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

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 7

Speedup: 1.0×

• 58.3% 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: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;t\_0 \leq 3.25 \cdot 10^{+268}:\\ \;\;\;\;e^{\mathsf{fma}\left(-z, \frac{z}{x + z}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{1 - \frac{z}{x}}{x}, -1\right), x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{z}{x \cdot \left(x \cdot x\right)}, -\frac{z}{x \cdot x}\right), \frac{z}{x}\right), \left(x \cdot x\right) \cdot \frac{1}{x + z}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 -2e-302)
     (exp (- x z))
     (if (<= t_0 3.25e+268)
       (exp
        (fma (- z) (/ z (+ x z)) (fma z (fma z (/ (- 1.0 (/ z x)) x) -1.0) x)))
       (exp
        (fma
         (- z)
         (fma z (fma z (/ z (* x (* x x))) (- (/ z (* x x)))) (/ z x))
         (* (* x x) (/ 1.0 (+ x z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (t_0 <= -2e-302) {
		tmp = exp((x - z));
	} else if (t_0 <= 3.25e+268) {
		tmp = exp(fma(-z, (z / (x + z)), fma(z, fma(z, ((1.0 - (z / x)) / x), -1.0), x)));
	} else {
		tmp = exp(fma(-z, fma(z, fma(z, (z / (x * (x * x))), -(z / (x * x))), (z / x)), ((x * x) * (1.0 / (x + z)))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (t_0 <= -2e-302)
		tmp = exp(Float64(x - z));
	elseif (t_0 <= 3.25e+268)
		tmp = exp(fma(Float64(-z), Float64(z / Float64(x + z)), fma(z, fma(z, Float64(Float64(1.0 - Float64(z / x)) / x), -1.0), x)));
	else
		tmp = exp(fma(Float64(-z), fma(z, fma(z, Float64(z / Float64(x * Float64(x * x))), Float64(-Float64(z / Float64(x * x)))), Float64(z / x)), Float64(Float64(x * x) * Float64(1.0 / Float64(x + z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-302], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 3.25e+268], N[Exp[N[((-z) * N[(z / N[(x + z), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(1.0 - N[(z / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[((-z) * N[(z * N[(z * N[(z / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(z / N[(x * x), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] + N[(z / x), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(1.0 / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (log.f64 y)) < -1.9999999999999999e-302

   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 98.4

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

   5. Applied rewrites 98.4%

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

   if -1.9999999999999999e-302 < (*.f64 y (log.f64 y)) < 3.25e268

   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 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 10.3%

      \[\leadsto e^{\mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(x, x \cdot x, \left(z \cdot z\right) \cdot z\right)}, \color{blue}{\mathsf{fma}\left(z, z - x, x \cdot x\right)}, -\frac{z \cdot z}{x + z}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.1%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 65.4%

       \[\leadsto e^{\mathsf{fma}\left(-z, \frac{z}{x + z}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{1 - \frac{z}{x}}{x}, -1\right), x\right)\right)} \]

   if 3.25e268 < (*.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 0

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

      1. lower--.f64 32.1

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

   5. Applied rewrites 32.1%

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

   7. Applied rewrites 0.6%

      \[\leadsto e^{\mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(x, x \cdot x, \left(z \cdot z\right) \cdot z\right)}, \color{blue}{\mathsf{fma}\left(z, z - x, x \cdot x\right)}, -\frac{z \cdot z}{x + z}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 39.7%

