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

Percentage Accurate: 100.0% → 100.0%

Time: 9.5s

Alternatives: 15

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "1.314275994942053939702710864978953837696e304")
  2. reg1 = (bf #e1.119533517885233904088243899662316772968e65)
  3. reg2 = (bf #e-2.159298382034281526581538818962611773596e103)

• 58.8% 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: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t\_0 \leq 100000:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 100000.0) (exp (- x z)) (exp (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (t_0 <= 100000.0) {
		tmp = exp((x - z));
	} else {
		tmp = exp((t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = y * log(y)
    if (t_0 <= 100000.0d0) then
        tmp = exp((x - z))
    else
        tmp = exp((t_0 - z))
    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 <= 100000.0) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp((t_0 - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if t_0 <= 100000.0:
		tmp = math.exp((x - z))
	else:
		tmp = math.exp((t_0 - z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (t_0 <= 100000.0)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(Float64(t_0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (t_0 <= 100000.0)
		tmp = exp((x - z));
	else
		tmp = exp((t_0 - z));
	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, 100000.0], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - z), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 1e5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1e5 < (*.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 x around 0 90.2%

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

Alternative 3: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-11}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-270}:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;x \leq 580:\\ \;\;\;\;{y}^{y}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e-11)
   (exp x)
   (if (<= x -6.2e-270) (exp (- z)) (if (<= x 580.0) (pow y y) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e-11) {
		tmp = exp(x);
	} else if (x <= -6.2e-270) {
		tmp = exp(-z);
	} else if (x <= 580.0) {
		tmp = pow(y, y);
	} else {
		tmp = exp(x);
	}
	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 (x <= (-3d-11)) then
        tmp = exp(x)
    else if (x <= (-6.2d-270)) then
        tmp = exp(-z)
    else if (x <= 580.0d0) then
        tmp = y ** y
    else
        tmp = exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e-11) {
		tmp = Math.exp(x);
	} else if (x <= -6.2e-270) {
		tmp = Math.exp(-z);
	} else if (x <= 580.0) {
		tmp = Math.pow(y, y);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3e-11:
		tmp = math.exp(x)
	elif x <= -6.2e-270:
		tmp = math.exp(-z)
	elif x <= 580.0:
		tmp = math.pow(y, y)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e-11)
		tmp = exp(x);
	elseif (x <= -6.2e-270)
		tmp = exp(Float64(-z));
	elseif (x <= 580.0)
		tmp = y ^ y;
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3e-11)
		tmp = exp(x);
	elseif (x <= -6.2e-270)
		tmp = exp(-z);
	elseif (x <= 580.0)
		tmp = y ^ y;
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e-11], N[Exp[x], $MachinePrecision], If[LessEqual[x, -6.2e-270], N[Exp[(-z)], $MachinePrecision], If[LessEqual[x, 580.0], N[Power[y, y], $MachinePrecision], N[Exp[x], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3e-11 or 580 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 84.8%

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

   if -3e-11 < x < -6.2e-270

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 76.5%

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

      1. neg-mul-1 76.5%

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

   5. Simplified 76.5%

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

   if -6.2e-270 < x < 580

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto e^{\color{blue}{\left(y \cdot \log y + x\right)} - z} \]

      2. associate--l+ 100.0%

         \[\leadsto e^{\color{blue}{y \cdot \log y + \left(x - z\right)}} \]

      3. exp-sum 85.7%

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

      4. *-commutative 85.7%

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

      5. exp-to-pow 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in z around 0 82.3%

