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

Percentage Accurate: 99.4% → 99.4%

Time: 11.4s

Alternatives: 7

Speedup: 1.0×

• 44.8% 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. Add Preprocessing

Alternative 2: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := e^{\frac{t \cdot t}{2}}\\ \mathbf{if}\;x \leq -0.0122 \lor \neg \left(x \leq 1.05 \cdot 10^{-69}\right):\\ \;\;\;\;t\_2 \cdot \left(0.5 \cdot \left(x \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (exp (/ (* t t) 2.0))))
   (if (or (<= x -0.0122) (not (<= x 1.05e-69)))
     (* t_2 (* 0.5 (* x t_1)))
     (* t_2 (* t_1 (- y))))))

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 ((x <= -0.0122) || !(x <= 1.05e-69)) {
		tmp = t_2 * (0.5 * (x * t_1));
	} else {
		tmp = t_2 * (t_1 * -y);
	}
	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 ((x <= (-0.0122d0)) .or. (.not. (x <= 1.05d-69))) then
        tmp = t_2 * (0.5d0 * (x * t_1))
    else
        tmp = t_2 * (t_1 * -y)
    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 ((x <= -0.0122) || !(x <= 1.05e-69)) {
		tmp = t_2 * (0.5 * (x * t_1));
	} else {
		tmp = t_2 * (t_1 * -y);
	}
	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 (x <= -0.0122) or not (x <= 1.05e-69):
		tmp = t_2 * (0.5 * (x * t_1))
	else:
		tmp = t_2 * (t_1 * -y)
	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 ((x <= -0.0122) || !(x <= 1.05e-69))
		tmp = Float64(t_2 * Float64(0.5 * Float64(x * t_1)));
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(-y)));
	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 ((x <= -0.0122) || ~((x <= 1.05e-69)))
		tmp = t_2 * (0.5 * (x * t_1));
	else
		tmp = t_2 * (t_1 * -y);
	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[x, -0.0122], N[Not[LessEqual[x, 1.05e-69]], $MachinePrecision]], N[(t$95$2 * N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * (-y)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0122000000000000008 or 1.05e-69 < x

   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 83.1%

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

      1. *-commutative 83.1%

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

      2. associate-*l* 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-*r* 83.1%

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

      5. associate-*l* 83.1%

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

   5. Simplified 83.1%

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

      1. pow1 83.1%

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

      2. associate-*r* 83.1%

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

      3. sqrt-prod 83.2%

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

      4. *-commutative 83.2%

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

   7. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{{\left(x \cdot \left(\sqrt{2 \cdot z} \cdot 0.5\right)\right)}^{1}} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. unpow1 83.2%

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

      2. associate-*r* 83.2%

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

      3. *-commutative 83.2%

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

   9. Simplified 83.2%

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

   if -0.0122000000000000008 < x < 1.05e-69

   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 83.3%

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

      1. mul-1-neg 83.3%

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

      2. associate-*l* 83.2%

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

      3. *-commutative 83.2%

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

      4. distribute-rgt-neg-in 83.2%

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

      5. distribute-rgt-neg-in 83.2%

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

   5. Simplified 83.2%

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

      1. distribute-rgt-neg-out 83.2%

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

      2. sqrt-prod 83.4%

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

      3. *-commutative 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. *-commutative 83.4%

