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

Percentage Accurate: 100.0% → 100.0%

Time: 11.3s

Alternatives: 22

Speedup: 1.0×

• 58.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+55}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 0.0005:\\ \;\;\;\;e^{0 - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -1e+55)
     (exp x)
     (if (<= t_0 0.0005) (exp (- 0.0 z)) (exp (fma y (log y) x))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y * log(y));
	double tmp;
	if (t_0 <= -1e+55) {
		tmp = exp(x);
	} else if (t_0 <= 0.0005) {
		tmp = exp((0.0 - z));
	} else {
		tmp = exp(fma(y, log(y), x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y * log(y)))
	tmp = 0.0
	if (t_0 <= -1e+55)
		tmp = exp(x);
	elseif (t_0 <= 0.0005)
		tmp = exp(Float64(0.0 - z));
	else
		tmp = exp(fma(y, log(y), x));
	end
	return 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, -1e+55], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 0.0005], N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(y * N[Log[y], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -1.00000000000000001e55

   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 97.5%

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

   if -1.00000000000000001e55 < (+.f64 x (*.f64 y (log.f64 y))) < 5.0000000000000001e-4

   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-sub0 N/A

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

      3. --lowering--.f64 99.0

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

   5. Simplified 99.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.0

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

   7. Applied egg-rr 99.0%

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

   if 5.0000000000000001e-4 < (+.f64 x (*.f64 y (log.f64 y)))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

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

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(y, \log y, x\right)}} \]

      4. log-lowering-log.f64 90.9

         \[\leadsto e^{\mathsf{fma}\left(y, \color{blue}{\log y}, x\right)} \]

   5. Simplified 90.9%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(y, \log y, x\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \cdot \log y \leq -1 \cdot 10^{+55}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x + y \cdot \log y \leq 0.0005:\\ \;\;\;\;e^{0 - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+55}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+50}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -1e+55) (exp x) (if (<= t_0 1e+50) (exp (- 0.0 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 <= -1e+55) {
		tmp = exp(x);
	} else if (t_0 <= 1e+50) {
		tmp = exp((0.0 - 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 <= (-1d+55)) then
        tmp = exp(x)
    else if (t_0 <= 1d+50) then
        tmp = exp((0.0d0 - 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 <= -1e+55) {
		tmp = Math.exp(x);
	} else if (t_0 <= 1e+50) {
		tmp = Math.exp((0.0 - 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 <= -1e+55:
		tmp = math.exp(x)
	elif t_0 <= 1e+50:
		tmp = math.exp((0.0 - 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 <= -1e+55)
		tmp = exp(x);
	elseif (t_0 <= 1e+50)
		tmp = exp(Float64(0.0 - 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 <= -1e+55)
		tmp = exp(x);
	elseif (t_0 <= 1e+50)
		tmp = exp((0.0 - 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, -1e+55], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 1e+50], N[Exp[N[(0.0 - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -1.00000000000000001e55

   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 97.5%

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

   if -1.00000000000000001e55 < (+.f64 x (*.f64 y (log.f64 y))) < 1.0000000000000001e50

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 92.9

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

   5. Simplified 92.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 92.9

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

   7. Applied egg-rr 92.9%

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

   if 1.0000000000000001e50 < (+.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 z around 0

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

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

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

      2. +-commutative N/A

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

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

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(y, \log y, x\right)}} \]

      4. log-lowering-log.f64 93.7

         \[\leadsto e^{\mathsf{fma}\left(y, \color{blue}{\log y}, x\right)} \]

   5. Simplified 93.7%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(y, \log y, x\right)}} \]

   6. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 76.7

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

   8. Simplified 76.7%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \cdot \log y \leq -1 \cdot 10^{+55}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x + y \cdot \log y \leq 10^{+50}:\\ \;\;\;\;e^{0 - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 200:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 200.0)
   (* (pow y y) (exp (- x z)))
   (exp (fma y (log y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 200.0) {
		tmp = pow(y, y) * exp((x - z));
	} else {
		tmp = exp(fma(y, log(y), x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * log(y)) <= 200.0)
		tmp = Float64((y ^ y) * exp(Float64(x - z)));
	else
		tmp = exp(fma(y, log(y), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision], 200.0], N[(N[Power[y, y], $MachinePrecision] * N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(y * N[Log[y], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 200

