Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A

Percentage Accurate: 99.4% → 99.4%

Time: 14.9s

Alternatives: 9

Speedup: 1.0×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
4. Add Preprocessing

Alternative 2: 42.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -1.18 \cdot 10^{+36}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{z} \cdot \left(x \cdot {2}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot t\_1\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{x \cdot \sqrt{z}}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= y -1.18e+36)
     (* t_1 (- y))
     (if (<= y 4.9e-101)
       (* (sqrt z) (* x (pow 2.0 -0.5)))
       (if (<= y 3.1e-5)
         (* (exp (/ (* t t) 2.0)) (* y t_1))
         (if (<= y 1.5e+113)
           (/ (* x (sqrt z)) (sqrt 2.0))
           (- (sqrt (* (* z 2.0) (pow y 2.0))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (y <= -1.18e+36) {
		tmp = t_1 * -y;
	} else if (y <= 4.9e-101) {
		tmp = sqrt(z) * (x * pow(2.0, -0.5));
	} else if (y <= 3.1e-5) {
		tmp = exp(((t * t) / 2.0)) * (y * t_1);
	} else if (y <= 1.5e+113) {
		tmp = (x * sqrt(z)) / sqrt(2.0);
	} else {
		tmp = -sqrt(((z * 2.0) * pow(y, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (y <= (-1.18d+36)) then
        tmp = t_1 * -y
    else if (y <= 4.9d-101) then
        tmp = sqrt(z) * (x * (2.0d0 ** (-0.5d0)))
    else if (y <= 3.1d-5) then
        tmp = exp(((t * t) / 2.0d0)) * (y * t_1)
    else if (y <= 1.5d+113) then
        tmp = (x * sqrt(z)) / sqrt(2.0d0)
    else
        tmp = -sqrt(((z * 2.0d0) * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (y <= -1.18e+36) {
		tmp = t_1 * -y;
	} else if (y <= 4.9e-101) {
		tmp = Math.sqrt(z) * (x * Math.pow(2.0, -0.5));
	} else if (y <= 3.1e-5) {
		tmp = Math.exp(((t * t) / 2.0)) * (y * t_1);
	} else if (y <= 1.5e+113) {
		tmp = (x * Math.sqrt(z)) / Math.sqrt(2.0);
	} else {
		tmp = -Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if y <= -1.18e+36:
		tmp = t_1 * -y
	elif y <= 4.9e-101:
		tmp = math.sqrt(z) * (x * math.pow(2.0, -0.5))
	elif y <= 3.1e-5:
		tmp = math.exp(((t * t) / 2.0)) * (y * t_1)
	elif y <= 1.5e+113:
		tmp = (x * math.sqrt(z)) / math.sqrt(2.0)
	else:
		tmp = -math.sqrt(((z * 2.0) * math.pow(y, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (y <= -1.18e+36)
		tmp = Float64(t_1 * Float64(-y));
	elseif (y <= 4.9e-101)
		tmp = Float64(sqrt(z) * Float64(x * (2.0 ^ -0.5)));
	elseif (y <= 3.1e-5)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * t_1));
	elseif (y <= 1.5e+113)
		tmp = Float64(Float64(x * sqrt(z)) / sqrt(2.0));
	else
		tmp = Float64(-sqrt(Float64(Float64(z * 2.0) * (y ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (y <= -1.18e+36)
		tmp = t_1 * -y;
	elseif (y <= 4.9e-101)
		tmp = sqrt(z) * (x * (2.0 ^ -0.5));
	elseif (y <= 3.1e-5)
		tmp = exp(((t * t) / 2.0)) * (y * t_1);
	elseif (y <= 1.5e+113)
		tmp = (x * sqrt(z)) / sqrt(2.0);
	else
		tmp = -sqrt(((z * 2.0) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.18e+36], N[(t$95$1 * (-y)), $MachinePrecision], If[LessEqual[y, 4.9e-101], N[(N[Sqrt[z], $MachinePrecision] * N[(x * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e-5], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+113], N[(N[(x * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.17999999999999997e36

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 89.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 89.2%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 89.2%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 89.2%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 89.2%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 89.2%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 89.2%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 89.2%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 89.2%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 89.2%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 2.1%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 2.1%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 89.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 89.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 89.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 89.4%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 89.4%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 89.4%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 89.4%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 89.4%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Taylor expanded in t around 0 49.7%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 49.7%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. associate-*l* 49.7%

          \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   12. Simplified 49.7%

       \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
   13. Step-by-step derivation

       1. sqrt-prod 49.9%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

       2. add0 49.9%

          \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]

   14. Applied egg-rr 49.9%

       \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]
   15. Step-by-step derivation

       1. add0 49.9%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   16. Simplified 49.9%

       \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   if -1.17999999999999997e36 < y < 4.9e-101

   1. Initial program 99.7%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.7%

         \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. flip-+ 66.8%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 66.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 66.8%

         \[\leadsto \frac{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      5. swap-sqr 58.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      6. add-sqr-sqrt 58.5%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 58.5%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      8. fma-neg 58.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot 2, \left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right), -0\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. pow2 58.5%

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}, -0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 58.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, \color{blue}{0}\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      11. fma-neg 58.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, -0\right)}} \cdot e^{\frac{t \cdot t}{2}} \]

      12. metadata-eval 58.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, \color{blue}{0}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      13. fma-define 58.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      14. add0 58.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 58.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. fma-undefine 58.5%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2} + 0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add0 58.5%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*l* 58.5%

         \[\leadsto \frac{\color{blue}{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      4. associate-/l* 69.7%

         \[\leadsto \color{blue}{\left(z \cdot \frac{2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-/l* 69.7%

         \[\leadsto \left(z \cdot \color{blue}{\left(2 \cdot \frac{{\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. associate-/r* 75.2%

         \[\leadsto \left(z \cdot \left(2 \cdot \color{blue}{\frac{\frac{{\left(x \cdot 0.5 - y\right)}^{2}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. unpow2 75.2%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-/l* 94.6%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{x \cdot 0.5 - y}{x \cdot 0.5 - y}}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-inverses 94.6%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\left(x \cdot 0.5 - y\right) \cdot \color{blue}{1}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. *-rgt-identity 94.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative 94.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{0.5 \cdot x} - y}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. *-commutative 94.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{\color{blue}{2 \cdot z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Simplified 94.6%

      \[\leadsto \color{blue}{\left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{2 \cdot z}}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around inf 86.6%

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. Taylor expanded in t around 0 52.8%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \]
   9. Step-by-step derivation

      1. div-inv 52.7%

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt{2}}\right)} \cdot \sqrt{z} \]

      2. pow1/2 52.7%

         \[\leadsto \left(x \cdot \frac{1}{\color{blue}{{2}^{0.5}}}\right) \cdot \sqrt{z} \]

      3. pow-flip 52.8%

         \[\leadsto \left(x \cdot \color{blue}{{2}^{\left(-0.5\right)}}\right) \cdot \sqrt{z} \]