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

   9. Simplified 83.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0122 \lor \neg \left(x \leq 1.05 \cdot 10^{-69}\right):\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.65e+192)
   (* 0.5 (* (* x (sqrt 2.0)) (sqrt z)))
   (if (<= x 4.8e-18)
     (* (exp (/ (* t t) 2.0)) (* (sqrt (* z 2.0)) (- y)))
     (* 0.5 (* x (* (sqrt 2.0) (sqrt z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.65e+192) {
		tmp = 0.5 * ((x * sqrt(2.0)) * sqrt(z));
	} else if (x <= 4.8e-18) {
		tmp = exp(((t * t) / 2.0)) * (sqrt((z * 2.0)) * -y);
	} else {
		tmp = 0.5 * (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 (x <= (-1.65d+192)) then
        tmp = 0.5d0 * ((x * sqrt(2.0d0)) * sqrt(z))
    else if (x <= 4.8d-18) then
        tmp = exp(((t * t) / 2.0d0)) * (sqrt((z * 2.0d0)) * -y)
    else
        tmp = 0.5d0 * (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 (x <= -1.65e+192) {
		tmp = 0.5 * ((x * Math.sqrt(2.0)) * Math.sqrt(z));
	} else if (x <= 4.8e-18) {
		tmp = Math.exp(((t * t) / 2.0)) * (Math.sqrt((z * 2.0)) * -y);
	} else {
		tmp = 0.5 * (x * (Math.sqrt(2.0) * Math.sqrt(z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.65e+192:
		tmp = 0.5 * ((x * math.sqrt(2.0)) * math.sqrt(z))
	elif x <= 4.8e-18:
		tmp = math.exp(((t * t) / 2.0)) * (math.sqrt((z * 2.0)) * -y)
	else:
		tmp = 0.5 * (x * (math.sqrt(2.0) * math.sqrt(z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.65e+192)
		tmp = Float64(0.5 * Float64(Float64(x * sqrt(2.0)) * sqrt(z)));
	elseif (x <= 4.8e-18)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(sqrt(Float64(z * 2.0)) * Float64(-y)));
	else
		tmp = Float64(0.5 * Float64(x * Float64(sqrt(2.0) * sqrt(z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.65e+192)
		tmp = 0.5 * ((x * sqrt(2.0)) * sqrt(z));
	elseif (x <= 4.8e-18)
		tmp = exp(((t * t) / 2.0)) * (sqrt((z * 2.0)) * -y);
	else
		tmp = 0.5 * (x * (sqrt(2.0) * sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.65e+192], N[(0.5 * N[(N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e-18], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000005e192

   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 95.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) + 0.5 \cdot \left(\left(x \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. +-commutative 95.9%

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

      2. mul-1-neg 95.9%

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

      3. unsub-neg 95.9%

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

      4. associate-*l* 95.9%

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

      5. *-commutative 95.9%

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

      6. associate-*l* 95.9%

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

      7. *-commutative 95.9%

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

      8. associate-*r* 95.9%

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

      9. *-commutative 95.9%

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

      10. distribute-rgt-out-- 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-*l* 99.6%

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

      13. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf 92.2%

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

      1. associate-*r* 92.2%

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

   8. Simplified 92.2%

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

   9. Taylor expanded in t around 0 67.8%

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

   if -1.65000000000000005e192 < x < 4.79999999999999988e-18

   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 74.5%

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

      1. mul-1-neg 74.5%

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

      2. associate-*l* 74.5%

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

      3. *-commutative 74.5%

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

      4. distribute-rgt-neg-in 74.5%

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

      5. distribute-rgt-neg-in 74.5%

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

   5. Simplified 74.5%

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

      1. distribute-rgt-neg-out 74.5%

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

      2. sqrt-prod 74.5%

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

      3. *-commutative 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. *-commutative 74.5%

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

   9. Simplified 74.5%

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

   if 4.79999999999999988e-18 < x

   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 99.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) + 0.5 \cdot \left(\left(x \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

      4. associate-*l* 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*l* 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r* 99.6%

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

      9. *-commutative 99.6%

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

      10. distribute-rgt-out-- 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-*l* 99.6%

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

      13. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf 84.0%

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

      1. associate-*r* 84.0%

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

   8. Simplified 84.0%

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

   9. Taylor expanded in t around 0 49.7%

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

       1. associate-*l* 49.8%

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

   11. Simplified 49.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \left(\left(x \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-79} \lor \neg \left(y \leq 3.6 \cdot 10^{+164}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.45e-79) (not (<= y 3.6e+164)))
   (* (sqrt (* z 2.0)) (- y))
   (* 0.5 (* x (* (sqrt 2.0) (sqrt z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.45e-79) || !(y <= 3.6e+164)) {
		tmp = sqrt((z * 2.0)) * -y;
	} else {
		tmp = 0.5 * (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 <= (-1.45d-79)) .or. (.not. (y <= 3.6d+164))) then
        tmp = sqrt((z * 2.0d0)) * -y
    else
        tmp = 0.5d0 * (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 <= -1.45e-79) || !(y <= 3.6e+164)) {
		tmp = Math.sqrt((z * 2.0)) * -y;
	} else {
		tmp = 0.5 * (x * (Math.sqrt(2.0) * Math.sqrt(z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.45e-79) or not (y <= 3.6e+164):
		tmp = math.sqrt((z * 2.0)) * -y
	else:
		tmp = 0.5 * (x * (math.sqrt(2.0) * math.sqrt(z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.45e-79) || !(y <= 3.6e+164))
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(-y));
	else
		tmp = Float64(0.5 * Float64(x * Float64(sqrt(2.0) * sqrt(z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.45e-79) || ~((y <= 3.6e+164)))
		tmp = sqrt((z * 2.0)) * -y;
	else
		tmp = 0.5 * (x * (sqrt(2.0) * sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.45e-79], N[Not[LessEqual[y, 3.6e+164]], $MachinePrecision]], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[(0.5 * N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e-79 or 3.5999999999999999e164 < 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 84.5%

