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

Percentage Accurate: 99.4% → 99.4%

Time: 14.2s

Alternatives: 4

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: 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%

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

   5. Simplified 87.9%

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

      1. pow1 87.9%

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

      2. sqrt-unprod 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. unpow1 88.1%

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

      2. *-commutative 88.1%

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

   9. Simplified 88.1%

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

   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. *-commutative 81.8%

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

      2. associate-*l* 81.8%

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

      3. *-commutative 81.8%

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

      4. associate-*r* 81.8%

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

      5. associate-*l* 81.8%

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

      6. *-commutative 81.8%

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

   5. Simplified 81.8%

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

      1. expm1-log1p-u 53.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{z} \cdot \left(0.5 \cdot x\right)\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. expm1-undefine 28.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(\sqrt{z} \cdot \left(0.5 \cdot x\right)\right)\right)} - 1\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*r* 28.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \left(0.5 \cdot x\right)}\right)} - 1\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. sqrt-unprod 28.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot z}} \cdot \left(0.5 \cdot x\right)\right)} - 1\right) \cdot e^{\frac{t \cdot t}{2}} \]

   7. Applied egg-rr 28.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x\right)\right)} - 1\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. sub-neg 28.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x\right)\right)} + \left(-1\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. metadata-eval 28.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x\right)\right)} + \color{blue}{-1}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. +-commutative 28.9%

         \[\leadsto \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x\right)\right)}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      4. log1p-undefine 28.9%

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

      5. rem-exp-log 57.8%

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

      6. associate-+r+ 81.9%

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

      7. metadata-eval 81.9%

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

      8. +-lft-identity 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. *-commutative 81.9%

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

      10. *-commutative 81.9%

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

   9. Simplified 81.9%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot x\right) \cdot \sqrt{z \cdot 2}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
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 3: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\frac{t \cdot t}{2}}\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{+211}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(y \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (exp (/ (* t t) 2.0))) (t_2 (sqrt (* z 2.0))))
   (if (<= x 9.5e+211) (* t_1 (* t_2 (- y))) (* t_1 (* y t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = exp(((t * t) / 2.0));
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (x <= 9.5e+211) {
		tmp = t_1 * (t_2 * -y);
	} else {
		tmp = t_1 * (y * t_2);
	}
	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 = exp(((t * t) / 2.0d0))
    t_2 = sqrt((z * 2.0d0))
    if (x <= 9.5d+211) then
        tmp = t_1 * (t_2 * -y)
    else
        tmp = t_1 * (y * t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.exp(((t * t) / 2.0));
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (x <= 9.5e+211) {
		tmp = t_1 * (t_2 * -y);
	} else {
		tmp = t_1 * (y * t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.exp(((t * t) / 2.0))
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if x <= 9.5e+211:
		tmp = t_1 * (t_2 * -y)
	else:
		tmp = t_1 * (y * t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = exp(Float64(Float64(t * t) / 2.0))
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (x <= 9.5e+211)
		tmp = Float64(t_1 * Float64(t_2 * Float64(-y)));
	else
		tmp = Float64(t_1 * Float64(y * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = exp(((t * t) / 2.0));
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (x <= 9.5e+211)
		tmp = t_1 * (t_2 * -y);
	else
		tmp = t_1 * (y * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 9.5e+211], N[(t$95$1 * N[(t$95$2 * (-y)), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(y * t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.4999999999999997e211

   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 66.4%

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

      1. mul-1-neg 66.4%

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

      2. associate-*l* 66.4%

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

   5. Simplified 66.4%

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

      1. pow1 66.4%

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

      2. sqrt-unprod 66.5%

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

   7. Applied egg-rr 66.5%

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

      1. unpow1 66.5%

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

      2. *-commutative 66.5%

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

   9. Simplified 66.5%

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

   if 9.4999999999999997e211 < 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. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 71.5%