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

   6. Taylor expanded in x around 0 82.2%

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

Alternative 4: 72.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-11} \lor \neg \left(x \leq 1.7 \cdot 10^{-28}\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3e-11) (not (<= x 1.7e-28))) (exp x) (exp (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e-11) || !(x <= 1.7e-28)) {
		tmp = exp(x);
	} else {
		tmp = exp(-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 ((x <= (-3d-11)) .or. (.not. (x <= 1.7d-28))) then
        tmp = exp(x)
    else
        tmp = exp(-z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e-11) || !(x <= 1.7e-28)) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.exp(-z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3e-11) or not (x <= 1.7e-28):
		tmp = math.exp(x)
	else:
		tmp = math.exp(-z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3e-11) || !(x <= 1.7e-28))
		tmp = exp(x);
	else
		tmp = exp(Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3e-11) || ~((x <= 1.7e-28)))
		tmp = exp(x);
	else
		tmp = exp(-z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3e-11], N[Not[LessEqual[x, 1.7e-28]], $MachinePrecision]], N[Exp[x], $MachinePrecision], N[Exp[(-z)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3e-11 or 1.7e-28 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.3%

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

   if -3e-11 < x < 1.7e-28

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 67.8%

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

      1. neg-mul-1 67.8%

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

   5. Simplified 67.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-11} \lor \neg \left(x \leq 1.7 \cdot 10^{-28}\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+95}:\\ \;\;\;\;1 + z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+81}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e+95)
   (+ 1.0 (* z (+ (* z (* z -0.16666666666666666)) -1.0)))
   (if (<= z 1.55e+81) (exp x) 0.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+95) {
		tmp = 1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0));
	} else if (z <= 1.55e+81) {
		tmp = exp(x);
	} else {
		tmp = 0.0;
	}
	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) :: tmp
    if (z <= (-3d+95)) then
        tmp = 1.0d0 + (z * ((z * (z * (-0.16666666666666666d0))) + (-1.0d0)))
    else if (z <= 1.55d+81) then
        tmp = exp(x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+95) {
		tmp = 1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0));
	} else if (z <= 1.55e+81) {
		tmp = Math.exp(x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3e+95:
		tmp = 1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0))
	elif z <= 1.55e+81:
		tmp = math.exp(x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e+95)
		tmp = Float64(1.0 + Float64(z * Float64(Float64(z * Float64(z * -0.16666666666666666)) + -1.0)));
	elseif (z <= 1.55e+81)
		tmp = exp(x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3e+95)
		tmp = 1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0));
	elseif (z <= 1.55e+81)
		tmp = exp(x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e+95], N[(1.0 + N[(z * N[(N[(z * N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e+81], N[Exp[x], $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if z < -2.99999999999999991e95

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 95.0%

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

      1. neg-mul-1 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in z around 0 92.6%

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(0.5 + -0.16666666666666666 \cdot z\right) - 1\right)} \]

   7. Taylor expanded in z around inf 92.6%

      \[\leadsto 1 + z \cdot \left(z \cdot \color{blue}{\left(-0.16666666666666666 \cdot z\right)} - 1\right) \]
   8. Step-by-step derivation

      1. *-commutative 92.6%

         \[\leadsto 1 + z \cdot \left(z \cdot \color{blue}{\left(z \cdot -0.16666666666666666\right)} - 1\right) \]

   9. Simplified 92.6%

      \[\leadsto 1 + z \cdot \left(z \cdot \color{blue}{\left(z \cdot -0.16666666666666666\right)} - 1\right) \]

   if -2.99999999999999991e95 < z < 1.55e81

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 62.0%

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

   if 1.55e81 < 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 69.4%

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

      1. neg-mul-1 69.4%

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

   5. Simplified 69.4%

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

   6. Taylor expanded in z around 0 1.5%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 1.5%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 1.5%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 1.5%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 1.5%

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

       1. neg-mul-1 1.5%

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

   11. Simplified 1.5%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 1.1%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 1.1%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 1.1%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 1.1%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 1.1%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 1.1%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 1.1%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 1.1%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 1.1%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 0.0%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 0.0%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 0.0%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 0.0%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 0.0%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 0.0%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 0.0%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 0.0%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 69.4%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 69.4%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 69.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+95}:\\ \;\;\;\;1 + z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+81}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{+63}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.7e+63) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.7e+63) {
		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 <= 5.7d+63) 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 <= 5.7e+63) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.7e+63:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.7e+63)
		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 <= 5.7e+63)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.7e+63], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.7000000000000002e63

