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

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 12

Speedup: 1.0×

• 59.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot y + x\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+121}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 10^{+44}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* (log y) y) x)))
   (if (<= t_0 -2e+121) (exp x) (if (<= t_0 1e+44) (exp (- z)) (pow y y)))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * y) + x;
	double tmp;
	if (t_0 <= -2e+121) {
		tmp = exp(x);
	} else if (t_0 <= 1e+44) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (log(y) * y) + x
    if (t_0 <= (-2d+121)) then
        tmp = exp(x)
    else if (t_0 <= 1d+44) then
        tmp = exp(-z)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.log(y) * y) + x;
	double tmp;
	if (t_0 <= -2e+121) {
		tmp = Math.exp(x);
	} else if (t_0 <= 1e+44) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(y) * y) + x
	tmp = 0
	if t_0 <= -2e+121:
		tmp = math.exp(x)
	elif t_0 <= 1e+44:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * y) + x)
	tmp = 0.0
	if (t_0 <= -2e+121)
		tmp = exp(x);
	elseif (t_0 <= 1e+44)
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(y) * y) + x;
	tmp = 0.0;
	if (t_0 <= -2e+121)
		tmp = exp(x);
	elseif (t_0 <= 1e+44)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+121], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 1e+44], N[Exp[(-z)], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -2.00000000000000007e121

   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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.2%

      \[\leadsto e^{x} \]

   if -2.00000000000000007e121 < (+.f64 x (*.f64 y (log.f64 y))) < 1.0000000000000001e44

   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. lower-neg.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if 1.0000000000000001e44 < (+.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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y + x \leq -2 \cdot 10^{+121}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\log y \cdot y + x \leq 10^{+44}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot y\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+138}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) y))) (if (<= t_0 2e+138) (exp (- x z)) (exp t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * y;
	double tmp;
	if (t_0 <= 2e+138) {
		tmp = exp((x - z));
	} else {
		tmp = exp(t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(y) * y
    if (t_0 <= 2d+138) then
        tmp = exp((x - z))
    else
        tmp = exp(t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(y) * y;
	double tmp;
	if (t_0 <= 2e+138) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp(t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * y
	tmp = 0
	if t_0 <= 2e+138:
		tmp = math.exp((x - z))
	else:
		tmp = math.exp(t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * y)
	tmp = 0.0
	if (t_0 <= 2e+138)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) * y;
	tmp = 0.0;
	if (t_0 <= 2e+138)
		tmp = exp((x - z));
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+138], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 2.0000000000000001e138

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 2.0000000000000001e138 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 88.9

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

   5. Applied rewrites 88.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 2 \cdot 10^{+138}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\log y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\log y \cdot y + x\right) - z\\ t_1 := \left(x \cdot x\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+81}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (* (log y) y) x) z)) (t_1 (* (* x x) 0.5)))
   (if (<= t_0 -5e+23) t_1 (if (<= t_0 1e+81) (+ 1.0 x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((log(y) * y) + x) - z;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -5e+23) {
		tmp = t_1;
	} else if (t_0 <= 1e+81) {
		tmp = 1.0 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((log(y) * y) + x) - z
    t_1 = (x * x) * 0.5d0
    if (t_0 <= (-5d+23)) then
        tmp = t_1
    else if (t_0 <= 1d+81) then
        tmp = 1.0d0 + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((Math.log(y) * y) + x) - z;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -5e+23) {
		tmp = t_1;
	} else if (t_0 <= 1e+81) {
		tmp = 1.0 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((math.log(y) * y) + x) - z
	t_1 = (x * x) * 0.5
	tmp = 0
	if t_0 <= -5e+23:
		tmp = t_1
	elif t_0 <= 1e+81:
		tmp = 1.0 + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(log(y) * y) + x) - z)
	t_1 = Float64(Float64(x * x) * 0.5)
	tmp = 0.0
	if (t_0 <= -5e+23)
		tmp = t_1;
	elseif (t_0 <= 1e+81)
		tmp = Float64(1.0 + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((log(y) * y) + x) - z;
	t_1 = (x * x) * 0.5;
	tmp = 0.0;
	if (t_0 <= -5e+23)
		tmp = t_1;
	elseif (t_0 <= 1e+81)
		tmp = 1.0 + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+23], t$95$1, If[LessEqual[t$95$0, 1e+81], N[(1.0 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -4.9999999999999999e23 or 9.99999999999999921e80 < (-.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 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.8%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 20.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto \frac{1}{2} \cdot {x}^{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 28.4%

       \[\leadsto \left(x \cdot x\right) \cdot 0.5 \]

   if -4.9999999999999999e23 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 9.99999999999999921e80

   1. Initial program 99.8%

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

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 87.0

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.9%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

      \[\leadsto 1 + x \]
   10. Step-by-step derivation

   11. Applied rewrites 52.5%

       \[\leadsto 1 + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot y + x\right) - z \leq -5 \cdot 10^{+23}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(\log y \cdot y + x\right) - z \leq 10^{+81}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(\log y \cdot y + x\right) - z} \leq 0:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp (- (+ (* (log y) y) x) z)) 0.0)
   (* (* x x) 0.5)
   (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp((((log(y) * y) + x) - z)) <= 0.0) {
		tmp = (x * x) * 0.5;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[N[(N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)) < 0.0

   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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 39.9

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.5%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 2.4%