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. exp-sum N/A

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

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

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

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

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

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

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

      9. --lowering--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if 200 < (*.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 z around 0

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

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

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

      2. +-commutative N/A

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

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

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(y, \log y, x\right)}} \]

      4. log-lowering-log.f64 92.4

         \[\leadsto e^{\mathsf{fma}\left(y, \color{blue}{\log y}, x\right)} \]

   5. Simplified 92.4%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(y, \log y, x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 30.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y \cdot \log y\right) - z\\ t_1 := \mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+120}:\\ \;\;\;\;1 - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (* y (log y))) z)) (t_1 (fma 0.5 (* z z) 0.0)))
   (if (<= t_0 -1e+55) t_1 (if (<= t_0 5e+120) (- 1.0 z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + (y * log(y))) - z;
	double t_1 = fma(0.5, (z * z), 0.0);
	double tmp;
	if (t_0 <= -1e+55) {
		tmp = t_1;
	} else if (t_0 <= 5e+120) {
		tmp = 1.0 - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + Float64(y * log(y))) - z)
	t_1 = fma(0.5, Float64(z * z), 0.0)
	tmp = 0.0
	if (t_0 <= -1e+55)
		tmp = t_1;
	elseif (t_0 <= 5e+120)
		tmp = Float64(1.0 - z);
	else
		tmp = t_1;
	end
	return 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[(0.5 * N[(z * z), $MachinePrecision] + 0.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+55], t$95$1, If[LessEqual[t$95$0, 5e+120], N[(1.0 - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -1.00000000000000001e55 or 5.00000000000000019e120 < (-.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 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-sub0 N/A

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

      3. --lowering--.f64 45.4

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

   5. Simplified 45.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 21.2

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

   8. Simplified 21.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2} + 0} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {z}^{2}, 0\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{z \cdot z}, 0\right) \]

       4. *-lowering-*.f64 27.0

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{z \cdot z}, 0\right) \]

   11. Simplified 27.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, z \cdot z, 0\right)} \]

   if -1.00000000000000001e55 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 5.00000000000000019e120

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 70.5

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

   5. Simplified 70.5%

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

   6. Taylor expanded in z around 0

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

      1. neg-mul-1 N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 47.1

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

   8. Simplified 47.1%

      \[\leadsto \color{blue}{1 - z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 31.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(x + y \cdot \log y\right) - z} \leq 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp (- (+ x (* y (log y))) z)) 1e-300)
   (fma 0.5 (* z z) 0.0)
   (fma z (* z 0.5) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(((x + (y * log(y))) - z)) <= 1e-300) {
		tmp = fma(0.5, (z * z), 0.0);
	} else {
		tmp = fma(z, (z * 0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(Float64(Float64(x + Float64(y * log(y))) - z)) <= 1e-300)
		tmp = fma(0.5, Float64(z * z), 0.0);
	else
		tmp = fma(z, Float64(z * 0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision], 1e-300], N[(0.5 * N[(z * z), $MachinePrecision] + 0.0), $MachinePrecision], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)) < 1.00000000000000003e-300

   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-sub0 N/A

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

      3. --lowering--.f64 57.9

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

   5. Simplified 57.9%

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

   6. Taylor expanded in z around 0

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

      1. +-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 2.3

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

   8. Simplified 2.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2} + 0} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {z}^{2}, 0\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{z \cdot z}, 0\right) \]

       4. *-lowering-*.f64 18.4

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{z \cdot z}, 0\right) \]

   11. Simplified 18.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, z \cdot z, 0\right)} \]

   if 1.00000000000000003e-300 < (exp.f64 (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z))

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 51.3

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

   5. Simplified 51.3%

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

   6. Taylor expanded in z around 0

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

      1. +-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 38.8

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

   8. Simplified 38.8%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.f64 38.5

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

   11. Simplified 38.5%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{0.5 \cdot z}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(x + y \cdot \log y\right) - z} \leq 10^{-300}:\\ \;\;\;\;\mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 2 \cdot 10^{+117}:\\ \;\;\;\;1 - z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(0.5, z, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ x (* y (log y))) z) 2e+117) (- 1.0 z) (* z (fma 0.5 z -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * log(y))) - z) <= 2e+117) {
		tmp = 1.0 - z;
	} else {
		tmp = z * fma(0.5, z, -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * log(y))) - z) <= 2e+117)
		tmp = Float64(1.0 - z);
	else
		tmp = Float64(z * fma(0.5, z, -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], 2e+117], N[(1.0 - z), $MachinePrecision], N[(z * N[(0.5 * z + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 2.0000000000000001e117