      4. metadata-eval 52.8%

         \[\leadsto \left(x \cdot {2}^{\color{blue}{-0.5}}\right) \cdot \sqrt{z} \]

   10. Applied egg-rr 52.8%

       \[\leadsto \color{blue}{\left(x \cdot {2}^{-0.5}\right)} \cdot \sqrt{z} \]

   if 4.9e-101 < y < 3.10000000000000014e-5

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.1%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 54.0%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 54.0%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 54.0%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 54.0%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \left(y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. sqrt-unprod 42.5%

         \[\leadsto \left(y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. sqr-neg 42.5%

         \[\leadsto \left(y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 42.5%

         \[\leadsto \left(y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqrt-prod 42.5%

         \[\leadsto \left(y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. add0 42.5%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 42.5%

      \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. add0 42.5%

         \[\leadsto \left(y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 42.5%

      \[\leadsto \left(y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   if 3.10000000000000014e-5 < y < 1.5e113

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.8%

         \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. flip-+ 74.2%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 74.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 74.2%

         \[\leadsto \frac{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      5. swap-sqr 70.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      6. add-sqr-sqrt 70.5%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 70.5%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      8. fma-neg 70.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot 2, \left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right), -0\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. pow2 70.5%

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}, -0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 70.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, \color{blue}{0}\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      11. fma-neg 70.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, -0\right)}} \cdot e^{\frac{t \cdot t}{2}} \]

      12. metadata-eval 70.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, \color{blue}{0}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      13. fma-define 70.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      14. add0 70.5%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 70.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. fma-undefine 70.5%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2} + 0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add0 70.5%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*l* 70.5%

         \[\leadsto \frac{\color{blue}{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      4. associate-/l* 73.9%

         \[\leadsto \color{blue}{\left(z \cdot \frac{2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-/l* 73.9%

         \[\leadsto \left(z \cdot \color{blue}{\left(2 \cdot \frac{{\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. associate-/r* 81.6%

         \[\leadsto \left(z \cdot \left(2 \cdot \color{blue}{\frac{\frac{{\left(x \cdot 0.5 - y\right)}^{2}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. unpow2 81.6%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-/l* 99.6%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{x \cdot 0.5 - y}{x \cdot 0.5 - y}}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-inverses 99.6%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\left(x \cdot 0.5 - y\right) \cdot \color{blue}{1}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. *-rgt-identity 99.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative 99.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{0.5 \cdot x} - y}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. *-commutative 99.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{\color{blue}{2 \cdot z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{2 \cdot z}}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around inf 69.9%

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. Taylor expanded in t around 0 37.0%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \]
   9. Step-by-step derivation

      1. associate-*l/ 37.0%

         \[\leadsto \color{blue}{\frac{x \cdot \sqrt{z}}{\sqrt{2}}} \]

   10. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{\frac{x \cdot \sqrt{z}}{\sqrt{2}}} \]

   if 1.5e113 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 91.7%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 91.8%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 91.8%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 91.8%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 91.7%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 91.7%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 91.7%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 91.7%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 91.7%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 0.6%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 0.6%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 91.8%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 91.8%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 91.8%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 91.9%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 91.9%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 91.9%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 91.9%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 91.9%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Taylor expanded in t around 0 50.6%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 50.6%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. associate-*l* 50.6%

          \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   12. Simplified 50.6%

       \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
   13. Step-by-step derivation

       1. rem-square-sqrt 50.6%

          \[\leadsto -\color{blue}{\sqrt{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \cdot \sqrt{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}} \]

       2. sqrt-unprod 68.2%

          \[\leadsto -\color{blue}{\sqrt{\left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)}} \]

       3. sqrt-prod 68.2%

          \[\leadsto -\sqrt{\left(y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot \left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)} \]

       4. *-commutative 68.2%

          \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{2 \cdot z} \cdot y\right)} \cdot \left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)} \]

       5. sqrt-prod 68.2%

          \[\leadsto -\sqrt{\left(\sqrt{2 \cdot z} \cdot y\right) \cdot \left(y \cdot \color{blue}{\sqrt{2 \cdot z}}\right)} \]

       6. *-commutative 68.2%

          \[\leadsto -\sqrt{\left(\sqrt{2 \cdot z} \cdot y\right) \cdot \color{blue}{\left(\sqrt{2 \cdot z} \cdot y\right)}} \]

       7. swap-sqr 64.1%

          \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z}\right) \cdot \left(y \cdot y\right)}} \]

       8. add-sqr-sqrt 64.1%

          \[\leadsto -\sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot \left(y \cdot y\right)} \]

       9. pow2 64.1%

          \[\leadsto -\sqrt{\left(2 \cdot z\right) \cdot \color{blue}{{y}^{2}}} \]

   14. Applied egg-rr 64.1%

       \[\leadsto -\color{blue}{\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+36}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-101}:\\ \;\;\;\;\sqrt{z} \cdot \left(x \cdot {2}^{-0.5}\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{x \cdot \sqrt{z}}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := e^{\frac{t \cdot t}{2}}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-42} \lor \neg \left(y \leq 3.7 \cdot 10^{+87}\right):\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\left(x \cdot 0.5\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (exp (/ (* t t) 2.0))))
   (if (or (<= y -1.32e-42) (not (<= y 3.7e+87)))
     (* t_2 (* t_1 (- y)))
     (* t_2 (* (* x 0.5) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = exp(((t * t) / 2.0));
	double tmp;
	if ((y <= -1.32e-42) || !(y <= 3.7e+87)) {
		tmp = t_2 * (t_1 * -y);
	} else {
		tmp = t_2 * ((x * 0.5) * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = exp(((t * t) / 2.0d0))
    if ((y <= (-1.32d-42)) .or. (.not. (y <= 3.7d+87))) then
        tmp = t_2 * (t_1 * -y)
    else
        tmp = t_2 * ((x * 0.5d0) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double t_2 = Math.exp(((t * t) / 2.0));
	double tmp;
	if ((y <= -1.32e-42) || !(y <= 3.7e+87)) {
		tmp = t_2 * (t_1 * -y);
	} else {
		tmp = t_2 * ((x * 0.5) * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	t_2 = math.exp(((t * t) / 2.0))
	tmp = 0
	if (y <= -1.32e-42) or not (y <= 3.7e+87):
		tmp = t_2 * (t_1 * -y)
	else:
		tmp = t_2 * ((x * 0.5) * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = exp(Float64(Float64(t * t) / 2.0))
	tmp = 0.0
	if ((y <= -1.32e-42) || !(y <= 3.7e+87))
		tmp = Float64(t_2 * Float64(t_1 * Float64(-y)));
	else
		tmp = Float64(t_2 * Float64(Float64(x * 0.5) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	t_2 = exp(((t * t) / 2.0));
	tmp = 0.0;
	if ((y <= -1.32e-42) || ~((y <= 3.7e+87)))
		tmp = t_2 * (t_1 * -y);
	else
		tmp = t_2 * ((x * 0.5) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -1.32e-42], N[Not[LessEqual[y, 3.7e+87]], $MachinePrecision]], N[(t$95$2 * N[(t$95$1 * (-y)), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000006e-42 or 3.70000000000000003e87 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 88.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 88.0%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 87.9%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 87.9%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 87.9%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 87.9%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 88.0%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 88.0%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 88.0%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 88.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 88.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 2.3%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 2.3%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 87.9%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 87.9%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 87.9%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 88.1%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 88.1%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 88.1%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 88.1%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 88.1%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 88.1%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   if -1.32000000000000006e-42 < y < 3.70000000000000003e87