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

      1. mul-1-neg 84.5%

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

      2. associate-*l* 84.5%

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

      3. *-commutative 84.5%

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

      4. distribute-rgt-neg-in 84.5%

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

      5. distribute-rgt-neg-in 84.5%

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

   5. Simplified 84.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 6.9%

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

      3. distribute-rgt-neg-out 6.9%

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

      4. sqrt-prod 6.9%

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

      5. distribute-rgt-neg-out 6.9%

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

      6. sqrt-prod 6.9%

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

      7. sqr-neg 6.9%

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

      8. add-sqr-sqrt 6.9%

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

      9. pow1/2 6.9%

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

      10. *-commutative 6.9%

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

   7. Applied egg-rr 6.9%

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

      1. unpow1/2 6.9%

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

      2. *-commutative 6.9%

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

   9. Simplified 6.9%

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

   10. Taylor expanded in t around 0 1.0%

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

       1. *-commutative 1.0%

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

       2. associate-*r* 1.0%

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

       3. *-commutative 1.0%

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

   12. Simplified 1.0%

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

       1. associate-*r* 1.0%

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

       2. sqrt-prod 1.0%

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

       3. add-sqr-sqrt 0.3%

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

       4. sqrt-unprod 32.1%

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

       5. sqr-neg 32.1%

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

       6. sqrt-unprod 34.8%

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

       7. add-sqr-sqrt 55.5%

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

       8. distribute-rgt-neg-out 55.5%

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

       9. sqrt-prod 55.4%

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

       10. associate-*r* 55.4%

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

       11. neg-sub0 55.4%

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

       12. *-commutative 55.4%

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

       13. *-commutative 55.4%

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

       14. associate-*l* 55.4%

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

       15. sqrt-prod 55.5%

           \[\leadsto 0 - y \cdot \color{blue}{\sqrt{z \cdot 2}} \]

       16. *-commutative 55.5%

           \[\leadsto 0 - y \cdot \sqrt{\color{blue}{2 \cdot z}} \]

   14. Applied egg-rr 55.5%

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

       1. neg-sub0 55.5%

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

       2. distribute-lft-neg-in 55.5%

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

   16. Simplified 55.5%

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

   if -1.45e-79 < y < 3.5999999999999999e164

   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 99.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) + 0.5 \cdot \left(\left(x \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. unsub-neg 99.7%

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

      4. associate-*l* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r* 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out-- 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-*l* 99.6%

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

      13. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf 81.7%

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

      1. associate-*r* 81.7%

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

   8. Simplified 81.7%

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

   9. Taylor expanded in t around 0 43.3%

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

       1. associate-*l* 43.3%

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

   11. Simplified 43.3%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-79} \lor \neg \left(y \leq 3.6 \cdot 10^{+164}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 31.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3200000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3200000.0)
   (* (sqrt (* z 2.0)) (- y))
   (sqrt (* (* z 2.0) (pow y 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3200000.0) {
		tmp = sqrt((z * 2.0)) * -y;
	} 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 (t <= 3200000.0d0) then
        tmp = sqrt((z * 2.0d0)) * -y
    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 (t <= 3200000.0) {
		tmp = Math.sqrt((z * 2.0)) * -y;
	} else {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 3200000.0:
		tmp = math.sqrt((z * 2.0)) * -y
	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 (t <= 3200000.0)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(-y));
	else
		tmp = 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 (t <= 3200000.0)
		tmp = sqrt((z * 2.0)) * -y;
	else
		tmp = sqrt(((z * 2.0) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3200000.0], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.2e6

