.5)), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+88], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[z, 580.0], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.50000000000000003e88

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 90.9

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

   5. Simplified 90.9%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 87.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

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

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 87.5

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

   11. Simplified 87.5%

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

   if -1.50000000000000003e88 < z < 580

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 39.2

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. associate-*l* N/A

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

       6. lft-mult-inverse N/A

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

       7. metadata-eval N/A

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

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

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. accelerator-lowering-fma.f64 39.1

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

   11. Simplified 39.1%

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

   if 580 < 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

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

   5. Simplified 34.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 12.7

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 12.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 42.3

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

   11. Simplified 42.3%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 580:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 15000000:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+88)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= z 15000000.0)
     (fma x (* x (* x 0.16666666666666666)) 1.0)
     (* (* x x) (* x 0.16666666666666666)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+88) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (z <= 15000000.0) {
		tmp = fma(x, (x * (x * 0.16666666666666666)), 1.0);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+88)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (z <= 15000000.0)
		tmp = fma(x, Float64(x * Float64(x * 0.16666666666666666)), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+88], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[z, 15000000.0], N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.50000000000000003e88

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 90.9

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

   5. Simplified 90.9%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 87.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

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

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 87.5

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

   11. Simplified 87.5%

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

   if -1.50000000000000003e88 < z < 1.5e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 62.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 39.2

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 39.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, 1\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x}, 1\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot x\right)}, 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot x\right)}, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \frac{1}{6}\right)}, 1\right) \]

       6. *-lowering-*.f64 39.1

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot 0.16666666666666666\right)}, 1\right) \]

   11. Simplified 39.1%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right)}, 1\right) \]

   if 1.5e7 < 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

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

   5. Simplified 34.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 12.7

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 12.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 42.3

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

   11. Simplified 42.3%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 15000000:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \left(x \cdot 0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+88)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= z 1.7e+39)
     (fma x (fma x 0.5 1.0) 1.0)
     (* (* x x) (* x 0.16666666666666666)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+88) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (z <= 1.7e+39) {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	} else {
		tmp = (x * x) * (x * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+88)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (z <= 1.7e+39)
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(x * 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+88], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[z, 1.7e+39], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.50000000000000003e88

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 90.9

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

   5. Simplified 90.9%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 87.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

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

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 87.5

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

   11. Simplified 87.5%

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

   if -1.50000000000000003e88 < z < 1.6999999999999999e39

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 38.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 38.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   if 1.6999999999999999e39 < 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

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

   5. Simplified 35.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 11.8

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

   8. Simplified 11.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 41.8

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

   11. Simplified 41.8%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 390:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.5e+88)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= z 390.0) (fma x (fma x 0.5 1.0) 1.0) (* (* x x) 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.5e+88) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (z <= 390.0) {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.5e+88)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (z <= 390.0)
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	else
		tmp = Float64(Float64(x * x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.5e+88], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[z, 390.0], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.50000000000000003e88

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 90.9

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

   5. Simplified 90.9%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 87.5

         \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   9. Taylor expanded in z around inf

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

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

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {z}^{3}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 87.5

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

   11. Simplified 87.5%

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

   if -1.50000000000000003e88 < 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

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

   5. Simplified 62.2%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 38.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 38.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   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

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

   5. Simplified 34.7%

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

   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. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 12.9

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 12.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 35.0

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

   11. Simplified 35.0%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 390:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;z \leq 700000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+137)
   (* (* z z) 0.5)
   (if (<= z 700000000.0) (fma x (fma x 0.5 1.0) 1.0) (* (* x x) 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+137) {
		tmp = (z * z) * 0.5;
	} else if (z <= 700000000.0) {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e+137)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (z <= 700000000.0)
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	else
		tmp = Float64(Float64(x * x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e+137], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[z, 700000000.0], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0000000000000001e137

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 89.7

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

   5. Simplified 89.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   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. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 82.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 82.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       3. *-lowering-*.f64 82.1

          \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot z\right)} \]

   11. Simplified 82.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if -2.0000000000000001e137 < z < 7e8

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto e^{\color{blue}{x}} \]
   4. Step-by-step derivation

   5. Simplified 61.1%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right) \]

      5. accelerator-lowering-fma.f64 37.6

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 37.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   if 7e8 < 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

      \[\leadsto e^{\color{blue}{x}} \]
   4. Step-by-step derivation

   5. Simplified 34.7%

      \[\leadsto e^{\color{blue}{x}} \]

   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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right) \]

