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

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 10

Speedup: 1.0×

• 58.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-17}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\log y \cdot y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e-17) (exp (- x z)) (exp (- (* (log y) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e-17) {
		tmp = exp((x - z));
	} else {
		tmp = exp(((log(y) * y) - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9.2d-17) 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 (y <= 9.2e-17) {
		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 y <= 9.2e-17:
		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 (y <= 9.2e-17)
		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 (y <= 9.2e-17)
		tmp = exp((x - z));
	else
		tmp = exp(((log(y) * y) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e-17], 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 y < 9.20000000000000035e-17

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 9.20000000000000035e-17 < y

   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 94.6

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

   5. Applied rewrites 94.6%

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

Alternative 3: 73.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 1.0) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 1.0) {
		tmp = exp(x);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y * log(y)) <= 1.0d0) then
        tmp = exp(x)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y * Math.log(y)) <= 1.0) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * math.log(y)) <= 1.0:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * log(y)) <= 1.0)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y * log(y)) <= 1.0)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision], 1.0], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 1

   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 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 65.7%

       \[\leadsto e^{x} \]

   if 1 < (*.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 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.3%

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

Alternative 4: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;z \leq -24:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-68}:\\ \;\;\;\;{y}^{y}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+66}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= z -24.0)
     t_0
     (if (<= z 5.5e-68) (pow y y) (if (<= z 2.7e+66) (exp x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (z <= -24.0) {
		tmp = t_0;
	} else if (z <= 5.5e-68) {
		tmp = pow(y, y);
	} else if (z <= 2.7e+66) {
		tmp = exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-z)
    if (z <= (-24.0d0)) then
        tmp = t_0
    else if (z <= 5.5d-68) then
        tmp = y ** y
    else if (z <= 2.7d+66) then
        tmp = exp(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.exp(-z);
	double tmp;
	if (z <= -24.0) {
		tmp = t_0;
	} else if (z <= 5.5e-68) {
		tmp = Math.pow(y, y);
	} else if (z <= 2.7e+66) {
		tmp = Math.exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp(-z)
	tmp = 0
	if z <= -24.0:
		tmp = t_0
	elif z <= 5.5e-68:
		tmp = math.pow(y, y)
	elif z <= 2.7e+66:
		tmp = math.exp(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(-z))
	tmp = 0.0
	if (z <= -24.0)
		tmp = t_0;
	elseif (z <= 5.5e-68)
		tmp = y ^ y;
	elseif (z <= 2.7e+66)
		tmp = exp(x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp(-z);
	tmp = 0.0;
	if (z <= -24.0)
		tmp = t_0;
	elseif (z <= 5.5e-68)
		tmp = y ^ y;
	elseif (z <= 2.7e+66)
		tmp = exp(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[(-z)], $MachinePrecision]}, If[LessEqual[z, -24.0], t$95$0, If[LessEqual[z, 5.5e-68], N[Power[y, y], $MachinePrecision], If[LessEqual[z, 2.7e+66], N[Exp[x], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -24 or 2.7e66 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -24 < z < 5.5000000000000003e-68

   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 91.1

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.9%

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

   if 5.5000000000000003e-68 < z < 2.7e66

   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 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 75.4%

       \[\leadsto e^{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+74}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.1e+74) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e+74) {
		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 (y <= 4.1d+74) 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 <= 4.1e+74) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.1e+74:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.1e+74)
		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 <= 4.1e+74)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.1e+74], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1e74

   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 93.7

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

   5. Applied rewrites 93.7%

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

   if 4.1e74 < 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 72.7

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.8%

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

Alternative 6: 52.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp x))

C

double code(double x, double y, double z) {
	return exp(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 = exp(x)
end function

Java

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

Python

def code(x, y, z):
	return math.exp(x)

Julia

function code(x, y, z)
	return exp(x)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Exp[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 68.1

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 55.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 45.8%

    \[\leadsto e^{x} \]
12. Add Preprocessing

Alternative 7: 31.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000235:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.000235)
   (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 <= -0.000235) {
		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 <= -0.000235)
		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, -0.000235], 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 < -2.34999999999999993e-4

   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 46.2

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 59.4%

       \[\leadsto e^{x} \]

   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 12.2%

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

   if -2.34999999999999993e-4 < 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 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 41.4%

       \[\leadsto e^{x} \]

   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 31.1%

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

Alternative 8: 27.6% 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 68.1

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 55.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 45.8%

    \[\leadsto e^{x} \]

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 23.2%

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

Alternative 9: 14.2% 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 68.1

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 55.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 45.8%

    \[\leadsto e^{x} \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 11.2%

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

Alternative 10: 13.9% 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 68.1

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 55.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 11.2%

    \[\leadsto 1 \]
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 2024324 
(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)))