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \left(\left(x \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. associate-*r* 81.8%

         \[\leadsto \color{blue}{\left(\left(0.5 \cdot \left(x \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. *-commutative 81.8%

         \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{2}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*r* 81.8%

         \[\leadsto \left(\sqrt{z} \cdot \color{blue}{\left(\left(0.5 \cdot x\right) \cdot \sqrt{2}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 81.8%

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{2}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. add0 81.8%

         \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{2}\right) + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. *-commutative 81.8%

         \[\leadsto \left(\sqrt{z} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(0.5 \cdot x\right)\right)} + 0\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*r* 81.8%

         \[\leadsto \left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(0.5 \cdot x\right)} + 0\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. sqrt-prod 81.9%

         \[\leadsto \left(\color{blue}{\sqrt{z \cdot 2}} \cdot \left(0.5 \cdot x\right) + 0\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. *-commutative 81.9%

         \[\leadsto \left(\sqrt{\color{blue}{2 \cdot z}} \cdot \left(0.5 \cdot x\right) + 0\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x\right) + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. add0 81.9%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 81.9%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-42} \lor \neg \left(y \leq 3.7 \cdot 10^{+87}\right):\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+205}:\\ \;\;\;\;\frac{x}{\sqrt{2}} \cdot \sqrt{z}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+256}:\\ \;\;\;\;x \cdot \frac{\sqrt{z}}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{z \cdot {x}^{2}}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.75e+205)
   (* (/ x (sqrt 2.0)) (sqrt z))
   (if (<= x 5.2e+153)
     (* (exp (/ (* t t) 2.0)) (* (sqrt (* z 2.0)) (- y)))
     (if (<= x 3.5e+256)
       (* x (/ (sqrt z) (sqrt 2.0)))
       (sqrt (/ (* z (pow x 2.0)) 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.75e+205) {
		tmp = (x / sqrt(2.0)) * sqrt(z);
	} else if (x <= 5.2e+153) {
		tmp = exp(((t * t) / 2.0)) * (sqrt((z * 2.0)) * -y);
	} else if (x <= 3.5e+256) {
		tmp = x * (sqrt(z) / sqrt(2.0));
	} else {
		tmp = sqrt(((z * pow(x, 2.0)) / 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.75d+205)) then
        tmp = (x / sqrt(2.0d0)) * sqrt(z)
    else if (x <= 5.2d+153) then
        tmp = exp(((t * t) / 2.0d0)) * (sqrt((z * 2.0d0)) * -y)
    else if (x <= 3.5d+256) then
        tmp = x * (sqrt(z) / sqrt(2.0d0))
    else
        tmp = sqrt(((z * (x ** 2.0d0)) / 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.75e+205) {
		tmp = (x / Math.sqrt(2.0)) * Math.sqrt(z);
	} else if (x <= 5.2e+153) {
		tmp = Math.exp(((t * t) / 2.0)) * (Math.sqrt((z * 2.0)) * -y);
	} else if (x <= 3.5e+256) {
		tmp = x * (Math.sqrt(z) / Math.sqrt(2.0));
	} else {
		tmp = Math.sqrt(((z * Math.pow(x, 2.0)) / 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.75e+205:
		tmp = (x / math.sqrt(2.0)) * math.sqrt(z)
	elif x <= 5.2e+153:
		tmp = math.exp(((t * t) / 2.0)) * (math.sqrt((z * 2.0)) * -y)
	elif x <= 3.5e+256:
		tmp = x * (math.sqrt(z) / math.sqrt(2.0))
	else:
		tmp = math.sqrt(((z * math.pow(x, 2.0)) / 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.75e+205)
		tmp = Float64(Float64(x / sqrt(2.0)) * sqrt(z));
	elseif (x <= 5.2e+153)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(sqrt(Float64(z * 2.0)) * Float64(-y)));
	elseif (x <= 3.5e+256)
		tmp = Float64(x * Float64(sqrt(z) / sqrt(2.0)));
	else
		tmp = sqrt(Float64(Float64(z * (x ^ 2.0)) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.75e+205)
		tmp = (x / sqrt(2.0)) * sqrt(z);
	elseif (x <= 5.2e+153)
		tmp = exp(((t * t) / 2.0)) * (sqrt((z * 2.0)) * -y);
	elseif (x <= 3.5e+256)
		tmp = x * (sqrt(z) / sqrt(2.0));
	else
		tmp = sqrt(((z * (x ^ 2.0)) / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.75e+205], N[(N[(x / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e+153], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+256], N[(x * N[(N[Sqrt[z], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.7499999999999999e205

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.9%

         \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. flip-+ 22.7%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 22.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 22.7%

         \[\leadsto \frac{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      5. swap-sqr 22.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      6. add-sqr-sqrt 22.7%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 22.7%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      8. fma-neg 22.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot 2, \left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right), -0\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. pow2 22.7%

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}, -0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 22.7%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, \color{blue}{0}\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      11. fma-neg 22.7%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, -0\right)}} \cdot e^{\frac{t \cdot t}{2}} \]

      12. metadata-eval 22.7%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, \color{blue}{0}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      13. fma-define 22.7%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      14. add0 22.7%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 22.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. fma-undefine 22.7%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2} + 0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add0 22.7%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*l* 22.7%

         \[\leadsto \frac{\color{blue}{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      4. associate-/l* 22.7%

         \[\leadsto \color{blue}{\left(z \cdot \frac{2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-/l* 22.7%

         \[\leadsto \left(z \cdot \color{blue}{\left(2 \cdot \frac{{\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. associate-/r* 72.7%

         \[\leadsto \left(z \cdot \left(2 \cdot \color{blue}{\frac{\frac{{\left(x \cdot 0.5 - y\right)}^{2}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. unpow2 72.7%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-/l* 84.2%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{x \cdot 0.5 - y}{x \cdot 0.5 - y}}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-inverses 84.2%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\left(x \cdot 0.5 - y\right) \cdot \color{blue}{1}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. *-rgt-identity 84.2%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative 84.2%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{0.5 \cdot x} - y}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. *-commutative 84.2%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{\color{blue}{2 \cdot z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Simplified 84.2%

      \[\leadsto \color{blue}{\left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{2 \cdot z}}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around inf 95.8%