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.0%

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

   if 5.7000000000000002e63 < y

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto e^{\color{blue}{\left(y \cdot \log y + x\right)} - z} \]

      2. associate--l+ 100.0%

         \[\leadsto e^{\color{blue}{y \cdot \log y + \left(x - z\right)}} \]

      3. exp-sum 64.2%

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

      4. *-commutative 64.2%

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

      5. exp-to-pow 64.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in z around 0 77.1%

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

   6. Taylor expanded in x around 0 83.7%

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

Alternative 7: 54.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3200:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+101}:\\ \;\;\;\;1 + z \cdot \left(z \cdot \left(0.5 + z \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3200.0)
   0.0
   (if (<= x 2.4e+101)
     (+ 1.0 (* z (+ (* z (+ 0.5 (* z -0.16666666666666666))) -1.0)))
     (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3200.0) {
		tmp = 0.0;
	} else if (x <= 2.4e+101) {
		tmp = 1.0 + (z * ((z * (0.5 + (z * -0.16666666666666666))) + -1.0));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	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 (x <= (-3200.0d0)) then
        tmp = 0.0d0
    else if (x <= 2.4d+101) then
        tmp = 1.0d0 + (z * ((z * (0.5d0 + (z * (-0.16666666666666666d0)))) + (-1.0d0)))
    else
        tmp = 1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3200.0) {
		tmp = 0.0;
	} else if (x <= 2.4e+101) {
		tmp = 1.0 + (z * ((z * (0.5 + (z * -0.16666666666666666))) + -1.0));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3200.0:
		tmp = 0.0
	elif x <= 2.4e+101:
		tmp = 1.0 + (z * ((z * (0.5 + (z * -0.16666666666666666))) + -1.0))
	else:
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3200.0)
		tmp = 0.0;
	elseif (x <= 2.4e+101)
		tmp = Float64(1.0 + Float64(z * Float64(Float64(z * Float64(0.5 + Float64(z * -0.16666666666666666))) + -1.0)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3200.0)
		tmp = 0.0;
	elseif (x <= 2.4e+101)
		tmp = 1.0 + (z * ((z * (0.5 + (z * -0.16666666666666666))) + -1.0));
	else
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3200.0], 0.0, If[LessEqual[x, 2.4e+101], N[(1.0 + N[(z * N[(N[(z * N[(0.5 + N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3200

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 35.8%

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

      1. neg-mul-1 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in z around 0 3.1%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 3.1%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 3.1%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 4.3%

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

       1. neg-mul-1 4.3%

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

   11. Simplified 4.3%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 51.7%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 51.7%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 51.7%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 51.7%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 32.4%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 23.8%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 47.6%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 47.6%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 80.0%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 80.0%

       \[\leadsto \color{blue}{0} \]

   if -3200 < x < 2.39999999999999988e101

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 63.3%

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

      1. neg-mul-1 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in z around 0 35.3%

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(0.5 + -0.16666666666666666 \cdot z\right) - 1\right)} \]

   if 2.39999999999999988e101 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 95.4%

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

   4. Taylor expanded in x around 0 95.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 95.4%

         \[\leadsto 1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right) \]