   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 56.5%

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

      1. mul-1-neg 56.5%

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

      2. associate-*l* 56.5%

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

      3. *-commutative 56.5%

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

      4. distribute-rgt-neg-in 56.5%

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

      5. distribute-rgt-neg-in 56.5%

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

   5. Simplified 56.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 8.6%

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

      3. distribute-rgt-neg-out 8.6%

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

      4. sqrt-prod 8.6%

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

      5. distribute-rgt-neg-out 8.6%

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

      6. sqrt-prod 8.6%

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

      7. sqr-neg 8.6%

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

      8. add-sqr-sqrt 8.6%

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

      9. pow1/2 8.6%

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

      10. *-commutative 8.6%

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

   7. Applied egg-rr 8.6%

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

      1. unpow1/2 8.6%

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

      2. *-commutative 8.6%

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

   9. Simplified 8.6%

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

   10. Taylor expanded in t around 0 3.0%

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

       1. *-commutative 3.0%

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

       2. associate-*r* 3.0%

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

       3. *-commutative 3.0%

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

   12. Simplified 3.0%

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

       1. associate-*r* 3.0%

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

       2. sqrt-prod 3.0%

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

       3. add-sqr-sqrt 1.1%

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

       4. sqrt-unprod 16.9%

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

       5. sqr-neg 16.9%

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

       6. sqrt-unprod 19.1%

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

       7. add-sqr-sqrt 35.7%

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

       8. distribute-rgt-neg-out 35.7%

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

       9. sqrt-prod 35.6%

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

       10. associate-*r* 35.6%

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

       11. neg-sub0 35.6%

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

       12. *-commutative 35.6%

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

       13. *-commutative 35.6%

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

       14. associate-*l* 35.6%

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

       15. sqrt-prod 35.7%

           \[\leadsto 0 - y \cdot \color{blue}{\sqrt{z \cdot 2}} \]

       16. *-commutative 35.7%

           \[\leadsto 0 - y \cdot \sqrt{\color{blue}{2 \cdot z}} \]

   14. Applied egg-rr 35.7%

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

       1. neg-sub0 35.7%

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

       2. distribute-lft-neg-in 35.7%

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

   16. Simplified 35.7%

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

   if 3.2e6 < t

   1. Initial program 100.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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* 64.3%

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

      3. *-commutative 64.3%

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

      4. distribute-rgt-neg-in 64.3%

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

      5. distribute-rgt-neg-in 64.3%

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

   5. Simplified 64.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 30.4%

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

      3. distribute-rgt-neg-out 30.4%

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

      4. sqrt-prod 30.4%

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

      5. distribute-rgt-neg-out 30.4%

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

      6. sqrt-prod 30.4%

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

      7. sqr-neg 30.4%

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

      8. add-sqr-sqrt 30.4%

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

      9. pow1/2 30.4%

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

      10. *-commutative 30.4%

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

   7. Applied egg-rr 30.4%

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

      1. unpow1/2 30.4%

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

      2. *-commutative 30.4%

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

   9. Simplified 30.4%

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

   10. Taylor expanded in t around 0 1.4%

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

       1. *-commutative 1.4%

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

       2. associate-*r* 1.4%

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

       3. *-commutative 1.4%

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

   12. Simplified 1.4%

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

       1. add-sqr-sqrt 1.0%

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

       2. sqrt-unprod 6.8%

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

       3. associate-*r* 6.8%

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

       4. sqrt-prod 6.8%

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

       5. associate-*r* 6.8%

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

       6. sqrt-prod 6.8%

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

       7. swap-sqr 12.0%

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

       8. add-sqr-sqrt 12.0%

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

       9. pow2 12.0%

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

   14. Applied egg-rr 12.0%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3200000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.7% 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 58.2%