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. Taylor expanded in t around 0 68.9%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \]

   if -1.7499999999999999e205 < x < 5.1999999999999998e153

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 74.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 74.4%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 74.4%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 74.4%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 74.4%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 74.4%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 74.4%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 74.4%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 74.4%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 74.4%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 8.3%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 8.3%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 74.4%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 74.4%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 74.4%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 74.5%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 74.5%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 74.5%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 74.5%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 74.5%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 74.5%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   if 5.1999999999999998e153 < x < 3.4999999999999998e256

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.8%

         \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. flip-+ 52.3%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 52.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 52.3%

         \[\leadsto \frac{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      5. swap-sqr 40.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      6. add-sqr-sqrt 40.3%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 40.3%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      8. fma-neg 40.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot 2, \left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right), -0\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. pow2 40.3%

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}, -0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 40.3%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, \color{blue}{0}\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      11. fma-neg 40.3%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, -0\right)}} \cdot e^{\frac{t \cdot t}{2}} \]

      12. metadata-eval 40.3%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, \color{blue}{0}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      13. fma-define 40.3%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      14. add0 40.3%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 40.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. fma-undefine 40.3%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2} + 0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add0 40.3%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*l* 40.3%

         \[\leadsto \frac{\color{blue}{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      4. associate-/l* 40.3%

         \[\leadsto \color{blue}{\left(z \cdot \frac{2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-/l* 40.3%

         \[\leadsto \left(z \cdot \color{blue}{\left(2 \cdot \frac{{\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. associate-/r* 48.7%

         \[\leadsto \left(z \cdot \left(2 \cdot \color{blue}{\frac{\frac{{\left(x \cdot 0.5 - y\right)}^{2}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. unpow2 48.7%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-/l* 95.6%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{x \cdot 0.5 - y}{x \cdot 0.5 - y}}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-inverses 95.6%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\left(x \cdot 0.5 - y\right) \cdot \color{blue}{1}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. *-rgt-identity 95.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative 95.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{0.5 \cdot x} - y}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. *-commutative 95.6%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{\color{blue}{2 \cdot z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Simplified 95.6%

      \[\leadsto \color{blue}{\left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{2 \cdot z}}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around inf 96.9%

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. Taylor expanded in t around 0 69.2%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \]
   9. Step-by-step derivation

      1. *-commutative 69.2%

         \[\leadsto \color{blue}{\sqrt{z} \cdot \frac{x}{\sqrt{2}}} \]

      2. clear-num 69.3%

         \[\leadsto \sqrt{z} \cdot \color{blue}{\frac{1}{\frac{\sqrt{2}}{x}}} \]

      3. un-div-inv 69.2%

         \[\leadsto \color{blue}{\frac{\sqrt{z}}{\frac{\sqrt{2}}{x}}} \]

   10. Applied egg-rr 69.2%

       \[\leadsto \color{blue}{\frac{\sqrt{z}}{\frac{\sqrt{2}}{x}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 69.3%

          \[\leadsto \color{blue}{\frac{\sqrt{z}}{\sqrt{2}} \cdot x} \]

   12. Simplified 69.3%

       \[\leadsto \color{blue}{\frac{\sqrt{z}}{\sqrt{2}} \cdot x} \]

   if 3.4999999999999998e256 < x

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.9%

         \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. flip-+ 54.4%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 54.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 54.4%

         \[\leadsto \frac{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      5. swap-sqr 54.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      6. add-sqr-sqrt 54.4%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 54.4%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      8. fma-neg 54.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot 2, \left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right), -0\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. pow2 54.4%

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}, -0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 54.4%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, \color{blue}{0}\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      11. fma-neg 54.4%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, -0\right)}} \cdot e^{\frac{t \cdot t}{2}} \]

      12. metadata-eval 54.4%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, \color{blue}{0}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      13. fma-define 54.4%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      14. add0 54.4%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. fma-undefine 54.4%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2} + 0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add0 54.4%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*l* 54.4%

         \[\leadsto \frac{\color{blue}{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      4. associate-/l* 54.4%

         \[\leadsto \color{blue}{\left(z \cdot \frac{2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-/l* 54.4%

         \[\leadsto \left(z \cdot \color{blue}{\left(2 \cdot \frac{{\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. associate-/r* 92.9%

         \[\leadsto \left(z \cdot \left(2 \cdot \color{blue}{\frac{\frac{{\left(x \cdot 0.5 - y\right)}^{2}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. unpow2 92.9%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-/l* 100.0%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{x \cdot 0.5 - y}{x \cdot 0.5 - y}}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-inverses 100.0%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\left(x \cdot 0.5 - y\right) \cdot \color{blue}{1}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. *-rgt-identity 100.0%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative 100.0%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{0.5 \cdot x} - y}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. *-commutative 100.0%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{\color{blue}{2 \cdot z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{2 \cdot z}}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. Taylor expanded in t around 0 50.3%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 50.4%

         \[\leadsto \color{blue}{\sqrt{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \cdot \sqrt{\frac{x}{\sqrt{2}} \cdot \sqrt{z}}} \]

      2. sqrt-unprod 92.9%

         \[\leadsto \color{blue}{\sqrt{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right) \cdot \left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)}} \]

      3. *-commutative 92.9%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z} \cdot \frac{x}{\sqrt{2}}\right)} \cdot \left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \]

      4. *-commutative 92.9%

         \[\leadsto \sqrt{\left(\sqrt{z} \cdot \frac{x}{\sqrt{2}}\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \frac{x}{\sqrt{2}}\right)}} \]

      5. swap-sqr 92.9%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\frac{x}{\sqrt{2}} \cdot \frac{x}{\sqrt{2}}\right)}} \]

      6. add-sqr-sqrt 92.9%

         \[\leadsto \sqrt{\color{blue}{z} \cdot \left(\frac{x}{\sqrt{2}} \cdot \frac{x}{\sqrt{2}}\right)} \]

      7. frac-times 92.9%

         \[\leadsto \sqrt{z \cdot \color{blue}{\frac{x \cdot x}{\sqrt{2} \cdot \sqrt{2}}}} \]

      8. pow2 92.9%

         \[\leadsto \sqrt{z \cdot \frac{\color{blue}{{x}^{2}}}{\sqrt{2} \cdot \sqrt{2}}} \]

      9. rem-square-sqrt 92.9%

         \[\leadsto \sqrt{z \cdot \frac{{x}^{2}}{\color{blue}{2}}} \]

   10. Applied egg-rr 92.9%

       \[\leadsto \color{blue}{\sqrt{z \cdot \frac{{x}^{2}}{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 92.9%

          \[\leadsto \sqrt{\color{blue}{\frac{z \cdot {x}^{2}}{2}}} \]