   6. Simplified 95.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3200:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+101}:\\ \;\;\;\;1 + z \cdot \left(z \cdot \left(0.5 + z \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2900000:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+102}:\\ \;\;\;\;1 + z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2900000.0)
   0.0
   (if (<= x 1.95e+102)
     (+ 1.0 (* z (+ (* z (* z -0.16666666666666666)) -1.0)))
     (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2900000.0) {
		tmp = 0.0;
	} else if (x <= 1.95e+102) {
		tmp = 1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	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 (x <= (-2900000.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.95d+102) then
        tmp = 1.0d0 + (z * ((z * (z * (-0.16666666666666666d0))) + (-1.0d0)))
    else
        tmp = 1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2900000.0) {
		tmp = 0.0;
	} else if (x <= 1.95e+102) {
		tmp = 1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2900000.0:
		tmp = 0.0
	elif x <= 1.95e+102:
		tmp = 1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0))
	else:
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2900000.0)
		tmp = 0.0;
	elseif (x <= 1.95e+102)
		tmp = Float64(1.0 + Float64(z * Float64(Float64(z * Float64(z * -0.16666666666666666)) + -1.0)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2900000.0)
		tmp = 0.0;
	elseif (x <= 1.95e+102)
		tmp = 1.0 + (z * ((z * (z * -0.16666666666666666)) + -1.0));
	else
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2900000.0], 0.0, If[LessEqual[x, 1.95e+102], N[(1.0 + N[(z * N[(N[(z * N[(z * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e6

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 35.8%

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

      1. neg-mul-1 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in z around 0 3.1%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 3.1%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 3.1%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 4.3%

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

       1. neg-mul-1 4.3%

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

   11. Simplified 4.3%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 51.7%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 51.7%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 51.7%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 51.7%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 32.4%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 23.8%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 47.6%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 47.6%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 80.0%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 80.0%

       \[\leadsto \color{blue}{0} \]

   if -2.9e6 < x < 1.9499999999999999e102

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 63.3%

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

      1. neg-mul-1 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in z around 0 35.3%

      \[\leadsto \color{blue}{1 + z \cdot \left(z \cdot \left(0.5 + -0.16666666666666666 \cdot z\right) - 1\right)} \]

   7. Taylor expanded in z around inf 35.2%

      \[\leadsto 1 + z \cdot \left(z \cdot \color{blue}{\left(-0.16666666666666666 \cdot z\right)} - 1\right) \]
   8. Step-by-step derivation

      1. *-commutative 35.2%

         \[\leadsto 1 + z \cdot \left(z \cdot \color{blue}{\left(z \cdot -0.16666666666666666\right)} - 1\right) \]

   9. Simplified 35.2%

      \[\leadsto 1 + z \cdot \left(z \cdot \color{blue}{\left(z \cdot -0.16666666666666666\right)} - 1\right) \]

   if 1.9499999999999999e102 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 95.4%

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

   4. Taylor expanded in x around 0 95.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 95.4%

         \[\leadsto 1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right) \]

   6. Simplified 95.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2900000:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+102}:\\ \;\;\;\;1 + z \cdot \left(z \cdot \left(z \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.5% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+116}:\\ \;\;\;\;1 + z \cdot \left(z \cdot 0.5 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -48.0)
   0.0
   (if (<= x 3.6e+116)
     (+ 1.0 (* z (+ (* z 0.5) -1.0)))
     (+ 1.0 (* x (+ 1.0 (* x 0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -48.0) {
		tmp = 0.0;
	} else if (x <= 3.6e+116) {
		tmp = 1.0 + (z * ((z * 0.5) + -1.0));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	}
	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 (x <= (-48.0d0)) then
        tmp = 0.0d0
    else if (x <= 3.6d+116) then
        tmp = 1.0d0 + (z * ((z * 0.5d0) + (-1.0d0)))
    else
        tmp = 1.0d0 + (x * (1.0d0 + (x * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -48.0) {
		tmp = 0.0;
	} else if (x <= 3.6e+116) {
		tmp = 1.0 + (z * ((z * 0.5) + -1.0));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -48.0:
		tmp = 0.0
	elif x <= 3.6e+116:
		tmp = 1.0 + (z * ((z * 0.5) + -1.0))
	else:
		tmp = 1.0 + (x * (1.0 + (x * 0.5)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -48.0)
		tmp = 0.0;
	elseif (x <= 3.6e+116)
		tmp = Float64(1.0 + Float64(z * Float64(Float64(z * 0.5) + -1.0)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -48.0)
		tmp = 0.0;
	elseif (x <= 3.6e+116)
		tmp = 1.0 + (z * ((z * 0.5) + -1.0));
	else
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -48.0], 0.0, If[LessEqual[x, 3.6e+116], N[(1.0 + N[(z * N[(N[(z * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -48

