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

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+17}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-32}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y (log y)))))
   (if (<= t_0 -5e+17) (exp x) (if (<= t_0 5e-32) (exp (- z)) (pow y y)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y * log(y));
	double tmp;
	if (t_0 <= -5e+17) {
		tmp = exp(x);
	} else if (t_0 <= 5e-32) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y * log(y))
    if (t_0 <= (-5d+17)) then
        tmp = exp(x)
    else if (t_0 <= 5d-32) then
        tmp = exp(-z)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y * Math.log(y));
	double tmp;
	if (t_0 <= -5e+17) {
		tmp = Math.exp(x);
	} else if (t_0 <= 5e-32) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y * math.log(y))
	tmp = 0
	if t_0 <= -5e+17:
		tmp = math.exp(x)
	elif t_0 <= 5e-32:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y * log(y)))
	tmp = 0.0
	if (t_0 <= -5e+17)
		tmp = exp(x);
	elseif (t_0 <= 5e-32)
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y * log(y));
	tmp = 0.0;
	if (t_0 <= -5e+17)
		tmp = exp(x);
	elseif (t_0 <= 5e-32)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+17], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 5e-32], 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))) < -5e17

   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 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 94.1%

       \[\leadsto e^{x} \]

   if -5e17 < (+.f64 x (*.f64 y (log.f64 y))) < 5e-32

   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. lower-neg.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 5e-32 < (+.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 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.6%

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

Alternative 3: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+16} \lor \neg \left(x \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\log y \cdot y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7e+16) (not (<= x 2e+15)))
   (exp (- x z))
   (exp (- (* (log y) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e+16) || !(x <= 2e+15)) {
		tmp = exp((x - z));
	} else {
		tmp = exp(((log(y) * y) - z));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3.7d+16)) .or. (.not. (x <= 2d+15))) then
        tmp = exp((x - z))
    else
        tmp = exp(((log(y) * y) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e+16) || !(x <= 2e+15)) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp(((Math.log(y) * y) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.7e+16) or not (x <= 2e+15):
		tmp = math.exp((x - z))
	else:
		tmp = math.exp(((math.log(y) * y) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7e+16) || !(x <= 2e+15))
		tmp = exp(Float64(x - z));
	else
		tmp = exp(Float64(Float64(log(y) * y) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.7e+16) || ~((x <= 2e+15)))
		tmp = exp((x - z));
	else
		tmp = exp(((log(y) * y) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.7e+16], N[Not[LessEqual[x, 2e+15]], $MachinePrecision]], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e16 or 2e15 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if -3.7e16 < x < 2e15

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+16} \lor \neg \left(x \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\log y \cdot y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 10^{+214}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ x (* y (log y))) z) 1e+214)
   (exp x)
   (fma (fma 0.5 x 1.0) x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * log(y))) - z) <= 1e+214) {
		tmp = exp(x);
	} 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(x + Float64(y * log(y))) - z) <= 1e+214)
		tmp = exp(x);
	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[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], 1e+214], N[Exp[x], $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) < 9.9999999999999995e213

   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 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 50.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 65.8%

       \[\leadsto e^{x} \]

   if 9.9999999999999995e213 < (-.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 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 40.1%

       \[\leadsto e^{x} \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 48.3%

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

Alternative 5: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -42000000000 \lor \neg \left(x \leq 680\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -42000000000.0) (not (<= x 680.0))) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -42000000000.0) || !(x <= 680.0)) {
		tmp = exp(x);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-42000000000.0d0)) .or. (.not. (x <= 680.0d0))) then
        tmp = exp(x)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -42000000000.0) || !(x <= 680.0)) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -42000000000.0) or not (x <= 680.0):
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -42000000000.0) || !(x <= 680.0))
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -42000000000.0) || ~((x <= 680.0)))
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -42000000000.0], N[Not[LessEqual[x, 680.0]], $MachinePrecision]], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e10 or 680 < 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 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 85.6%

       \[\leadsto e^{x} \]

   if -4.2e10 < x < 680

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 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 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -42000000000 \lor \neg \left(x \leq 680\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+46}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.4e+46) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.4e+46) {
		tmp = exp((x - z));
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.4d+46) 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 (y <= 3.4e+46) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.4e+46:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.4e+46)
		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 (y <= 3.4e+46)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.4e+46], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3999999999999998e46

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if 3.3999999999999998e46 < 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 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.9%

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

Alternative 7: 31.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.0)
   (fma (fma 0.5 x 1.0) x 1.0)
   (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 (x <= -3.0) {
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	} 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 (x <= -3.0)
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	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[x, -3.0], N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $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 x < -3

   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 45.5

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

   5. Applied rewrites 45.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 74.6%

       \[\leadsto e^{x} \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 11.4%

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

   if -3 < 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 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 54.0%

       \[\leadsto e^{x} \]

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 45.4%

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

Alternative 8: 28.3% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (fma 0.5 x 1.0) x 1.0))

C

double code(double x, double y, double z) {
	return fma(fma(0.5, x, 1.0), x, 1.0);
}

Julia

function code(x, y, z)
	return fma(fma(0.5, x, 1.0), x, 1.0)
end

Wolfram

code[x_, y_, z_] := N[(N[(0.5 * x + 1.0), $MachinePrecision] * 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 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 73.8

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

5. Applied rewrites 73.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 56.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 59.3%

    \[\leadsto e^{x} \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 33.0%

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

Alternative 9: 14.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 73.8

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

5. Applied rewrites 73.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 56.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 59.3%

    \[\leadsto e^{x} \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 17.7%

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

Alternative 10: 14.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 73.8

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

5. Applied rewrites 73.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 56.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 17.3%

    \[\leadsto 1 \]
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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 2024363 
(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)))