   12. Simplified 92.9%

       \[\leadsto \color{blue}{\sqrt{\frac{z \cdot {x}^{2}}{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+205}:\\ \;\;\;\;\frac{x}{\sqrt{2}} \cdot \sqrt{z}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+153}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+256}:\\ \;\;\;\;x \cdot \frac{\sqrt{z}}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{z \cdot {x}^{2}}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{z} \cdot \left(x \cdot {2}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.6e+35)
   (* (sqrt (* z 2.0)) (- y))
   (if (<= y 1.4e+113)
     (* (sqrt z) (* x (pow 2.0 -0.5)))
     (- (sqrt (* (* z 2.0) (pow y 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.6e+35) {
		tmp = sqrt((z * 2.0)) * -y;
	} else if (y <= 1.4e+113) {
		tmp = sqrt(z) * (x * pow(2.0, -0.5));
	} else {
		tmp = -sqrt(((z * 2.0) * pow(y, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.6d+35)) then
        tmp = sqrt((z * 2.0d0)) * -y
    else if (y <= 1.4d+113) then
        tmp = sqrt(z) * (x * (2.0d0 ** (-0.5d0)))
    else
        tmp = -sqrt(((z * 2.0d0) * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.6e+35) {
		tmp = Math.sqrt((z * 2.0)) * -y;
	} else if (y <= 1.4e+113) {
		tmp = Math.sqrt(z) * (x * Math.pow(2.0, -0.5));
	} else {
		tmp = -Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.6e+35:
		tmp = math.sqrt((z * 2.0)) * -y
	elif y <= 1.4e+113:
		tmp = math.sqrt(z) * (x * math.pow(2.0, -0.5))
	else:
		tmp = -math.sqrt(((z * 2.0) * math.pow(y, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.6e+35)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(-y));
	elseif (y <= 1.4e+113)
		tmp = Float64(sqrt(z) * Float64(x * (2.0 ^ -0.5)));
	else
		tmp = Float64(-sqrt(Float64(Float64(z * 2.0) * (y ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.6e+35)
		tmp = sqrt((z * 2.0)) * -y;
	elseif (y <= 1.4e+113)
		tmp = sqrt(z) * (x * (2.0 ^ -0.5));
	else
		tmp = -sqrt(((z * 2.0) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.6e+35], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[y, 1.4e+113], N[(N[Sqrt[z], $MachinePrecision] * N[(x * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -1.59999999999999991e35

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 89.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 89.2%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 89.2%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 89.2%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 89.2%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 89.2%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 89.2%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 89.2%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 89.2%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 89.2%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 2.1%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 2.1%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 89.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 89.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 89.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 89.4%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 89.4%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 89.4%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 89.4%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 89.4%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Taylor expanded in t around 0 49.7%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 49.7%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. associate-*l* 49.7%

          \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   12. Simplified 49.7%

       \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
   13. Step-by-step derivation

       1. sqrt-prod 49.9%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

       2. add0 49.9%

          \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]

   14. Applied egg-rr 49.9%

       \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]
   15. Step-by-step derivation

       1. add0 49.9%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   16. Simplified 49.9%

       \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   if -1.59999999999999991e35 < y < 1.39999999999999999e113

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.8%

         \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. flip-+ 70.9%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 70.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 70.9%

         \[\leadsto \frac{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      5. swap-sqr 64.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      6. add-sqr-sqrt 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 64.8%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      8. fma-neg 64.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot 2, \left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right), -0\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. pow2 64.8%

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}, -0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, \color{blue}{0}\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      11. fma-neg 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, -0\right)}} \cdot e^{\frac{t \cdot t}{2}} \]

      12. metadata-eval 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, \color{blue}{0}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      13. fma-define 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      14. add0 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. fma-undefine 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2} + 0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add0 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*l* 64.8%

         \[\leadsto \frac{\color{blue}{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      4. associate-/l* 73.4%

         \[\leadsto \color{blue}{\left(z \cdot \frac{2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-/l* 73.4%

         \[\leadsto \left(z \cdot \color{blue}{\left(2 \cdot \frac{{\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. associate-/r* 80.5%

         \[\leadsto \left(z \cdot \left(2 \cdot \color{blue}{\frac{\frac{{\left(x \cdot 0.5 - y\right)}^{2}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. unpow2 80.5%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-/l* 96.4%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{x \cdot 0.5 - y}{x \cdot 0.5 - y}}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-inverses 96.4%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\left(x \cdot 0.5 - y\right) \cdot \color{blue}{1}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. *-rgt-identity 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{0.5 \cdot x} - y}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. *-commutative 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{\color{blue}{2 \cdot z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Simplified 96.4%

      \[\leadsto \color{blue}{\left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{2 \cdot z}}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around inf 79.0%

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. Taylor expanded in t around 0 44.3%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \]
   9. Step-by-step derivation

      1. div-inv 44.3%

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt{2}}\right)} \cdot \sqrt{z} \]

      2. pow1/2 44.3%

         \[\leadsto \left(x \cdot \frac{1}{\color{blue}{{2}^{0.5}}}\right) \cdot \sqrt{z} \]

      3. pow-flip 44.3%

         \[\leadsto \left(x \cdot \color{blue}{{2}^{\left(-0.5\right)}}\right) \cdot \sqrt{z} \]

      4. metadata-eval 44.3%

         \[\leadsto \left(x \cdot {2}^{\color{blue}{-0.5}}\right) \cdot \sqrt{z} \]

   10. Applied egg-rr 44.3%

       \[\leadsto \color{blue}{\left(x \cdot {2}^{-0.5}\right)} \cdot \sqrt{z} \]

   if 1.39999999999999999e113 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 91.7%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 91.8%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 91.8%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 91.8%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 91.7%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 91.7%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 91.7%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 91.7%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 91.7%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 0.6%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 0.6%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 91.8%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 91.8%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 91.8%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 91.9%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 91.9%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 91.9%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 91.9%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 91.9%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Taylor expanded in t around 0 50.6%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 50.6%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. associate-*l* 50.6%

          \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   12. Simplified 50.6%

       \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
   13. Step-by-step derivation

       1. rem-square-sqrt 50.6%

          \[\leadsto -\color{blue}{\sqrt{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \cdot \sqrt{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}} \]

       2. sqrt-unprod 68.2%

          \[\leadsto -\color{blue}{\sqrt{\left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)}} \]

       3. sqrt-prod 68.2%

          \[\leadsto -\sqrt{\left(y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot \left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)} \]

       4. *-commutative 68.2%

          \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{2 \cdot z} \cdot y\right)} \cdot \left(y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)} \]

       5. sqrt-prod 68.2%

          \[\leadsto -\sqrt{\left(\sqrt{2 \cdot z} \cdot y\right) \cdot \left(y \cdot \color{blue}{\sqrt{2 \cdot z}}\right)} \]

       6. *-commutative 68.2%

          \[\leadsto -\sqrt{\left(\sqrt{2 \cdot z} \cdot y\right) \cdot \color{blue}{\left(\sqrt{2 \cdot z} \cdot y\right)}} \]

       7. swap-sqr 64.1%

          \[\leadsto -\sqrt{\color{blue}{\left(\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z}\right) \cdot \left(y \cdot y\right)}} \]