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 64.1

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

   5. Simplified 64.1%

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

   6. Taylor expanded in z around 0

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

      1. neg-mul-1 N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 27.3

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

   8. Simplified 27.3%

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

   if 2.0000000000000001e117 < (-.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 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-sub0 N/A

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

      3. --lowering--.f64 39.6

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

   5. Simplified 39.6%

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

   6. Taylor expanded in z around 0

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

      1. +-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 31.4

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

   8. Simplified 31.4%

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

   9. Taylor expanded in z around inf

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

   11. Simplified 30.7%

       \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(0.5, z, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), 4, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 4.45 \cdot 10^{+102}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.02e+86)
   (fma
    z
    (fma (* z (fma (* z (* z z)) -0.004629629629629629 0.125)) 4.0 -1.0)
    1.0)
   (if (<= z 4.45e+102)
     (exp x)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.02e+86) {
		tmp = fma(z, fma((z * fma((z * (z * z)), -0.004629629629629629, 0.125)), 4.0, -1.0), 1.0);
	} else if (z <= 4.45e+102) {
		tmp = exp(x);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.02e+86)
		tmp = fma(z, fma(Float64(z * fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125)), 4.0, -1.0), 1.0);
	elseif (z <= 4.45e+102)
		tmp = exp(x);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.02e+86], N[(z * N[(N[(z * N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision]), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 4.45e+102], N[Exp[x], $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.01999999999999996e86

   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-sub0 N/A

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

      3. --lowering--.f64 90.4

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

   5. Simplified 90.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 88.2

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

   8. Simplified 88.2%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 19.6%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 90.4%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{4}, -1\right), 1\right) \]

   if -1.01999999999999996e86 < z < 4.4499999999999999e102

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.0%

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

   if 4.4499999999999999e102 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 74.8

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

   5. Simplified 74.8%

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

   6. Taylor expanded in z around 0

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

      1. +-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 1.2

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

   8. Simplified 1.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.2%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 72.6

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

   13. Simplified 72.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), 4, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 4.45 \cdot 10^{+102}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{+47}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 8.4e+47) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.4e+47) {
		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 <= 8.4d+47) 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 <= 8.4e+47) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.4e+47:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.4e+47)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.4e+47)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.4e+47], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.4e47

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 67.0%

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

   if 8.4e47 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

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

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(y, \log y, x\right)}} \]

      4. log-lowering-log.f64 94.3

         \[\leadsto e^{\mathsf{fma}\left(y, \color{blue}{\log y}, x\right)} \]