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

   10. Taylor expanded in z around 0

       \[\leadsto e^{\mathsf{fma}\left(\mathsf{neg}\left(z\right), z \cdot \color{blue}{\left(z \cdot \left(\frac{z}{{x}^{3}} - \frac{1}{{x}^{2}}\right) + \frac{1}{x}\right)}, \left(x \cdot x\right) \cdot \frac{1}{x + z}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 84.6%

       \[\leadsto e^{\mathsf{fma}\left(-z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{z}{x \cdot \left(x \cdot x\right)}, \frac{z}{x \cdot \left(-x\right)}\right)}, \frac{z}{x}\right), \left(x \cdot x\right) \cdot \frac{1}{x + z}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq -2 \cdot 10^{-302}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;y \cdot \log y \leq 3.25 \cdot 10^{+268}:\\ \;\;\;\;e^{\mathsf{fma}\left(-z, \frac{z}{x + z}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{1 - \frac{z}{x}}{x}, -1\right), x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{z}{x \cdot \left(x \cdot x\right)}, -\frac{z}{x \cdot x}\right), \frac{z}{x}\right), \left(x \cdot x\right) \cdot \frac{1}{x + z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 5 \cdot 10^{+29}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(y, \log y, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) 5e+29) (exp (- x z)) (exp (fma y (log y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= 5e+29) {
		tmp = exp((x - z));
	} else {
		tmp = exp(fma(y, log(y), x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * log(y)) <= 5e+29)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(fma(y, log(y), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision], 5e+29], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(y * N[Log[y], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 5.0000000000000001e29

   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 97.9

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

   5. Applied rewrites 97.9%

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

   if 5.0000000000000001e29 < (*.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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 4: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t\_0 \leq 3.5 \cdot 10^{+119}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y)))) (if (<= t_0 3.5e+119) (exp (- x z)) (exp t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (t_0 <= 3.5e+119) {
		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 = y * log(y)
    if (t_0 <= 3.5d+119) 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 = y * Math.log(y);
	double tmp;
	if (t_0 <= 3.5e+119) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp(t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if t_0 <= 3.5e+119:
		tmp = math.exp((x - z))
	else:
		tmp = math.exp(t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (t_0 <= 3.5e+119)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (t_0 <= 3.5e+119)
		tmp = exp((x - z));
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 3.5e+119], 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)) < 3.5000000000000001e119

   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 3.5000000000000001e119 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 91.2

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

   5. Applied rewrites 91.2%

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

Alternative 5: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq -2 \cdot 10^{-302}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-z, \frac{z}{x + z}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{1 - \frac{z}{x}}{x}, -1\right), x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (log y)) -2e-302)
   (exp (- x z))
   (exp
    (fma (- z) (/ z (+ x z)) (fma z (fma z (/ (- 1.0 (/ z x)) x) -1.0) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * log(y)) <= -2e-302) {
		tmp = exp((x - z));
	} else {
		tmp = exp(fma(-z, (z / (x + z)), fma(z, fma(z, ((1.0 - (z / x)) / x), -1.0), x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * log(y)) <= -2e-302)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(fma(Float64(-z), Float64(z / Float64(x + z)), fma(z, fma(z, Float64(Float64(1.0 - Float64(z / x)) / x), -1.0), x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision], -2e-302], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[((-z) * N[(z / N[(x + z), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z * N[(N[(1.0 - N[(z / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < -1.9999999999999999e-302

   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 98.4

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

   5. Applied rewrites 98.4%

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

   if -1.9999999999999999e-302 < (*.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 0

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

      1. lower--.f64 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 9.3%

      \[\leadsto e^{\mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(x, x \cdot x, \left(z \cdot z\right) \cdot z\right)}, \color{blue}{\mathsf{fma}\left(z, z - x, x \cdot x\right)}, -\frac{z \cdot z}{x + z}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.3%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 62.1%

       \[\leadsto e^{\mathsf{fma}\left(-z, \frac{z}{x + z}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{1 - \frac{z}{x}}{x}, -1\right), x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 79.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 78.3

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

5. Applied rewrites 78.3%

   \[\leadsto e^{\color{blue}{x - z}} \]
6. Add Preprocessing

Alternative 7: 52.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{-z} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 50.0

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

5. Applied rewrites 50.0%

   \[\leadsto e^{\color{blue}{-z}} \]
6. 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 2024220 
(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)))