       8. add-sqr-sqrt 64.1%

          \[\leadsto -\sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot \left(y \cdot y\right)} \]

       9. pow2 64.1%

          \[\leadsto -\sqrt{\left(2 \cdot z\right) \cdot \color{blue}{{y}^{2}}} \]

   14. Applied egg-rr 64.1%

       \[\leadsto -\color{blue}{\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{z} \cdot \left(x \cdot {2}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+36} \lor \neg \left(y \leq 1.5 \cdot 10^{+113}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(x \cdot {2}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1e+36) (not (<= y 1.5e+113)))
   (* (sqrt (* z 2.0)) (- y))
   (* (sqrt z) (* x (pow 2.0 -0.5)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1e+36) || !(y <= 1.5e+113)) {
		tmp = sqrt((z * 2.0)) * -y;
	} else {
		tmp = sqrt(z) * (x * pow(2.0, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1d+36)) .or. (.not. (y <= 1.5d+113))) then
        tmp = sqrt((z * 2.0d0)) * -y
    else
        tmp = sqrt(z) * (x * (2.0d0 ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1e+36) || !(y <= 1.5e+113)) {
		tmp = Math.sqrt((z * 2.0)) * -y;
	} else {
		tmp = Math.sqrt(z) * (x * Math.pow(2.0, -0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1e+36) or not (y <= 1.5e+113):
		tmp = math.sqrt((z * 2.0)) * -y
	else:
		tmp = math.sqrt(z) * (x * math.pow(2.0, -0.5))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1e+36) || !(y <= 1.5e+113))
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(-y));
	else
		tmp = Float64(sqrt(z) * Float64(x * (2.0 ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1e+36) || ~((y <= 1.5e+113)))
		tmp = sqrt((z * 2.0)) * -y;
	else
		tmp = sqrt(z) * (x * (2.0 ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1e+36], N[Not[LessEqual[y, 1.5e+113]], $MachinePrecision]], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(x * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000004e36 or 1.5e113 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 90.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 90.3%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 90.3%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 90.3%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 90.3%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 90.3%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 90.3%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 90.3%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 90.3%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 90.3%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 1.5%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 1.5%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 90.3%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 90.3%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 90.3%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 90.5%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 90.5%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 90.5%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 90.5%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 90.5%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Taylor expanded in t around 0 50.1%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 50.1%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. associate-*l* 50.1%

          \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   12. Simplified 50.1%

       \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
   13. Step-by-step derivation

       1. sqrt-prod 50.3%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

       2. add0 50.3%

          \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]

   14. Applied egg-rr 50.3%

       \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]
   15. Step-by-step derivation

       1. add0 50.3%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   16. Simplified 50.3%

       \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   if -1.00000000000000004e36 < y < 1.5e113

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.8%

         \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. flip-+ 70.9%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 70.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 70.9%

         \[\leadsto \frac{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      5. swap-sqr 64.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      6. add-sqr-sqrt 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 64.8%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      8. fma-neg 64.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot 2, \left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right), -0\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. pow2 64.8%

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}, -0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, \color{blue}{0}\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      11. fma-neg 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, -0\right)}} \cdot e^{\frac{t \cdot t}{2}} \]

      12. metadata-eval 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, \color{blue}{0}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      13. fma-define 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      14. add0 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. fma-undefine 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2} + 0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add0 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*l* 64.8%

         \[\leadsto \frac{\color{blue}{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      4. associate-/l* 73.4%

         \[\leadsto \color{blue}{\left(z \cdot \frac{2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-/l* 73.4%

         \[\leadsto \left(z \cdot \color{blue}{\left(2 \cdot \frac{{\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. associate-/r* 80.5%

         \[\leadsto \left(z \cdot \left(2 \cdot \color{blue}{\frac{\frac{{\left(x \cdot 0.5 - y\right)}^{2}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. unpow2 80.5%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-/l* 96.4%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{x \cdot 0.5 - y}{x \cdot 0.5 - y}}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-inverses 96.4%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\left(x \cdot 0.5 - y\right) \cdot \color{blue}{1}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. *-rgt-identity 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{0.5 \cdot x} - y}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. *-commutative 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{\color{blue}{2 \cdot z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Simplified 96.4%

      \[\leadsto \color{blue}{\left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{2 \cdot z}}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around inf 79.0%

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. Taylor expanded in t around 0 44.3%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \]
   9. Step-by-step derivation

      1. div-inv 44.3%

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt{2}}\right)} \cdot \sqrt{z} \]

      2. pow1/2 44.3%

         \[\leadsto \left(x \cdot \frac{1}{\color{blue}{{2}^{0.5}}}\right) \cdot \sqrt{z} \]

      3. pow-flip 44.3%

         \[\leadsto \left(x \cdot \color{blue}{{2}^{\left(-0.5\right)}}\right) \cdot \sqrt{z} \]

      4. metadata-eval 44.3%

         \[\leadsto \left(x \cdot {2}^{\color{blue}{-0.5}}\right) \cdot \sqrt{z} \]

   10. Applied egg-rr 44.3%

       \[\leadsto \color{blue}{\left(x \cdot {2}^{-0.5}\right)} \cdot \sqrt{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+36} \lor \neg \left(y \leq 1.5 \cdot 10^{+113}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(x \cdot {2}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+35} \lor \neg \left(y \leq 2.7 \cdot 10^{+113}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{2}} \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.55e+35) (not (<= y 2.7e+113)))
   (* (sqrt (* z 2.0)) (- y))
   (* (/ x (sqrt 2.0)) (sqrt z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.55e+35) || !(y <= 2.7e+113)) {
		tmp = sqrt((z * 2.0)) * -y;
	} else {
		tmp = (x / sqrt(2.0)) * sqrt(z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-2.55d+35)) .or. (.not. (y <= 2.7d+113))) then
        tmp = sqrt((z * 2.0d0)) * -y
    else
        tmp = (x / sqrt(2.0d0)) * sqrt(z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.55e+35) || !(y <= 2.7e+113)) {
		tmp = Math.sqrt((z * 2.0)) * -y;
	} else {
		tmp = (x / Math.sqrt(2.0)) * Math.sqrt(z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.55e+35) or not (y <= 2.7e+113):
		tmp = math.sqrt((z * 2.0)) * -y
	else:
		tmp = (x / math.sqrt(2.0)) * math.sqrt(z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.55e+35) || !(y <= 2.7e+113))
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(-y));
	else
		tmp = Float64(Float64(x / sqrt(2.0)) * sqrt(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -2.55e+35) || ~((y <= 2.7e+113)))
		tmp = sqrt((z * 2.0)) * -y;
	else
		tmp = (x / sqrt(2.0)) * sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.55e+35], N[Not[LessEqual[y, 2.7e+113]], $MachinePrecision]], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[(N[(x / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.55000000000000009e35 or 2.70000000000000011e113 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 90.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 90.3%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 90.3%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 90.3%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 90.3%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 90.3%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 90.3%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 90.3%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 90.3%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 90.3%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 90.3%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 1.5%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 1.5%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 90.3%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 90.3%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 90.3%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 90.5%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 90.5%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 90.5%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 90.5%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 90.5%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Taylor expanded in t around 0 50.1%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 50.1%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. associate-*l* 50.1%