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

   12. Taylor expanded in x around inf

       \[\leadsto \frac{1}{2} \cdot {x}^{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 27.0%

       \[\leadsto \left(x \cdot x\right) \cdot 0.5 \]

   if 0.0 < (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 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 38.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\log y \cdot y + x\right) - z} \leq 0:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 2 \cdot 10^{+138}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (log y) y) 2e+138) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(y) * y) <= 2e+138) {
		tmp = exp((x - 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) :: tmp
    if ((log(y) * y) <= 2d+138) then
        tmp = exp((x - z))
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((Math.log(y) * y) <= 2e+138) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.log(y) * y) <= 2e+138:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(y) * y) <= 2e+138)
		tmp = exp(Float64(x - z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((log(y) * y) <= 2e+138)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision], 2e+138], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 2.0000000000000001e138

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 2.0000000000000001e138 < (*.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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 2 \cdot 10^{+138}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq -1 \cdot 10^{-302}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (log y) y) -1e-302) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(y) * y) <= -1e-302) {
		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 ((log(y) * y) <= (-1d-302)) 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 ((Math.log(y) * y) <= -1e-302) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.log(y) * y) <= -1e-302:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(y) * y) <= -1e-302)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((log(y) * y) <= -1e-302)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision], -1e-302], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < -9.9999999999999996e-303

   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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.4%

      \[\leadsto e^{x} \]

   if -9.9999999999999996e-303 < (*.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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq -1 \cdot 10^{-302}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\log y \cdot y + x\right) - z \leq -5 \cdot 10^{+23}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ (* (log y) y) x) z) -5e+23)
   (* (* x x) 0.5)
   (fma (fma 0.5 x 1.0) x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((log(y) * y) + x) - z) <= -5e+23) {
		tmp = (x * x) * 0.5;
	} else {
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(log(y) * y) + x) - z) <= -5e+23)
		tmp = Float64(Float64(x * x) * 0.5);
	else
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], -5e+23], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -4.9999999999999999e23

   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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 39.9

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.5%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 2.4%

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

   12. Taylor expanded in x around inf

       \[\leadsto \frac{1}{2} \cdot {x}^{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 27.0%

       \[\leadsto \left(x \cdot x\right) \cdot 0.5 \]

   if -4.9999999999999999e23 < (-.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 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 34.9%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot y + x\right) - z \leq -5 \cdot 10^{+23}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-164}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-186}:\\ \;\;\;\;\left(\left(-x\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.16666666666666666, x, -0.5 - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.02e-164)
   (exp x)
   (if (<= x 1.16e-186)
     (* (* (- x) x) (fma -0.16666666666666666 x (- -0.5 (/ 1.0 x))))
     (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e-164) {
		tmp = exp(x);
	} else if (x <= 1.16e-186) {
		tmp = (-x * x) * fma(-0.16666666666666666, x, (-0.5 - (1.0 / x)));
	} else {
		tmp = exp(x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.02e-164)
		tmp = exp(x);
	elseif (x <= 1.16e-186)
		tmp = Float64(Float64(Float64(-x) * x) * fma(-0.16666666666666666, x, Float64(-0.5 - Float64(1.0 / x))));
	else
		tmp = exp(x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.02e-164], N[Exp[x], $MachinePrecision], If[LessEqual[x, 1.16e-186], N[(N[((-x) * x), $MachinePrecision] * N[(-0.16666666666666666 * x + N[(-0.5 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[x], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02e-164 or 1.15999999999999995e-186 < x

   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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.5%

      \[\leadsto e^{x} \]

   if -1.02e-164 < x < 1.15999999999999995e-186

   1. Initial program 99.8%

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

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.0%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 11.0%

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

   12. Taylor expanded in x around -inf

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

   14. Applied rewrites 30.3%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x, -0.5 - \frac{1}{x}\right) \cdot \left(\left(-x\right) \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-164}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-186}:\\ \;\;\;\;\left(\left(-x\right) \cdot x\right) \cdot \mathsf{fma}\left(-0.16666666666666666, x, -0.5 - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.8% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.2e-61)
   (fma (fma 0.5 x 1.0) x 1.0)
   (* (* (fma 0.16666666666666666 x 0.5) x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.2e-61) {
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	} else {
		tmp = (fma(0.16666666666666666, x, 0.5) * x) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.2e-61)
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, x, 0.5) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.2e-61], N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.20000000000000009e-61

   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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 31.7%

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

   if 2.20000000000000009e-61 < z

   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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 19.6%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 39.8%

       \[\leadsto \left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot x\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 14.8% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ 1 + x \end{array} \]

FPCore

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

C

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

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 + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(1.0 + x), $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 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. exp-to-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 71.0

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

5. Applied rewrites 71.0%

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

6. Taylor expanded in y around 0

   \[\leadsto e^{x} \]
7. Step-by-step derivation

8. Applied rewrites 51.6%

   \[\leadsto e^{x} \]

9. Taylor expanded in x around 0

   \[\leadsto 1 + x \]
10. Step-by-step derivation

11. Applied rewrites 11.5%

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

Alternative 12: 14.6% 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 z around 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. exp-to-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 71.0

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

5. Applied rewrites 71.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 52.5%

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

9. Taylor expanded in y around 0

   \[\leadsto 1 \]
10. Step-by-step derivation

11. Applied rewrites 11.1%

    \[\leadsto 1 \]
12. 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 2024276 
(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)))