      5. accelerator-lowering-fma.f64 12.9

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 12.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

       3. *-lowering-*.f64 35.0

          \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   11. Simplified 35.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;z \leq 700000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;z \leq 1200:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+137)
   (* (* z z) 0.5)
   (if (<= z 1200.0) (fma x (* x 0.5) 1.0) (* (* x x) 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+137) {
		tmp = (z * z) * 0.5;
	} else if (z <= 1200.0) {
		tmp = fma(x, (x * 0.5), 1.0);
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e+137)
		tmp = Float64(Float64(z * z) * 0.5);
	elseif (z <= 1200.0)
		tmp = fma(x, Float64(x * 0.5), 1.0);
	else
		tmp = Float64(Float64(x * x) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e+137], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[z, 1200.0], N[(x * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.0000000000000001e137

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 89.7

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 89.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   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. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 82.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 82.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       3. *-lowering-*.f64 82.1

          \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot z\right)} \]

   11. Simplified 82.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if -2.0000000000000001e137 < z < 1200

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto e^{\color{blue}{x}} \]
   4. Step-by-step derivation

   5. Simplified 61.1%

      \[\leadsto e^{\color{blue}{x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right) \]

      5. accelerator-lowering-fma.f64 37.6

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 37.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x}, 1\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 37.5

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.5 \cdot x}, 1\right) \]

   11. Simplified 37.5%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.5 \cdot x}, 1\right) \]

   if 1200 < 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

      \[\leadsto e^{\color{blue}{x}} \]
   4. Step-by-step derivation

   5. Simplified 34.7%

      \[\leadsto e^{\color{blue}{x}} \]

   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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right) \]

      5. accelerator-lowering-fma.f64 12.9

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 12.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

       3. *-lowering-*.f64 35.0

          \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   11. Simplified 35.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{elif}\;z \leq 1200:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+137) (* (* z z) 0.5) (* (* x x) 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+137) {
		tmp = (z * z) * 0.5;
	} else {
		tmp = (x * x) * 0.5;
	}
	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 <= (-2d+137)) then
        tmp = (z * z) * 0.5d0
    else
        tmp = (x * x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+137) {
		tmp = (z * z) * 0.5;
	} else {
		tmp = (x * x) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e+137:
		tmp = (z * z) * 0.5
	else:
		tmp = (x * x) * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e+137)
		tmp = Float64(Float64(z * z) * 0.5);
	else
		tmp = Float64(Float64(x * x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2e+137)
		tmp = (z * z) * 0.5;
	else
		tmp = (x * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e+137], N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e137

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. neg-lowering-neg.f64 89.7

         \[\leadsto e^{\color{blue}{-z}} \]

   5. Simplified 89.7%

      \[\leadsto e^{\color{blue}{-z}} \]

   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. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1}{2} \cdot z - 1\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot z - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot z + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{1}{2} \cdot z + \color{blue}{-1}, 1\right) \]

      5. accelerator-lowering-fma.f64 82.1

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(0.5, z, -1\right)}, 1\right) \]

   8. Simplified 82.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)} \]

   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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

       3. *-lowering-*.f64 82.1

          \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot z\right)} \]

   11. Simplified 82.1%

       \[\leadsto \color{blue}{0.5 \cdot \left(z \cdot z\right)} \]

   if -2.0000000000000001e137 < 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

      \[\leadsto e^{\color{blue}{x}} \]
   4. Step-by-step derivation

   5. Simplified 52.6%

      \[\leadsto e^{\color{blue}{x}} \]

   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. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, 1\right) \]

      5. accelerator-lowering-fma.f64 29.6

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right) \]

   8. Simplified 29.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]

   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

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

       3. *-lowering-*.f64 26.5

          \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   11. Simplified 26.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 14.4% accurate, 53.0× 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

   \[\leadsto e^{\color{blue}{x}} \]
4. Step-by-step derivation

5. Simplified 50.1%

   \[\leadsto e^{\color{blue}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 11.6

      \[\leadsto \color{blue}{1 + x} \]

8. Simplified 11.6%

   \[\leadsto \color{blue}{1 + x} \]

9. Final simplification 11.6%

   \[\leadsto x + 1 \]
10. Add Preprocessing

Alternative 19: 14.2% accurate, 212.0× 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

   \[\leadsto e^{\color{blue}{x}} \]
4. Step-by-step derivation

5. Simplified 50.1%

   \[\leadsto e^{\color{blue}{x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 11.3%

   \[\leadsto \color{blue}{1} \]
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 2024199 
(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)))