   5. Simplified 94.3%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(y, \log y, x\right)}} \]

   6. Taylor expanded in x around 0

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

      1. pow-lowering-pow.f64 86.7

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

   8. Simplified 86.7%

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

Alternative 10: 55.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right)\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(t\_1, \mathsf{fma}\left(z, -1.3333333333333333, 4\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{-231}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(t\_1, \frac{36 + \frac{\frac{972}{z \cdot z} + -108}{z}}{z \cdot z}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))
        (t_1 (* z (fma (* z (* z z)) -0.004629629629629629 0.125))))
   (if (<= z -4.4e+51)
     (fma z (fma t_1 (fma z -1.3333333333333333 4.0) -1.0) 1.0)
     (if (<= z 1.06e-231)
       t_0
       (if (<= z 3.9e-47)
         (fma
          z
          (fma
           t_1
           (/ (+ 36.0 (/ (+ (/ 972.0 (* z z)) -108.0) z)) (* z z))
           -1.0)
          1.0)
         (if (<= z 1.6e+80)
           t_0
           (/
            1.0
            (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = z * fma((z * (z * z)), -0.004629629629629629, 0.125);
	double tmp;
	if (z <= -4.4e+51) {
		tmp = fma(z, fma(t_1, fma(z, -1.3333333333333333, 4.0), -1.0), 1.0);
	} else if (z <= 1.06e-231) {
		tmp = t_0;
	} else if (z <= 3.9e-47) {
		tmp = fma(z, fma(t_1, ((36.0 + (((972.0 / (z * z)) + -108.0) / z)) / (z * z)), -1.0), 1.0);
	} else if (z <= 1.6e+80) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0)
	t_1 = Float64(z * fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125))
	tmp = 0.0
	if (z <= -4.4e+51)
		tmp = fma(z, fma(t_1, fma(z, -1.3333333333333333, 4.0), -1.0), 1.0);
	elseif (z <= 1.06e-231)
		tmp = t_0;
	elseif (z <= 3.9e-47)
		tmp = fma(z, fma(t_1, Float64(Float64(36.0 + Float64(Float64(Float64(972.0 / Float64(z * z)) + -108.0) / z)) / Float64(z * z)), -1.0), 1.0);
	elseif (z <= 1.6e+80)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.4e+51], N[(z * N[(t$95$1 * N[(z * -1.3333333333333333 + 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1.06e-231], t$95$0, If[LessEqual[z, 3.9e-47], N[(z * N[(t$95$1 * N[(N[(36.0 + N[(N[(N[(972.0 / N[(z * z), $MachinePrecision]), $MachinePrecision] + -108.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 1.6e+80], t$95$0, N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.39999999999999984e51

   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-sub0 N/A

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

      3. --lowering--.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 77.5

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

   8. Simplified 77.5%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 19.6%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{4 + \frac{-4}{3} \cdot z}, -1\right), 1\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{\frac{-4}{3} \cdot z + 4}, -1\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{z \cdot \frac{-4}{3}} + 4, -1\right), 1\right) \]

       3. accelerator-lowering-fma.f64 87.4

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\mathsf{fma}\left(z, -1.3333333333333333, 4\right)}, -1\right), 1\right) \]

   13. Simplified 87.4%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\mathsf{fma}\left(z, -1.3333333333333333, 4\right)}, -1\right), 1\right) \]

   if -4.39999999999999984e51 < z < 1.0600000000000001e-231 or 3.89999999999999978e-47 < z < 1.59999999999999995e80

   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 43.2

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

   8. Simplified 43.2%

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

   if 1.0600000000000001e-231 < z < 3.89999999999999978e-47

   1. Initial program 99.8%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 18.0

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

   5. Simplified 18.0%

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

   6. Taylor expanded in z around 0

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

      1. +-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 18.0

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

   8. Simplified 18.0%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 18.0%

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

   11. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{\frac{\left(36 + \frac{972}{{z}^{3}}\right) - 108 \cdot \frac{1}{z}}{{z}^{2}}}, -1\right), 1\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{\frac{\left(36 + \frac{972}{{z}^{3}}\right) - 108 \cdot \frac{1}{z}}{{z}^{2}}}, -1\right), 1\right) \]

   13. Simplified 64.2%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\frac{36 + \frac{\frac{972}{z \cdot z} + -108}{z}}{z \cdot z}}, -1\right), 1\right) \]

   if 1.59999999999999995e80 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 75.4

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

   5. Simplified 75.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 1.2

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

   8. Simplified 1.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.2%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 71.0

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

   13. Simplified 71.0%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), \mathsf{fma}\left(z, -1.3333333333333333, 4\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{-231}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), \frac{36 + \frac{\frac{972}{z \cdot z} + -108}{z}}{z \cdot z}, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), \mathsf{fma}\left(z, -1.3333333333333333, 4\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4e+51)
   (fma
    z
    (fma
     (* z (fma (* z (* z z)) -0.004629629629629629 0.125))
     (fma z -1.3333333333333333 4.0)
     -1.0)
    1.0)
   (if (<= z 6.3e+81)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e+51) {
		tmp = fma(z, fma((z * fma((z * (z * z)), -0.004629629629629629, 0.125)), fma(z, -1.3333333333333333, 4.0), -1.0), 1.0);
	} else if (z <= 6.3e+81) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4e+51)
		tmp = fma(z, fma(Float64(z * fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125)), fma(z, -1.3333333333333333, 4.0), -1.0), 1.0);
	elseif (z <= 6.3e+81)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4e+51], N[(z * N[(N[(z * N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision]), $MachinePrecision] * N[(z * -1.3333333333333333 + 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 6.3e+81], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.39999999999999983e51

