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

Percentage Accurate: 99.3% → 99.8%

Time: 18.2s

Alternatives: 11

Speedup: 1.0×

• 45.0% 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: 99.3% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt((exp((t * t)) * (2.0 * z))) * ((x * 0.5) - 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((exp((t * t)) * (2.0d0 * z))) * ((x * 0.5d0) - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sqr-neg 99.4%

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

   2. associate-/l* 99.4%

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

   3. distribute-frac-neg 99.4%

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

   4. exp-neg 99.5%

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

   5. associate-*r/ 99.5%

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

   6. *-rgt-identity 99.5%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. associate-*r/ 99.8%

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

   10. exp-neg 99.8%

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

   11. distribute-frac-neg 99.8%

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

   12. associate-/l* 99.8%

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

   13. sqr-neg 99.8%

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

   14. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.3%

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

   2. expm1-udef 80.4%

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

   3. pow-exp 80.4%

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

   4. sqrt-unprod 80.4%

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

   5. associate-*l* 80.4%

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

5. Applied egg-rr 80.4%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

   1. pow-exp 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t \cdot \left(y - x \cdot 0.5\right)\right)\\ t_2 := x \cdot 0.5 - y\\ t_3 := 0.5 \cdot \left(t \cdot t\right)\\ t_4 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 4 \cdot 10^{+32}:\\ \;\;\;\;t_4 \cdot \left(0.5 \cdot \left(x + t \cdot \left(t \cdot t_2\right)\right) - y\right)\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+107}:\\ \;\;\;\;t_4 \cdot \left(0.5 \cdot \frac{x \cdot x - t_1 \cdot t_1}{x + t_1} - y\right)\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;t_4 \cdot \left(0.5 \cdot \left(x + t \cdot \left(x \cdot \left(0.5 \cdot t\right)\right)\right) - y\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+279}:\\ \;\;\;\;t_4 \cdot \left(y \cdot \frac{t_3 \cdot t_3 + -1}{1 - t_3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (* t (- y (* x 0.5)))))
        (t_2 (- (* x 0.5) y))
        (t_3 (* 0.5 (* t t)))
        (t_4 (sqrt (* 2.0 z))))
   (if (<= (* t t) 4e+32)
     (* t_4 (- (* 0.5 (+ x (* t (* t t_2)))) y))
     (if (<= (* t t) 2e+107)
       (* t_4 (- (* 0.5 (/ (- (* x x) (* t_1 t_1)) (+ x t_1))) y))
       (if (<= (* t t) 2e+121)
         (* t_4 (- (* 0.5 (+ x (* t (* x (* 0.5 t))))) y))
         (if (<= (* t t) 5e+279)
           (* t_4 (* y (/ (+ (* t_3 t_3) -1.0) (- 1.0 t_3))))
           (* t_2 (sqrt (* (* 2.0 z) (+ (* t t) 1.0))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t * (t * (y - (x * 0.5)));
	double t_2 = (x * 0.5) - y;
	double t_3 = 0.5 * (t * t);
	double t_4 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 4e+32) {
		tmp = t_4 * ((0.5 * (x + (t * (t * t_2)))) - y);
	} else if ((t * t) <= 2e+107) {
		tmp = t_4 * ((0.5 * (((x * x) - (t_1 * t_1)) / (x + t_1))) - y);
	} else if ((t * t) <= 2e+121) {
		tmp = t_4 * ((0.5 * (x + (t * (x * (0.5 * t))))) - y);
	} else if ((t * t) <= 5e+279) {
		tmp = t_4 * (y * (((t_3 * t_3) + -1.0) / (1.0 - t_3)));