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 35.8%

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

      1. neg-mul-1 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in z around 0 3.1%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 3.1%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 3.1%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 4.3%

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

       1. neg-mul-1 4.3%

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

   11. Simplified 4.3%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 51.7%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 51.7%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 51.7%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 51.7%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 32.4%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 23.8%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 47.6%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 47.6%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 80.0%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 80.0%

       \[\leadsto \color{blue}{0} \]

   if -48 < x < 3.59999999999999971e116

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 63.0%

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

      1. neg-mul-1 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in z around 0 35.7%

      \[\leadsto \color{blue}{1 + z \cdot \left(0.5 \cdot z - 1\right)} \]

   if 3.59999999999999971e116 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 95.0%

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

   4. Taylor expanded in x around 0 73.8%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + 0.5 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 73.8%

         \[\leadsto 1 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right) \]

   6. Simplified 73.8%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+116}:\\ \;\;\;\;1 + z \cdot \left(z \cdot 0.5 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+118}:\\ \;\;\;\;1 + z \cdot \left(z \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -720.0)
   0.0
   (if (<= x 6.2e+118)
     (+ 1.0 (* z (* z 0.5)))
     (+ 1.0 (* x (+ 1.0 (* x 0.5)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -720.0) {
		tmp = 0.0;
	} else if (x <= 6.2e+118) {
		tmp = 1.0 + (z * (z * 0.5));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	}
	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 (x <= (-720.0d0)) then
        tmp = 0.0d0
    else if (x <= 6.2d+118) then
        tmp = 1.0d0 + (z * (z * 0.5d0))
    else
        tmp = 1.0d0 + (x * (1.0d0 + (x * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -720.0) {
		tmp = 0.0;
	} else if (x <= 6.2e+118) {
		tmp = 1.0 + (z * (z * 0.5));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -720.0:
		tmp = 0.0
	elif x <= 6.2e+118:
		tmp = 1.0 + (z * (z * 0.5))
	else:
		tmp = 1.0 + (x * (1.0 + (x * 0.5)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -720.0)
		tmp = 0.0;
	elseif (x <= 6.2e+118)
		tmp = Float64(1.0 + Float64(z * Float64(z * 0.5)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -720.0)
		tmp = 0.0;
	elseif (x <= 6.2e+118)
		tmp = 1.0 + (z * (z * 0.5));
	else
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -720.0], 0.0, If[LessEqual[x, 6.2e+118], N[(1.0 + N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -720

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 35.8%

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

      1. neg-mul-1 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in z around 0 3.1%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 3.1%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 3.1%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 4.3%

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

       1. neg-mul-1 4.3%

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

   11. Simplified 4.3%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 51.7%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 51.7%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 51.7%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 51.7%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 32.4%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 23.8%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 47.6%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 47.6%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 80.0%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 80.0%

       \[\leadsto \color{blue}{0} \]

   if -720 < x < 6.19999999999999973e118

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 63.0%

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

      1. neg-mul-1 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in z around 0 35.7%

      \[\leadsto \color{blue}{1 + z \cdot \left(0.5 \cdot z - 1\right)} \]

   7. Taylor expanded in z around inf 35.3%

      \[\leadsto 1 + z \cdot \color{blue}{\left(0.5 \cdot z\right)} \]
   8. Step-by-step derivation

      1. *-commutative 35.3%

         \[\leadsto 1 + z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   9. Simplified 35.3%

      \[\leadsto 1 + z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   if 6.19999999999999973e118 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 95.0%

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

   4. Taylor expanded in x around 0 73.8%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + 0.5 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 73.8%

         \[\leadsto 1 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right) \]

