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

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 11

Speedup: 1.0×

• 58.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot y + x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+46}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+29}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* (log y) y) x)))
   (if (<= t_0 -5e+46) (exp x) (if (<= t_0 5e+29) (exp (- z)) (pow y y)))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * y) + x;
	double tmp;
	if (t_0 <= -5e+46) {
		tmp = exp(x);
	} else if (t_0 <= 5e+29) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (log(y) * y) + x
    if (t_0 <= (-5d+46)) then
        tmp = exp(x)
    else if (t_0 <= 5d+29) then
        tmp = exp(-z)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.log(y) * y) + x;
	double tmp;
	if (t_0 <= -5e+46) {
		tmp = Math.exp(x);
	} else if (t_0 <= 5e+29) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(y) * y) + x
	tmp = 0
	if t_0 <= -5e+46:
		tmp = math.exp(x)
	elif t_0 <= 5e+29:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * y) + x)
	tmp = 0.0
	if (t_0 <= -5e+46)
		tmp = exp(x);
	elseif (t_0 <= 5e+29)
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(y) * y) + x;
	tmp = 0.0;
	if (t_0 <= -5e+46)
		tmp = exp(x);
	elseif (t_0 <= 5e+29)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+46], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 5e+29], 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))) < -5.0000000000000002e46

   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 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.5%

      \[\leadsto e^{x} \]

   if -5.0000000000000002e46 < (+.f64 x (*.f64 y (log.f64 y))) < 5.0000000000000001e29

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if 5.0000000000000001e29 < (+.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 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y + x \leq -5 \cdot 10^{+46}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\log y \cdot y + x \leq 5 \cdot 10^{+29}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 34.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\log y \cdot y + x\right) - z\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (* (log y) y) x) z)))
   (if (<= t_0 -4e+20)
     (* (* x x) 0.5)
     (if (<= t_0 1e+149)
       (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)
       (fma (fma 0.5 x 1.0) x 1.0)))))

C

double code(double x, double y, double z) {
	double t_0 = ((log(y) * y) + x) - z;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (x * x) * 0.5;
	} else if (t_0 <= 1e+149) {
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else {
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(log(y) * y) + x) - z)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(x * x) * 0.5);
	elseif (t_0 <= 1e+149)
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	else
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e+149], N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -4e20

   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 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.9%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 2.2%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 12.6%

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

   if -4e20 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1.00000000000000005e149

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 91.2

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.2%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.6%

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

   if 1.00000000000000005e149 < (-.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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 37.1%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 40.5%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot y + x\right) - z \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(\log y \cdot y + x\right) - z \leq 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 31.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\log y \cdot y + x\right) - z\\ t_1 := \left(x \cdot x\right) \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (* (log y) y) x) z)) (t_1 (* (* x x) 0.5)))
   (if (<= t_0 -1e+28) t_1 (if (<= t_0 5e+61) (+ 1.0 x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((log(y) * y) + x) - z;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -1e+28) {
		tmp = t_1;
	} else if (t_0 <= 5e+61) {
		tmp = 1.0 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((log(y) * y) + x) - z
    t_1 = (x * x) * 0.5d0
    if (t_0 <= (-1d+28)) then
        tmp = t_1
    else if (t_0 <= 5d+61) then
        tmp = 1.0d0 + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((Math.log(y) * y) + x) - z;
	double t_1 = (x * x) * 0.5;
	double tmp;
	if (t_0 <= -1e+28) {
		tmp = t_1;
	} else if (t_0 <= 5e+61) {
		tmp = 1.0 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((math.log(y) * y) + x) - z
	t_1 = (x * x) * 0.5
	tmp = 0
	if t_0 <= -1e+28:
		tmp = t_1
	elif t_0 <= 5e+61:
		tmp = 1.0 + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(log(y) * y) + x) - z)
	t_1 = Float64(Float64(x * x) * 0.5)
	tmp = 0.0
	if (t_0 <= -1e+28)
		tmp = t_1;
	elseif (t_0 <= 5e+61)
		tmp = Float64(1.0 + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((log(y) * y) + x) - z;
	t_1 = (x * x) * 0.5;
	tmp = 0.0;
	if (t_0 <= -1e+28)
		tmp = t_1;
	elseif (t_0 <= 5e+61)
		tmp = 1.0 + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+28], t$95$1, If[LessEqual[t$95$0, 5e+61], N[(1.0 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -9.99999999999999958e27 or 5.00000000000000018e61 < (-.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 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.7%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.4%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 25.0%

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

   if -9.99999999999999958e27 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 5.00000000000000018e61

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.3%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 61.2%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot y + x\right) - z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;\left(\log y \cdot y + x\right) - z \leq 5 \cdot 10^{+61}:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 45.0

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.9%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 2.2%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 12.6%

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

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

   1. Initial program 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 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.5%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 41.9%

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

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, x, 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 41.9%

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

4. Final simplification 33.1%

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

Alternative 6: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 50:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (log y) y) 50.0) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(y) * y) <= 50.0) {
		tmp = exp((x - z));
	} else {
		tmp = pow(y, y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((log(y) * y) <= 50.0d0) then
        tmp = exp((x - z))
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((Math.log(y) * y) <= 50.0) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.log(y) * y) <= 50.0:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(y) * y) <= 50.0)
		tmp = exp(Float64(x - z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((log(y) * y) <= 50.0)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision], 50.0], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 50

   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 50 < (*.f64 y (log.f64 y))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.8%

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

4. Final simplification 90.0%

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

Alternative 7: 52.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log y \cdot y + x \leq 5 \cdot 10^{+209}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (* (log y) y) x) 5e+209) (exp x) (fma (* 0.5 x) x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((log(y) * y) + x) <= 5e+209) {
		tmp = exp(x);
	} else {
		tmp = fma((0.5 * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(log(y) * y) + x) <= 5e+209)
		tmp = exp(x);
	else
		tmp = fma(Float64(0.5 * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], 5e+209], N[Exp[x], $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < 4.99999999999999964e209

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.3%

      \[\leadsto e^{x} \]

   if 4.99999999999999964e209 < (+.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 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 37.0%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 46.4%

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

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot x, x, 1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 46.4%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y + x \leq 5 \cdot 10^{+209}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\log y \cdot y + x\right) - z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (+ (* (log y) y) x) z) -1e+28)
   (* (* x x) 0.5)
   (fma (fma 0.5 x 1.0) x 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((log(y) * y) + x) - z) <= -1e+28) {
		tmp = (x * x) * 0.5;
	} else {
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(log(y) * y) + x) - z) <= -1e+28)
		tmp = Float64(Float64(x * x) * 0.5);
	else
		tmp = fma(fma(0.5, x, 1.0), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], -1e+28], N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < -9.99999999999999958e27

   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 44.3

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.5%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 2.2%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 12.8%

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

   if -9.99999999999999958e27 < (-.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 80.2

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.8%

      \[\leadsto e^{x} \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 41.7%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot y + x\right) - z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 1.35) (exp x) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35) {
		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 <= 1.35d0) 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 <= 1.35) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.35:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.35)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3500000000000001

   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 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.5%

      \[\leadsto e^{x} \]

   if 1.3500000000000001 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.2%

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

Alternative 10: 14.3% 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 69.5

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

5. Applied rewrites 69.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 55.2%

   \[\leadsto e^{x} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 15.6%

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

Alternative 11: 14.0% 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 69.5

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

5. Applied rewrites 69.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 51.3%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 15.6%

    \[\leadsto 1 \]
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 2024277 
(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)))