          \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   12. Simplified 50.1%

       \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
   13. Step-by-step derivation

       1. sqrt-prod 50.3%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

       2. add0 50.3%

          \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]

   14. Applied egg-rr 50.3%

       \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]
   15. Step-by-step derivation

       1. add0 50.3%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   16. Simplified 50.3%

       \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   if -2.55000000000000009e35 < y < 2.70000000000000011e113

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.8%

         \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. flip-+ 70.9%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 70.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 70.9%

         \[\leadsto \frac{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      5. swap-sqr 64.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)} - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      6. add-sqr-sqrt 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - 0 \cdot 0}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      7. metadata-eval 64.8%

         \[\leadsto \frac{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right) - \color{blue}{0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      8. fma-neg 64.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot 2, \left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right), -0\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      9. pow2 64.8%

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, \color{blue}{{\left(x \cdot 0.5 - y\right)}^{2}}, -0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, \color{blue}{0}\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} - 0} \cdot e^{\frac{t \cdot t}{2}} \]

      11. fma-neg 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, -0\right)}} \cdot e^{\frac{t \cdot t}{2}} \]

      12. metadata-eval 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\mathsf{fma}\left(x \cdot 0.5 - y, \sqrt{z \cdot 2}, \color{blue}{0}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      13. fma-define 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2} + 0}} \cdot e^{\frac{t \cdot t}{2}} \]

      14. add0 64.8%

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot 2, {\left(x \cdot 0.5 - y\right)}^{2}, 0\right)}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}} \cdot e^{\frac{t \cdot t}{2}} \]
   5. Step-by-step derivation

      1. fma-undefine 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2} + 0}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add0 64.8%

         \[\leadsto \frac{\color{blue}{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*l* 64.8%

         \[\leadsto \frac{\color{blue}{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot e^{\frac{t \cdot t}{2}} \]

      4. associate-/l* 73.4%

         \[\leadsto \color{blue}{\left(z \cdot \frac{2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-/l* 73.4%

         \[\leadsto \left(z \cdot \color{blue}{\left(2 \cdot \frac{{\left(x \cdot 0.5 - y\right)}^{2}}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. associate-/r* 80.5%

         \[\leadsto \left(z \cdot \left(2 \cdot \color{blue}{\frac{\frac{{\left(x \cdot 0.5 - y\right)}^{2}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. unpow2 80.5%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)}}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-/l* 96.4%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{x \cdot 0.5 - y}{x \cdot 0.5 - y}}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. *-inverses 96.4%

         \[\leadsto \left(z \cdot \left(2 \cdot \frac{\left(x \cdot 0.5 - y\right) \cdot \color{blue}{1}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. *-rgt-identity 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{x \cdot 0.5 - y}}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{\color{blue}{0.5 \cdot x} - y}{\sqrt{z \cdot 2}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. *-commutative 96.4%

          \[\leadsto \left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{\color{blue}{2 \cdot z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Simplified 96.4%

      \[\leadsto \color{blue}{\left(z \cdot \left(2 \cdot \frac{0.5 \cdot x - y}{\sqrt{2 \cdot z}}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around inf 79.0%

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{2}} \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   8. Taylor expanded in t around 0 44.3%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{2}} \cdot \sqrt{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+35} \lor \neg \left(y \leq 2.7 \cdot 10^{+113}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{2}} \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{{\left(z \cdot 2\right)}^{1.5}}{z \cdot \left(-2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 5e+144)
   (* (sqrt (* z 2.0)) (- y))
   (* y (/ (pow (* z 2.0) 1.5) (* z (- 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5e+144) {
		tmp = sqrt((z * 2.0)) * -y;
	} else {
		tmp = y * (pow((z * 2.0), 1.5) / (z * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 5d+144) then
        tmp = sqrt((z * 2.0d0)) * -y
    else
        tmp = y * (((z * 2.0d0) ** 1.5d0) / (z * -2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 5e+144) {
		tmp = Math.sqrt((z * 2.0)) * -y;
	} else {
		tmp = y * (Math.pow((z * 2.0), 1.5) / (z * -2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 5e+144:
		tmp = math.sqrt((z * 2.0)) * -y
	else:
		tmp = y * (math.pow((z * 2.0), 1.5) / (z * -2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 5e+144)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(-y));
	else
		tmp = Float64(y * Float64((Float64(z * 2.0) ^ 1.5) / Float64(z * Float64(-2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 5e+144)
		tmp = sqrt((z * 2.0)) * -y;
	else
		tmp = y * (((z * 2.0) ^ 1.5) / (z * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 5e+144], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[(y * N[(N[Power[N[(z * 2.0), $MachinePrecision], 1.5], $MachinePrecision] / N[(z * (-2.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.9999999999999999e144

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 64.3%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 64.2%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 64.2%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 64.2%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 64.3%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 64.3%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 64.3%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 64.3%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 64.3%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 10.7%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 10.7%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 64.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 64.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 64.2%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 64.3%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 64.3%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 64.3%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 64.3%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 64.3%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Taylor expanded in t around 0 27.0%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 27.0%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. associate-*l* 27.0%

          \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   12. Simplified 27.0%

       \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
   13. Step-by-step derivation

       1. sqrt-prod 27.1%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

       2. add0 27.1%

          \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]

   14. Applied egg-rr 27.1%

       \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]
   15. Step-by-step derivation

       1. add0 27.1%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   16. Simplified 27.1%

       \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   if 4.9999999999999999e144 < z

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.5%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. associate-*l* 54.5%

         \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 54.5%

         \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. distribute-rgt-neg-in 54.5%

         \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Simplified 54.5%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   6. Step-by-step derivation

      1. associate-*r* 54.5%

         \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-rgt-neg-out 54.5%

         \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. neg-sub0 54.5%

         \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. add-sqr-sqrt 54.5%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. sqr-neg 54.5%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 17.2%

         \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. associate-*r* 17.2%

         \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. sqrt-unprod 54.5%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqr-neg 54.5%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. add-sqr-sqrt 54.5%

          \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. sqrt-prod 54.6%

          \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 54.6%

         \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. distribute-lft-neg-in 54.6%

         \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. *-commutative 54.6%

         \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 54.6%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Taylor expanded in t around 0 39.2%

       \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 39.2%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. associate-*l* 39.2%

          \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   12. Simplified 39.2%

       \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
   13. Step-by-step derivation

       1. sqrt-prod 39.3%

          \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

       2. add0 39.3%

          \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]

       3. flip3-+ 49.4%

          \[\leadsto -y \cdot \color{blue}{\frac{{\left(\sqrt{2 \cdot z}\right)}^{3} + {0}^{3}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)}} \]