   6. Simplified 73.8%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 32.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -42:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+21}:\\ \;\;\;\;1 - z\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -42.0) 0.0 (if (<= x 2.05e+21) (- 1.0 z) 0.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -42.0) {
		tmp = 0.0;
	} else if (x <= 2.05e+21) {
		tmp = 1.0 - z;
	} else {
		tmp = 0.0;
	}
	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) :: tmp
    if (x <= (-42.0d0)) then
        tmp = 0.0d0
    else if (x <= 2.05d+21) then
        tmp = 1.0d0 - z
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -42.0) {
		tmp = 0.0;
	} else if (x <= 2.05e+21) {
		tmp = 1.0 - z;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -42.0:
		tmp = 0.0
	elif x <= 2.05e+21:
		tmp = 1.0 - z
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -42.0)
		tmp = 0.0;
	elseif (x <= 2.05e+21)
		tmp = Float64(1.0 - z);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -42.0)
		tmp = 0.0;
	elseif (x <= 2.05e+21)
		tmp = 1.0 - z;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -42.0], 0.0, If[LessEqual[x, 2.05e+21], N[(1.0 - z), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -42 or 2.05e21 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 37.6%

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

      1. neg-mul-1 37.6%

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

   5. Simplified 37.6%

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

   6. Taylor expanded in z around 0 3.1%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 3.1%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 3.1%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 3.3%

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

       1. neg-mul-1 3.3%

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

   11. Simplified 3.3%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 39.2%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 39.2%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 39.2%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 39.2%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 39.2%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 39.2%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 39.2%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 39.2%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 39.2%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 29.4%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 41.3%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 41.3%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 11.9%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 23.7%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 23.7%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 23.7%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 23.7%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 45.9%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 45.9%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 45.9%

       \[\leadsto \color{blue}{0} \]

   if -42 < x < 2.05e21

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 64.9%

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

      1. neg-mul-1 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in z around 0 24.7%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 24.7%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 24.7%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 24.7%

      \[\leadsto \color{blue}{1 - z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 32.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.031:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.031) 0.0 (if (<= x 3.2e+21) (+ x 1.0) 0.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.031) {
		tmp = 0.0;
	} else if (x <= 3.2e+21) {
		tmp = x + 1.0;
	} else {
		tmp = 0.0;
	}
	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) :: tmp
    if (x <= (-0.031d0)) then
        tmp = 0.0d0
    else if (x <= 3.2d+21) then
        tmp = x + 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.031) {
		tmp = 0.0;
	} else if (x <= 3.2e+21) {
		tmp = x + 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.031:
		tmp = 0.0
	elif x <= 3.2e+21:
		tmp = x + 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.031)
		tmp = 0.0;
	elseif (x <= 3.2e+21)
		tmp = Float64(x + 1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.031)
		tmp = 0.0;
	elseif (x <= 3.2e+21)
		tmp = x + 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.031], 0.0, If[LessEqual[x, 3.2e+21], N[(x + 1.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -0.031 or 3.2e21 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 37.7%

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

      1. neg-mul-1 37.7%

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

   5. Simplified 37.7%

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

   6. Taylor expanded in z around 0 3.2%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 3.2%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 3.2%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 3.2%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 3.4%

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

       1. neg-mul-1 3.4%

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

   11. Simplified 3.4%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 39.2%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 39.2%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 39.2%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 39.2%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 39.2%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 39.2%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 39.2%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 39.2%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 39.2%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 29.5%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 41.4%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 41.4%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 11.9%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 23.8%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 23.8%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 23.8%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 23.8%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 45.9%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 45.9%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 45.9%

       \[\leadsto \color{blue}{0} \]

   if -0.031 < x < 3.2e21

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 26.3%

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

   4. Taylor expanded in x around 0 24.5%

      \[\leadsto \color{blue}{1 + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.031:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;x + 1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.3% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + z \cdot \left(z \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -700.0) 0.0 (+ 1.0 (* z (* z 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -700.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (z * (z * 0.5));
	}
	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 (x <= (-700.0d0)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + (z * (z * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -700.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (z * (z * 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -700.0:
		tmp = 0.0
	else:
		tmp = 1.0 + (z * (z * 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -700.0)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(z * Float64(z * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -700.0)
		tmp = 0.0;
	else
		tmp = 1.0 + (z * (z * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -700.0], 0.0, N[(1.0 + N[(z * N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -700