   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-sub0 N/A

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

      3. --lowering--.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 77.5

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

   8. Simplified 77.5%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 19.6%

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

   11. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{4 + \frac{-4}{3} \cdot z}, -1\right), 1\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{\frac{-4}{3} \cdot z + 4}, -1\right), 1\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), \frac{-1}{216}, \frac{1}{8}\right) \cdot z, \color{blue}{z \cdot \frac{-4}{3}} + 4, -1\right), 1\right) \]

       3. accelerator-lowering-fma.f64 87.4

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\mathsf{fma}\left(z, -1.3333333333333333, 4\right)}, -1\right), 1\right) \]

   13. Simplified 87.4%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{\mathsf{fma}\left(z, -1.3333333333333333, 4\right)}, -1\right), 1\right) \]

   if -5.39999999999999983e51 < z < 6.3000000000000004e81

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 60.5%

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

   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 40.9

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

   8. Simplified 40.9%

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

   if 6.3000000000000004e81 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 75.4

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

   5. Simplified 75.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 1.2

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

   8. Simplified 1.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.2%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 71.0

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

   13. Simplified 71.0%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), \mathsf{fma}\left(z, -1.3333333333333333, 4\right), -1\right), 1\right)\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), 4, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+55)
   (fma
    z
    (fma (* z (fma (* z (* z z)) -0.004629629629629629 0.125)) 4.0 -1.0)
    1.0)
   (if (<= z 6.2e+81)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+55) {
		tmp = fma(z, fma((z * fma((z * (z * z)), -0.004629629629629629, 0.125)), 4.0, -1.0), 1.0);
	} else if (z <= 6.2e+81) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e+55)
		tmp = fma(z, fma(Float64(z * fma(Float64(z * Float64(z * z)), -0.004629629629629629, 0.125)), 4.0, -1.0), 1.0);
	elseif (z <= 6.2e+81)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e+55], N[(z * N[(N[(z * N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.004629629629629629 + 0.125), $MachinePrecision]), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[z, 6.2e+81], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.00000000000000002e55

   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-sub0 N/A

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

      3. --lowering--.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 77.5

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

   8. Simplified 77.5%

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

      1. *-commutative N/A

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

      2. flip3-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   10. Applied egg-rr 19.6%

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

   11. Taylor expanded in z around 0

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

   13. Simplified 83.6%

       \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right) \cdot z, \color{blue}{4}, -1\right), 1\right) \]

   if -2.00000000000000002e55 < z < 6.2e81

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 60.5%

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

   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 40.9

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

   8. Simplified 40.9%

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

   if 6.2e81 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 75.4

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

   5. Simplified 75.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 1.2

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

   8. Simplified 1.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.2%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 71.0

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

   13. Simplified 71.0%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z \cdot \mathsf{fma}\left(z \cdot \left(z \cdot z\right), -0.004629629629629629, 0.125\right), 4, -1\right), 1\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+95}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.18e+95)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= z 6.2e+81)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (/ 1.0 (fma z (fma z (fma z 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.18e+95) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (z <= 6.2e+81) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.18e+95)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (z <= 6.2e+81)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(z, fma(z, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.18e+95], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[z, 6.2e+81], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.17999999999999998e95

   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-sub0 N/A

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

      3. --lowering--.f64 90.4

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

   5. Simplified 90.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 88.2

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

   8. Simplified 88.2%

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

   9. 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 88.2

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

   11. Simplified 88.2%

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

   if -1.17999999999999998e95 < z < 6.2e81

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 60.8%

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

   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 40.1

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

   8. Simplified 40.1%

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

   if 6.2e81 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 75.4

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

   5. Simplified 75.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 1.2

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

   8. Simplified 1.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.2%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 71.0

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

   13. Simplified 71.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+95}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+96}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.3e+96)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= z 1.95e+142)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
     (/ 1.0 (fma z (fma z 0.5 1.0) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e+96) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (z <= 1.95e+142) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(z, fma(z, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.3e+96)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (z <= 1.95e+142)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(1.0 / fma(z, fma(z, 0.5, 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.3e+96], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[z, 1.95e+142], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 / N[(z * N[(z * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.30000000000000015e96