       4. pow3 49.4%

          \[\leadsto -y \cdot \frac{\color{blue}{\left(\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z}\right) \cdot \sqrt{2 \cdot z}} + {0}^{3}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       5. add-sqr-sqrt 49.4%

          \[\leadsto -y \cdot \frac{\color{blue}{\left(2 \cdot z\right)} \cdot \sqrt{2 \cdot z} + {0}^{3}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       6. metadata-eval 49.4%

          \[\leadsto -y \cdot \frac{\left(2 \cdot z\right) \cdot \sqrt{2 \cdot z} + \color{blue}{0}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       7. add0 49.4%

          \[\leadsto -y \cdot \frac{\color{blue}{\left(2 \cdot z\right) \cdot \sqrt{2 \cdot z}}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       8. pow1 49.4%

          \[\leadsto -y \cdot \frac{\color{blue}{{\left(2 \cdot z\right)}^{1}} \cdot \sqrt{2 \cdot z}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       9. pow1/2 49.4%

          \[\leadsto -y \cdot \frac{{\left(2 \cdot z\right)}^{1} \cdot \color{blue}{{\left(2 \cdot z\right)}^{0.5}}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       10. pow-prod-up 49.4%

           \[\leadsto -y \cdot \frac{\color{blue}{{\left(2 \cdot z\right)}^{\left(1 + 0.5\right)}}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       11. metadata-eval 49.4%

           \[\leadsto -y \cdot \frac{{\left(2 \cdot z\right)}^{\color{blue}{1.5}}}{\sqrt{2 \cdot z} \cdot \sqrt{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       12. add-sqr-sqrt 49.4%

           \[\leadsto -y \cdot \frac{{\left(2 \cdot z\right)}^{1.5}}{\color{blue}{2 \cdot z} + \left(0 \cdot 0 - \sqrt{2 \cdot z} \cdot 0\right)} \]

       13. metadata-eval 49.4%

           \[\leadsto -y \cdot \frac{{\left(2 \cdot z\right)}^{1.5}}{2 \cdot z + \left(\color{blue}{0} - \sqrt{2 \cdot z} \cdot 0\right)} \]

   14. Applied egg-rr 49.4%

       \[\leadsto -y \cdot \color{blue}{\frac{{\left(2 \cdot z\right)}^{1.5}}{2 \cdot z + \left(0 - \sqrt{2 \cdot z} \cdot 0\right)}} \]
   15. Step-by-step derivation

       1. sub0-neg 49.4%

          \[\leadsto -y \cdot \frac{{\left(2 \cdot z\right)}^{1.5}}{2 \cdot z + \color{blue}{\left(-\sqrt{2 \cdot z} \cdot 0\right)}} \]

       2. mul0-rgt 49.4%

          \[\leadsto -y \cdot \frac{{\left(2 \cdot z\right)}^{1.5}}{2 \cdot z + \left(-\color{blue}{0}\right)} \]

       3. sub-neg 49.4%

          \[\leadsto -y \cdot \frac{{\left(2 \cdot z\right)}^{1.5}}{\color{blue}{2 \cdot z - 0}} \]

       4. --rgt-identity 49.4%

          \[\leadsto -y \cdot \frac{{\left(2 \cdot z\right)}^{1.5}}{\color{blue}{2 \cdot z}} \]

   16. Simplified 49.4%

       \[\leadsto -y \cdot \color{blue}{\frac{{\left(2 \cdot z\right)}^{1.5}}{2 \cdot z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{{\left(z \cdot 2\right)}^{1.5}}{z \cdot \left(-2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{z \cdot 2} \cdot \left(-y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (sqrt (* z 2.0)) (- y)))

C

double code(double x, double y, double z, double t) {
	return sqrt((z * 2.0)) * -y;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((z * 2.0d0)) * -y
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.sqrt((z * 2.0)) * -y;
}

Python

def code(x, y, z, t):
	return math.sqrt((z * 2.0)) * -y

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(z * 2.0)) * Float64(-y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = sqrt((z * 2.0)) * -y;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 62.1%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
4. Step-by-step derivation

   1. mul-1-neg 62.1%

      \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   2. associate-*l* 62.1%

      \[\leadsto \left(-\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. distribute-rgt-neg-in 62.1%

      \[\leadsto \color{blue}{\left(y \cdot \left(-\sqrt{2} \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   4. distribute-rgt-neg-in 62.1%

      \[\leadsto \left(y \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

5. Simplified 62.1%

   \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
6. Step-by-step derivation

   1. associate-*r* 62.1%

      \[\leadsto \color{blue}{\left(\left(y \cdot \sqrt{2}\right) \cdot \left(-\sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   2. distribute-rgt-neg-out 62.1%

      \[\leadsto \color{blue}{\left(-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. neg-sub0 62.1%

      \[\leadsto \color{blue}{\left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   4. add-sqr-sqrt 62.1%

      \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. sqr-neg 62.1%

      \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. sqrt-unprod 0.0%

      \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. add-sqr-sqrt 12.1%

      \[\leadsto \left(0 - \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   8. associate-*r* 12.1%

      \[\leadsto \left(0 - \color{blue}{y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. add-sqr-sqrt 0.0%

      \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-\sqrt{z}} \cdot \sqrt{-\sqrt{z}}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. sqrt-unprod 62.1%

       \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   11. sqr-neg 62.1%

       \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   12. add-sqr-sqrt 62.1%

       \[\leadsto \left(0 - y \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{z}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   13. sqrt-prod 62.2%

       \[\leadsto \left(0 - y \cdot \color{blue}{\sqrt{2 \cdot z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

7. Applied egg-rr 62.2%

   \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
8. Step-by-step derivation

   1. neg-sub0 62.2%

      \[\leadsto \color{blue}{\left(-y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   2. distribute-lft-neg-in 62.2%

      \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   3. *-commutative 62.2%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

9. Simplified 62.2%

   \[\leadsto \color{blue}{\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

10. Taylor expanded in t around 0 29.7%

    \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
11. Step-by-step derivation

    1. mul-1-neg 29.7%

       \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

    2. associate-*l* 29.7%

       \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{-y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]
13. Step-by-step derivation

    1. sqrt-prod 29.8%

       \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

    2. add0 29.8%

       \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]

14. Applied egg-rr 29.8%

    \[\leadsto -y \cdot \color{blue}{\left(\sqrt{2 \cdot z} + 0\right)} \]
15. Step-by-step derivation

    1. add0 29.8%

       \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

16. Simplified 29.8%

    \[\leadsto -y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

17. Final simplification 29.8%

    \[\leadsto \sqrt{z \cdot 2} \cdot \left(-y\right) \]
18. Add Preprocessing

Developer target: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(exp(1.0), ((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (exp(1.0d0) ** ((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.pow(Math.exp(1.0), ((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.exp(1.0), ((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * (exp(1.0) ^ Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(1.0) ^ ((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[1.0], $MachinePrecision], N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024046 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))