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 35.8%

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

      1. neg-mul-1 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in z around 0 3.1%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 3.1%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 3.1%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 4.3%

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

       1. neg-mul-1 4.3%

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

   11. Simplified 4.3%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 51.7%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 51.7%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 51.7%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 51.7%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 51.7%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 51.7%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 32.4%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 56.2%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 23.8%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 47.6%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 47.6%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 47.6%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 80.0%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 80.0%

       \[\leadsto \color{blue}{0} \]

   if -700 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 57.0%

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

      1. neg-mul-1 57.0%

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

   5. Simplified 57.0%

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

   6. Taylor expanded in z around 0 31.8%

      \[\leadsto \color{blue}{1 + z \cdot \left(0.5 \cdot z - 1\right)} \]

   7. Taylor expanded in z around inf 31.5%

      \[\leadsto 1 + z \cdot \color{blue}{\left(0.5 \cdot z\right)} \]
   8. Step-by-step derivation

      1. *-commutative 31.5%

         \[\leadsto 1 + z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]

   9. Simplified 31.5%

      \[\leadsto 1 + z \cdot \color{blue}{\left(z \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 32.4% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -42:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -42.0) 0.0 (if (<= x 3.2e+21) 1.0 0.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -42.0) {
		tmp = 0.0;
	} else if (x <= 3.2e+21) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	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) :: tmp
    if (x <= (-42.0d0)) then
        tmp = 0.0d0
    else if (x <= 3.2d+21) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -42.0) {
		tmp = 0.0;
	} else if (x <= 3.2e+21) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -42.0:
		tmp = 0.0
	elif x <= 3.2e+21:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -42.0)
		tmp = 0.0;
	elseif (x <= 3.2e+21)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -42.0)
		tmp = 0.0;
	elseif (x <= 3.2e+21)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -42.0], 0.0, If[LessEqual[x, 3.2e+21], 1.0, 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -42 or 3.2e21 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 37.9%

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

      1. neg-mul-1 37.9%

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

   5. Simplified 37.9%

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

   6. Taylor expanded in z around 0 3.1%

      \[\leadsto \color{blue}{1 + -1 \cdot z} \]
   7. Step-by-step derivation

      1. mul-1-neg 3.1%

         \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

      2. unsub-neg 3.1%

         \[\leadsto \color{blue}{1 - z} \]

   8. Simplified 3.1%

      \[\leadsto \color{blue}{1 - z} \]

   9. Taylor expanded in z around inf 3.3%

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

       1. neg-mul-1 3.3%

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

   11. Simplified 3.3%

       \[\leadsto \color{blue}{-z} \]
   12. Step-by-step derivation

       1. add-log-exp 39.5%

          \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

       2. exp-neg 39.5%

          \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

       3. add-sqr-sqrt 39.5%

          \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

       4. associate-/r* 39.5%

          \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

       5. metadata-eval 39.5%

          \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       6. sqrt-div 39.5%

          \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

       7. exp-neg 39.5%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       8. pow1 39.5%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       9. pow1 39.5%

          \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

       10. add-sqr-sqrt 29.7%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

       11. sqrt-unprod 41.7%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

       12. sqr-neg 41.7%

           \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

       13. sqrt-unprod 12.0%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

       14. add-sqr-sqrt 23.9%

           \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

       15. pow1 23.9%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

       16. pow1 23.9%

           \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

       17. log-div 23.9%

           \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

   13. Applied egg-rr 46.2%

       \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
   14. Step-by-step derivation