   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-sub0 N/A

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

      3. --lowering--.f64 90.4

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

   5. Simplified 90.4%

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

   6. Taylor expanded in z around 0

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

      1. +-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 88.2

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

   8. Simplified 88.2%

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

   9. 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 88.2

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

   11. Simplified 88.2%

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

   if -2.30000000000000015e96 < z < 1.95e142

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 60.5%

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

   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 39.5

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

   8. Simplified 39.5%

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

   if 1.95e142 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 79.6

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

   5. Simplified 79.6%

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

   6. Taylor expanded in z around 0

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

      1. +-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 1.2

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

   8. Simplified 1.2%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   10. Applied egg-rr 1.2%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 73.4

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

   13. Simplified 73.4%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, 1\right), 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+96}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+39}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e+39)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= x 6.8e+84)
     (fma z (fma z (fma z -0.16666666666666666 0.5) -1.0) 1.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+39) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (x <= 6.8e+84) {
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e+39)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (x <= 6.8e+84)
		tmp = fma(z, fma(z, fma(z, -0.16666666666666666, 0.5), -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e+39], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[x, 6.8e+84], N[(z * N[(z * N[(z * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999992e39

   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-sub0 N/A

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

      3. --lowering--.f64 37.1

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

   5. Simplified 37.1%

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

   6. Taylor expanded in z around 0

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

      1. +-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 14.1

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

   8. Simplified 14.1%

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

   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 40.6

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

   11. Simplified 40.6%

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

   if -1.79999999999999992e39 < x < 6.7999999999999996e84

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 64.9

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

   5. Simplified 64.9%

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

   6. Taylor expanded in z around 0

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

      1. +-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 39.7

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

   8. Simplified 39.7%

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

   if 6.7999999999999996e84 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 93.8%

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

   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 86.2

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

   8. Simplified 86.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+39}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+39)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= x 4.4e+84)
     (fma z (fma 0.5 z -1.0) 1.0)
     (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e+39) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (x <= 4.4e+84) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e+39)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (x <= 4.4e+84)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e+39], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[x, 4.4e+84], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e39

   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-sub0 N/A

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

      3. --lowering--.f64 37.1

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

   5. Simplified 37.1%

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

   6. Taylor expanded in z around 0

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

      1. +-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 14.1

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

   8. Simplified 14.1%

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

   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 40.6

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

   11. Simplified 40.6%

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

   if -1.5e39 < x < 4.3999999999999997e84

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 64.9

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

   5. Simplified 64.9%

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

   6. Taylor expanded in z around 0

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

      1. +-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 39.5

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

   8. Simplified 39.5%

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

   if 4.3999999999999997e84 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 93.8%

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

   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 86.2

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

   8. Simplified 86.2%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+39)
   (* (* z (* z z)) -0.16666666666666666)
   (if (<= x 3.6e+108)
     (fma z (fma 0.5 z -1.0) 1.0)
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e+39) {
		tmp = (z * (z * z)) * -0.16666666666666666;
	} else if (x <= 3.6e+108) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e+39)
		tmp = Float64(Float64(z * Float64(z * z)) * -0.16666666666666666);
	elseif (x <= 3.6e+108)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e+39], N[(N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[x, 3.6e+108], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e39

   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-sub0 N/A

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

      3. --lowering--.f64 37.1

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

   5. Simplified 37.1%

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

   6. Taylor expanded in z around 0

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

      1. +-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 14.1

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

   8. Simplified 14.1%

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

   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 40.6

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

   11. Simplified 40.6%

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

   if -1.5e39 < x < 3.6e108

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 63.2

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

   5. Simplified 63.2%

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

   6. Taylor expanded in z around 0

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

      1. +-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 38.7

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

   8. Simplified 38.7%

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

   if 3.6e108 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 93.0%

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

   6. Taylor expanded in x around 0

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

      1. +-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 70.9

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

   8. Simplified 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;\left(z \cdot \left(z \cdot z\right)\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(0.5, z, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+39)
   (fma 0.5 (* z z) 0.0)
   (if (<= x 3.6e+108)
     (fma z (fma 0.5 z -1.0) 1.0)
     (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e+39) {
		tmp = fma(0.5, (z * z), 0.0);
	} else if (x <= 3.6e+108) {
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e+39)
		tmp = fma(0.5, Float64(z * z), 0.0);
	elseif (x <= 3.6e+108)
		tmp = fma(z, fma(0.5, z, -1.0), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e+39], N[(0.5 * N[(z * z), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[x, 3.6e+108], N[(z * N[(0.5 * z + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e39