	} else {
		tmp = t_2 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = t * (t * (y - (x * 0.5d0)))
    t_2 = (x * 0.5d0) - y
    t_3 = 0.5d0 * (t * t)
    t_4 = sqrt((2.0d0 * z))
    if ((t * t) <= 4d+32) then
        tmp = t_4 * ((0.5d0 * (x + (t * (t * t_2)))) - y)
    else if ((t * t) <= 2d+107) then
        tmp = t_4 * ((0.5d0 * (((x * x) - (t_1 * t_1)) / (x + t_1))) - y)
    else if ((t * t) <= 2d+121) then
        tmp = t_4 * ((0.5d0 * (x + (t * (x * (0.5d0 * t))))) - y)
    else if ((t * t) <= 5d+279) then
        tmp = t_4 * (y * (((t_3 * t_3) + (-1.0d0)) / (1.0d0 - t_3)))
    else
        tmp = t_2 * sqrt(((2.0d0 * z) * ((t * t) + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = t * (t * (y - (x * 0.5)));
	double t_2 = (x * 0.5) - y;
	double t_3 = 0.5 * (t * t);
	double t_4 = Math.sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 4e+32) {
		tmp = t_4 * ((0.5 * (x + (t * (t * t_2)))) - y);
	} else if ((t * t) <= 2e+107) {
		tmp = t_4 * ((0.5 * (((x * x) - (t_1 * t_1)) / (x + t_1))) - y);
	} else if ((t * t) <= 2e+121) {
		tmp = t_4 * ((0.5 * (x + (t * (x * (0.5 * t))))) - y);
	} else if ((t * t) <= 5e+279) {
		tmp = t_4 * (y * (((t_3 * t_3) + -1.0) / (1.0 - t_3)));
	} else {
		tmp = t_2 * Math.sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t * (t * (y - (x * 0.5)))
	t_2 = (x * 0.5) - y
	t_3 = 0.5 * (t * t)
	t_4 = math.sqrt((2.0 * z))
	tmp = 0
	if (t * t) <= 4e+32:
		tmp = t_4 * ((0.5 * (x + (t * (t * t_2)))) - y)
	elif (t * t) <= 2e+107:
		tmp = t_4 * ((0.5 * (((x * x) - (t_1 * t_1)) / (x + t_1))) - y)
	elif (t * t) <= 2e+121:
		tmp = t_4 * ((0.5 * (x + (t * (x * (0.5 * t))))) - y)
	elif (t * t) <= 5e+279:
		tmp = t_4 * (y * (((t_3 * t_3) + -1.0) / (1.0 - t_3)))
	else:
		tmp = t_2 * math.sqrt(((2.0 * z) * ((t * t) + 1.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(t * Float64(t * Float64(y - Float64(x * 0.5))))
	t_2 = Float64(Float64(x * 0.5) - y)
	t_3 = Float64(0.5 * Float64(t * t))
	t_4 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 4e+32)
		tmp = Float64(t_4 * Float64(Float64(0.5 * Float64(x + Float64(t * Float64(t * t_2)))) - y));
	elseif (Float64(t * t) <= 2e+107)
		tmp = Float64(t_4 * Float64(Float64(0.5 * Float64(Float64(Float64(x * x) - Float64(t_1 * t_1)) / Float64(x + t_1))) - y));
	elseif (Float64(t * t) <= 2e+121)
		tmp = Float64(t_4 * Float64(Float64(0.5 * Float64(x + Float64(t * Float64(x * Float64(0.5 * t))))) - y));
	elseif (Float64(t * t) <= 5e+279)
		tmp = Float64(t_4 * Float64(y * Float64(Float64(Float64(t_3 * t_3) + -1.0) / Float64(1.0 - t_3))));
	else
		tmp = Float64(t_2 * sqrt(Float64(Float64(2.0 * z) * Float64(Float64(t * t) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t * (t * (y - (x * 0.5)));
	t_2 = (x * 0.5) - y;
	t_3 = 0.5 * (t * t);
	t_4 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((t * t) <= 4e+32)
		tmp = t_4 * ((0.5 * (x + (t * (t * t_2)))) - y);
	elseif ((t * t) <= 2e+107)
		tmp = t_4 * ((0.5 * (((x * x) - (t_1 * t_1)) / (x + t_1))) - y);
	elseif ((t * t) <= 2e+121)
		tmp = t_4 * ((0.5 * (x + (t * (x * (0.5 * t))))) - y);
	elseif ((t * t) <= 5e+279)
		tmp = t_4 * (y * (((t_3 * t_3) + -1.0) / (1.0 - t_3)));
	else
		tmp = t_2 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(t * N[(y - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 4e+32], N[(t$95$4 * N[(N[(0.5 * N[(x + N[(t * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 2e+107], N[(t$95$4 * N[(N[(0.5 * N[(N[(N[(x * x), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 2e+121], N[(t$95$4 * N[(N[(0.5 * N[(x + N[(t * N[(x * N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 5e+279], N[(t$95$4 * N[(y * N[(N[(N[(t$95$3 * t$95$3), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 t t) < 4.00000000000000021e32