       1. +-inverses 46.2%

          \[\leadsto \color{blue}{0} \]

   15. Simplified 46.2%

       \[\leadsto \color{blue}{0} \]

   if -42 < x < 3.2e21

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 26.9%

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

   4. Taylor expanded in x around 0 24.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 30.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around inf 52.1%

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

   1. neg-mul-1 52.1%

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

5. Simplified 52.1%

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

6. Taylor expanded in z around 0 14.6%

   \[\leadsto \color{blue}{1 + -1 \cdot z} \]
7. Step-by-step derivation

   1. mul-1-neg 14.6%

      \[\leadsto 1 + \color{blue}{\left(-z\right)} \]

   2. unsub-neg 14.6%

      \[\leadsto \color{blue}{1 - z} \]

8. Simplified 14.6%

   \[\leadsto \color{blue}{1 - z} \]

9. Taylor expanded in z around inf 3.2%

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

    1. neg-mul-1 3.2%

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

11. Simplified 3.2%

    \[\leadsto \color{blue}{-z} \]
12. Step-by-step derivation

    1. add-log-exp 34.7%

       \[\leadsto \color{blue}{\log \left(e^{-z}\right)} \]

    2. exp-neg 34.7%

       \[\leadsto \log \color{blue}{\left(\frac{1}{e^{z}}\right)} \]

    3. add-sqr-sqrt 34.7%

       \[\leadsto \log \left(\frac{1}{\color{blue}{\sqrt{e^{z}} \cdot \sqrt{e^{z}}}}\right) \]

    4. associate-/r* 34.7%

       \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right)} \]

    5. metadata-eval 34.7%

       \[\leadsto \log \left(\frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{e^{z}}}}{\sqrt{e^{z}}}\right) \]

    6. sqrt-div 34.7%

       \[\leadsto \log \left(\frac{\color{blue}{\sqrt{\frac{1}{e^{z}}}}}{\sqrt{e^{z}}}\right) \]

    7. exp-neg 34.7%

       \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

    8. pow1 34.7%

       \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{-z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

    9. pow1 34.7%

       \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{-z}}}}{\sqrt{e^{z}}}\right) \]

    10. add-sqr-sqrt 29.8%

        \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}}{\sqrt{e^{z}}}\right) \]

    11. sqrt-unprod 35.7%

        \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}}{\sqrt{e^{z}}}\right) \]

    12. sqr-neg 35.7%

        \[\leadsto \log \left(\frac{\sqrt{e^{\sqrt{\color{blue}{z \cdot z}}}}}{\sqrt{e^{z}}}\right) \]

    13. sqrt-unprod 5.8%

        \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}}{\sqrt{e^{z}}}\right) \]

    14. add-sqr-sqrt 11.7%

        \[\leadsto \log \left(\frac{\sqrt{e^{\color{blue}{z}}}}{\sqrt{e^{z}}}\right) \]

    15. pow1 11.7%

        \[\leadsto \log \left(\frac{\sqrt{\color{blue}{{\left(e^{z}\right)}^{1}}}}{\sqrt{e^{z}}}\right) \]

    16. pow1 11.7%

        \[\leadsto \log \left(\frac{\sqrt{\color{blue}{e^{z}}}}{\sqrt{e^{z}}}\right) \]

    17. log-div 11.7%

        \[\leadsto \color{blue}{\log \left(\sqrt{e^{z}}\right) - \log \left(\sqrt{e^{z}}\right)} \]

13. Applied egg-rr 28.7%

    \[\leadsto \color{blue}{0.5 \cdot z - 0.5 \cdot z} \]
14. Step-by-step derivation

    1. +-inverses 28.7%

       \[\leadsto \color{blue}{0} \]

15. Simplified 28.7%

    \[\leadsto \color{blue}{0} \]
16. 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 2024146 
(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)))