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

      3. *-commutative 71.5%

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

      4. *-commutative 71.5%

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

      5. swap-sqr 71.5%

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

      6. add-sqr-sqrt 71.5%

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

      7. pow2 71.5%

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

   4. Applied egg-rr 71.5%

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

      1. sqrt-prod 71.5%

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

      2. sqrt-prod 71.5%

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

      3. *-commutative 71.5%

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

      4. add-sqr-sqrt 71.5%

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

      5. *-commutative 71.5%

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

      6. sqrt-pow1 99.8%

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

      7. metadata-eval 99.8%

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

      8. pow1 99.8%

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

      9. associate-*l* 99.7%

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

      10. *-commutative 99.7%

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

      11. sqrt-prod 99.7%

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

      12. pow1/2 99.7%

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

      13. sqrt-pow1 99.7%

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

      14. *-commutative 99.7%

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

      15. metadata-eval 99.7%

          \[\leadsto \left({\left(2 \cdot z\right)}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2} \cdot \sqrt{z}} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(0.5 \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   7. Taylor expanded in x around 0 18.5%

      \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\left(-1 \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot y\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg 18.5%

         \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\left(-{\left(2 \cdot z\right)}^{0.25} \cdot y\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. *-commutative 18.5%

         \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(-\color{blue}{y \cdot {\left(2 \cdot z\right)}^{0.25}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. distribute-rgt-neg-in 18.5%

         \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\left(y \cdot \left(-{\left(2 \cdot z\right)}^{0.25}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. Simplified 18.5%

      \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\left(y \cdot \left(-{\left(2 \cdot z\right)}^{0.25}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   10. Step-by-step derivation

       1. pow1 18.5%

          \[\leadsto \color{blue}{{\left({\left(2 \cdot z\right)}^{0.25} \cdot \left(y \cdot \left(-{\left(2 \cdot z\right)}^{0.25}\right)\right)\right)}^{1}} \cdot e^{\frac{t \cdot t}{2}} \]

   11. Applied egg-rr 40.5%

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

       1. unpow1 40.5%

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

       2. *-commutative 40.5%

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

   13. Simplified 40.5%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+211}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 12.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $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. Step-by-step derivation

   1. add-sqr-sqrt 51.0%

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

   2. sqrt-unprod 40.3%

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

   3. *-commutative 40.3%

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

   4. *-commutative 40.3%

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

   5. swap-sqr 36.1%

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

   6. add-sqr-sqrt 36.2%

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

   7. pow2 36.2%

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

4. Applied egg-rr 36.2%

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

   1. sqrt-prod 38.7%

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

   2. sqrt-prod 38.7%

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

   3. *-commutative 38.7%

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

   4. add-sqr-sqrt 38.7%

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

   5. *-commutative 38.7%

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

   6. sqrt-pow1 99.6%

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

   7. metadata-eval 99.6%

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

   8. pow1 99.6%

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

   9. associate-*l* 99.6%

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

   10. *-commutative 99.6%

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

   11. sqrt-prod 99.7%

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

   12. pow1/2 99.7%

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

   13. sqrt-pow1 99.7%

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

   14. *-commutative 99.7%

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

   15. metadata-eval 99.7%

       \[\leadsto \left({\left(2 \cdot z\right)}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2} \cdot \sqrt{z}} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(0.5 \cdot x - y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

7. Taylor expanded in x around 0 62.1%

   \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\left(-1 \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot y\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
8. Step-by-step derivation

   1. mul-1-neg 62.1%

      \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\left(-{\left(2 \cdot z\right)}^{0.25} \cdot y\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. *-commutative 62.1%

      \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(-\color{blue}{y \cdot {\left(2 \cdot z\right)}^{0.25}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. distribute-rgt-neg-in 62.1%

      \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\left(y \cdot \left(-{\left(2 \cdot z\right)}^{0.25}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

9. Simplified 62.1%

   \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\left(y \cdot \left(-{\left(2 \cdot z\right)}^{0.25}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
10. Step-by-step derivation

    1. pow1 62.1%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot z\right)}^{0.25} \cdot \left(y \cdot \left(-{\left(2 \cdot z\right)}^{0.25}\right)\right)\right)}^{1}} \cdot e^{\frac{t \cdot t}{2}} \]

11. Applied egg-rr 12.1%

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

    1. unpow1 12.1%

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

    2. *-commutative 12.1%

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

13. Simplified 12.1%

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

14. Final simplification 12.1%

    \[\leadsto e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \sqrt{z \cdot 2}\right) \]
15. 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))))