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-frac-neg 99.7%

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

      4. exp-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. *-rgt-identity 99.7%

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

      10. associate-*r/ 99.7%

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

      11. exp-neg 99.7%

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

      12. distribute-frac-neg 99.7%

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

      13. associate-/l* 99.7%

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

      14. sqr-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 95.6%

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

      1. distribute-lft-out 95.6%

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

      2. unpow2 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in t around 0 95.6%

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

      1. unpow2 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r* 95.6%

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

      4. *-commutative 95.6%

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

   9. Simplified 95.6%

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

   if 4.00000000000000021e32 < (*.f64 t t) < 1.9999999999999999e107

   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. Step-by-step derivation

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. exp-neg 100.0%

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

      12. distribute-frac-neg 100.0%

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

      13. associate-/l* 100.0%

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

      14. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 12.0%

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

      1. distribute-lft-out 12.0%

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

      2. unpow2 12.0%

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

   6. Simplified 12.0%

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

      1. flip-+ 41.7%

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

      2. associate-*l* 41.7%

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

      3. *-commutative 41.7%

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

      4. associate-*l* 41.7%

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

      5. *-commutative 41.7%

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

      6. associate-*l* 41.7%

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

      7. *-commutative 41.7%

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

   8. Applied egg-rr 41.7%

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

   if 1.9999999999999999e107 < (*.f64 t t) < 2.00000000000000007e121

   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. Step-by-step derivation

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. exp-neg 100.0%

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

      12. distribute-frac-neg 100.0%

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

      13. associate-/l* 100.0%

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

      14. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 76.5%

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

      1. distribute-lft-out 76.5%

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

      2. unpow2 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in t around 0 76.5%

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

      1. unpow2 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*r* 76.5%

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

      4. *-commutative 76.5%

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

   9. Simplified 76.5%

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

   10. Taylor expanded in x around inf 76.5%

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

       1. associate-*r* 76.5%

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

   12. Simplified 76.5%

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

   if 2.00000000000000007e121 < (*.f64 t t) < 5.0000000000000002e279

   1. Initial program 97.3%

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

      1. sqr-neg 97.3%

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

      2. associate-/l* 97.3%

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

      3. distribute-frac-neg 97.3%

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

      4. exp-neg 97.3%

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

      5. associate-*r/ 97.3%

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

      6. *-rgt-identity 97.3%

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

      7. *-commutative 97.3%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. exp-neg 100.0%

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

      12. distribute-frac-neg 100.0%

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

      13. associate-/l* 100.0%

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

      14. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 46.1%

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

      1. distribute-lft-out 46.1%

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

      2. unpow2 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in y around -inf 32.6%

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

      1. associate-*r* 32.6%

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

      2. neg-mul-1 32.6%

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

      3. *-commutative 32.6%

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

      4. unpow2 32.6%

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

   9. Simplified 32.6%

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

       1. flip-+ 91.9%

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

       2. metadata-eval 91.9%

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

       3. *-commutative 91.9%

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

       4. *-commutative 91.9%

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

       5. *-commutative 91.9%

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

   11. Applied egg-rr 91.9%

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

   if 5.0000000000000002e279 < (*.f64 t 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. Step-by-step derivation

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. pow-exp 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. associate-*l* 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. unpow2 98.5%