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

   1. mul-1-neg 58.2%

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

   2. associate-*l* 58.2%

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

   3. *-commutative 58.2%

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

   4. distribute-rgt-neg-in 58.2%

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

   5. distribute-rgt-neg-in 58.2%

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

5. Simplified 58.2%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 13.4%

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

   3. distribute-rgt-neg-out 13.4%

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

   4. sqrt-prod 13.4%

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

   5. distribute-rgt-neg-out 13.4%

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

   6. sqrt-prod 13.4%

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

   7. sqr-neg 13.4%

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

   8. add-sqr-sqrt 13.4%

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

   9. pow1/2 13.4%

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

   10. *-commutative 13.4%

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

7. Applied egg-rr 13.4%

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

   1. unpow1/2 13.4%

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

   2. *-commutative 13.4%

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

9. Simplified 13.4%

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

10. Taylor expanded in t around 0 2.6%

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

    1. *-commutative 2.6%

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

    2. associate-*r* 2.6%

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

    3. *-commutative 2.6%

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

12. Simplified 2.6%

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

    1. associate-*r* 2.6%

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

    2. sqrt-prod 2.6%

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

    3. add-sqr-sqrt 1.1%

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

    4. sqrt-unprod 15.5%

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

    5. sqr-neg 15.5%

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

    6. sqrt-unprod 15.5%

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

    7. add-sqr-sqrt 29.1%

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

    8. distribute-rgt-neg-out 29.1%

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

    9. sqrt-prod 29.1%

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

    10. associate-*r* 29.1%

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

    11. neg-sub0 29.1%

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

    12. *-commutative 29.1%

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

    13. *-commutative 29.1%

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

    14. associate-*l* 29.1%

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

    15. sqrt-prod 29.1%

        \[\leadsto 0 - y \cdot \color{blue}{\sqrt{z \cdot 2}} \]

    16. *-commutative 29.1%

        \[\leadsto 0 - y \cdot \sqrt{\color{blue}{2 \cdot z}} \]

14. Applied egg-rr 29.1%

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

    1. neg-sub0 29.1%

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

    2. distribute-lft-neg-in 29.1%

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

16. Simplified 29.1%

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

17. Final simplification 29.1%

    \[\leadsto \sqrt{z \cdot 2} \cdot \left(-y\right) \]
18. Add Preprocessing

Alternative 7: 2.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \sqrt{z \cdot 2} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return y * sqrt((z * 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 = y * sqrt((z * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(y * N[Sqrt[N[(z * 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. Taylor expanded in x around 0 58.2%

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

   1. mul-1-neg 58.2%

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

   2. associate-*l* 58.2%

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

   3. *-commutative 58.2%

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

   4. distribute-rgt-neg-in 58.2%

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

   5. distribute-rgt-neg-in 58.2%

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

5. Simplified 58.2%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 13.4%

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

   3. distribute-rgt-neg-out 13.4%

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

   4. sqrt-prod 13.4%

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

   5. distribute-rgt-neg-out 13.4%

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

   6. sqrt-prod 13.4%

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

   7. sqr-neg 13.4%

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

   8. add-sqr-sqrt 13.4%

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

   9. pow1/2 13.4%

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

   10. *-commutative 13.4%

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

7. Applied egg-rr 13.4%

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

   1. unpow1/2 13.4%

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

   2. *-commutative 13.4%

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

9. Simplified 13.4%

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

10. Taylor expanded in t around 0 2.6%

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

    1. *-commutative 2.6%

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

    2. associate-*r* 2.6%

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

    3. *-commutative 2.6%

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

12. Simplified 2.6%

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

    1. associate-*r* 2.6%

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

    2. sqrt-prod 2.6%

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

    3. add-sqr-sqrt 1.1%

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

    4. sqrt-unprod 15.5%

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

    5. sqr-neg 15.5%

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

    6. sqrt-unprod 15.5%

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

    7. add-sqr-sqrt 29.1%

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

    8. expm1-log1p-u 19.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)\right)} \]

    9. expm1-undefine 12.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot z} \cdot \left(-y\right)\right)} - 1} \]

14. Applied egg-rr 2.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \sqrt{2 \cdot z}\right)} - 1} \]
15. Step-by-step derivation

    1. log1p-undefine 2.6%

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

    2. rem-exp-log 2.8%

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

    3. +-commutative 2.8%

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

    4. associate--l+ 2.6%

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

    5. metadata-eval 2.6%

       \[\leadsto y \cdot \sqrt{2 \cdot z} + \color{blue}{0} \]

    6. +-rgt-identity 2.6%

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

16. Simplified 2.6%

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

17. Final simplification 2.6%

    \[\leadsto y \cdot \sqrt{z \cdot 2} \]
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 2024119 
(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))))