   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-sub0 N/A

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

      3. --lowering--.f64 37.1

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

   5. Simplified 37.1%

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

   6. Taylor expanded in z around 0

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

      1. +-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 12.3

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

   8. Simplified 12.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2} + 0} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {z}^{2}, 0\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{z \cdot z}, 0\right) \]

       4. *-lowering-*.f64 33.5

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{z \cdot z}, 0\right) \]

   11. Simplified 33.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, z \cdot z, 0\right)} \]

   if -1.5e39 < x < 3.6e108

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 63.2

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

   5. Simplified 63.2%

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

   6. Taylor expanded in z around 0

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

      1. +-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 38.7

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

   8. Simplified 38.7%

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

   if 3.6e108 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 93.0%

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

   6. Taylor expanded in x around 0

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

      1. +-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 70.9

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

   8. Simplified 70.9%

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

Alternative 19: 40.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.65e+39)
   (fma 0.5 (* z z) 0.0)
   (if (<= x 3.6e+108) (fma z (* z 0.5) 1.0) (fma x (fma x 0.5 1.0) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.65e+39) {
		tmp = fma(0.5, (z * z), 0.0);
	} else if (x <= 3.6e+108) {
		tmp = fma(z, (z * 0.5), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.65e+39)
		tmp = fma(0.5, Float64(z * z), 0.0);
	elseif (x <= 3.6e+108)
		tmp = fma(z, Float64(z * 0.5), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.65e+39], N[(0.5 * N[(z * z), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[x, 3.6e+108], N[(z * N[(z * 0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6500000000000001e39

   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-sub0 N/A

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

      3. --lowering--.f64 37.1

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

   5. Simplified 37.1%

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

   6. Taylor expanded in z around 0

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

      1. +-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 12.3

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

   8. Simplified 12.3%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2}} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot {z}^{2} + 0} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {z}^{2}, 0\right)} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{z \cdot z}, 0\right) \]

       4. *-lowering-*.f64 33.5

          \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{z \cdot z}, 0\right) \]

   11. Simplified 33.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, z \cdot z, 0\right)} \]

   if -1.6500000000000001e39 < x < 3.6e108

   1. Initial program 99.9%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   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-sub0 N/A

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

      3. --lowering--.f64 63.2

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

   5. Simplified 63.2%

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

   6. Taylor expanded in z around 0

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

      1. +-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 38.7

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

   8. Simplified 38.7%

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

   9. Taylor expanded in z around inf

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

       1. *-lowering-*.f64 38.3

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

   11. Simplified 38.3%

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

   if 3.6e108 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 93.0%

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

   6. Taylor expanded in x around 0

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

      1. +-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 70.9

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

   8. Simplified 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(0.5, z \cdot z, 0\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 14.3% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ 1 - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 1.0 z))

C

double code(double x, double y, double z) {
	return 1.0 - z;
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return 1.0 - z;
}

Python

def code(x, y, z):
	return 1.0 - z

Julia

function code(x, y, z)
	return Float64(1.0 - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 - z;
end

Wolfram

code[x_, y_, z_] := N[(1.0 - z), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 53.0

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

5. Simplified 53.0%

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

6. Taylor expanded in z around 0

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

   1. neg-mul-1 N/A

      \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

   2. unsub-neg N/A

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

   3. --lowering--.f64 16.6

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

8. Simplified 16.6%

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

Alternative 21: 14.3% 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 52.3%

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

6. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 16.6

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

8. Simplified 16.6%

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

9. Final simplification 16.6%

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

Alternative 22: 14.1% 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 52.3%

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

6. Taylor expanded in x around 0

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

8. Simplified 16.4%

   \[\leadsto \color{blue}{1} \]
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 2024196 
(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)))