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

   10. Simplified 98.5%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \color{blue}{\left(t \cdot t + 1\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 4 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(x + t \cdot \left(t \cdot \left(x \cdot 0.5 - y\right)\right)\right) - y\right)\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \frac{x \cdot x - \left(t \cdot \left(t \cdot \left(y - x \cdot 0.5\right)\right)\right) \cdot \left(t \cdot \left(t \cdot \left(y - x \cdot 0.5\right)\right)\right)}{x + t \cdot \left(t \cdot \left(y - x \cdot 0.5\right)\right)} - y\right)\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(x + t \cdot \left(x \cdot \left(0.5 \cdot t\right)\right)\right) - y\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(y \cdot \frac{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right) + -1}{1 - 0.5 \cdot \left(t \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \]

Alternative 3: 87.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot \left(t \cdot t\right)\\ t_2 := x \cdot 0.5 - y\\ t_3 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;t_3 \cdot \left(0.5 \cdot \left(x + t \cdot \left(t \cdot t_2\right)\right) - y\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+279}:\\ \;\;\;\;t_3 \cdot \left(y \cdot \frac{t_1 \cdot t_1 + -1}{1 - t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 0.5 (* t t))) (t_2 (- (* x 0.5) y)) (t_3 (sqrt (* 2.0 z))))
   (if (<= (* t t) 2e+121)
     (* t_3 (- (* 0.5 (+ x (* t (* t t_2)))) y))
     (if (<= (* t t) 5e+279)
       (* t_3 (* y (/ (+ (* t_1 t_1) -1.0) (- 1.0 t_1))))
       (* t_2 (sqrt (* (* 2.0 z) (+ (* t t) 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * (t * t);
	double t_2 = (x * 0.5) - y;
	double t_3 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 2e+121) {
		tmp = t_3 * ((0.5 * (x + (t * (t * t_2)))) - y);
	} else if ((t * t) <= 5e+279) {
		tmp = t_3 * (y * (((t_1 * t_1) + -1.0) / (1.0 - t_1)));
	} else {
		tmp = t_2 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = 0.5d0 * (t * t)
    t_2 = (x * 0.5d0) - y
    t_3 = sqrt((2.0d0 * z))
    if ((t * t) <= 2d+121) then
        tmp = t_3 * ((0.5d0 * (x + (t * (t * t_2)))) - y)
    else if ((t * t) <= 5d+279) then
        tmp = t_3 * (y * (((t_1 * t_1) + (-1.0d0)) / (1.0d0 - t_1)))
    else
        tmp = t_2 * sqrt(((2.0d0 * z) * ((t * t) + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 0.5 * (t * t);
	double t_2 = (x * 0.5) - y;
	double t_3 = Math.sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 2e+121) {
		tmp = t_3 * ((0.5 * (x + (t * (t * t_2)))) - y);
	} else if ((t * t) <= 5e+279) {
		tmp = t_3 * (y * (((t_1 * t_1) + -1.0) / (1.0 - t_1)));
	} else {
		tmp = t_2 * Math.sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 0.5 * (t * t)
	t_2 = (x * 0.5) - y
	t_3 = math.sqrt((2.0 * z))
	tmp = 0
	if (t * t) <= 2e+121:
		tmp = t_3 * ((0.5 * (x + (t * (t * t_2)))) - y)
	elif (t * t) <= 5e+279:
		tmp = t_3 * (y * (((t_1 * t_1) + -1.0) / (1.0 - t_1)))
	else:
		tmp = t_2 * math.sqrt(((2.0 * z) * ((t * t) + 1.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(0.5 * Float64(t * t))
	t_2 = Float64(Float64(x * 0.5) - y)
	t_3 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 2e+121)
		tmp = Float64(t_3 * Float64(Float64(0.5 * Float64(x + Float64(t * Float64(t * t_2)))) - y));
	elseif (Float64(t * t) <= 5e+279)
		tmp = Float64(t_3 * Float64(y * Float64(Float64(Float64(t_1 * t_1) + -1.0) / Float64(1.0 - t_1))));
	else
		tmp = Float64(t_2 * sqrt(Float64(Float64(2.0 * z) * Float64(Float64(t * t) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 0.5 * (t * t);
	t_2 = (x * 0.5) - y;
	t_3 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((t * t) <= 2e+121)
		tmp = t_3 * ((0.5 * (x + (t * (t * t_2)))) - y);
	elseif ((t * t) <= 5e+279)
		tmp = t_3 * (y * (((t_1 * t_1) + -1.0) / (1.0 - t_1)));
	else
		tmp = t_2 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2e+121], N[(t$95$3 * N[(N[(0.5 * N[(x + N[(t * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 5e+279], N[(t$95$3 * N[(y * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 2.00000000000000007e121

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-frac-neg 99.7%

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

      4. exp-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. *-rgt-identity 99.7%

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

      10. associate-*r/ 99.7%

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

      11. exp-neg 99.7%

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

      12. distribute-frac-neg 99.7%

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

      13. associate-/l* 99.7%

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

      14. sqr-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 86.9%

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

      1. distribute-lft-out 86.9%

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

      2. unpow2 86.9%

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

   6. Simplified 86.9%

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

   7. Taylor expanded in t around 0 86.9%

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

      1. unpow2 86.9%

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

      2. *-commutative 86.9%

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

      3. associate-*r* 86.9%

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

      4. *-commutative 86.9%

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

   9. Simplified 86.9%

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

   if 2.00000000000000007e121 < (*.f64 t t) < 5.0000000000000002e279

   1. Initial program 97.3%

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

      1. sqr-neg 97.3%

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

      2. associate-/l* 97.3%

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

      3. distribute-frac-neg 97.3%

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

      4. exp-neg 97.3%

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

      5. associate-*r/ 97.3%

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

      6. *-rgt-identity 97.3%

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

      7. *-commutative 97.3%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. exp-neg 100.0%

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

      12. distribute-frac-neg 100.0%

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

      13. associate-/l* 100.0%

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

      14. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 46.1%

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

      1. distribute-lft-out 46.1%

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

      2. unpow2 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in y around -inf 32.6%

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

      1. associate-*r* 32.6%

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

      2. neg-mul-1 32.6%

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

      3. *-commutative 32.6%

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

      4. unpow2 32.6%

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

   9. Simplified 32.6%

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

       1. flip-+ 91.9%

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

       2. metadata-eval 91.9%

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

       3. *-commutative 91.9%

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

       4. *-commutative 91.9%

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

       5. *-commutative 91.9%

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

   11. Applied egg-rr 91.9%

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

   if 5.0000000000000002e279 < (*.f64 t 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. Step-by-step derivation

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. pow-exp 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. associate-*l* 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. unpow2 98.5%

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

   10. Simplified 98.5%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \color{blue}{\left(t \cdot t + 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(x + t \cdot \left(t \cdot \left(x \cdot 0.5 - y\right)\right)\right) - y\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(y \cdot \frac{\left(0.5 \cdot \left(t \cdot t\right)\right) \cdot \left(0.5 \cdot \left(t \cdot t\right)\right) + -1}{1 - 0.5 \cdot \left(t \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \]

Alternative 4: 87.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;2 \cdot z \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(x + t \cdot \left(t \cdot t_1\right)\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* 2.0 z) 5e+73)
     (* (sqrt (* 2.0 z)) (- (* 0.5 (+ x (* t (* t t_1)))) y))
     (* t_1 (sqrt (* (* 2.0 z) (+ (* t t) 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((2.0 * z) <= 5e+73) {
		tmp = sqrt((2.0 * z)) * ((0.5 * (x + (t * (t * t_1)))) - y);
	} else {
		tmp = t_1 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if ((2.0d0 * z) <= 5d+73) then
        tmp = sqrt((2.0d0 * z)) * ((0.5d0 * (x + (t * (t * t_1)))) - y)
    else
        tmp = t_1 * sqrt(((2.0d0 * z) * ((t * t) + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((2.0 * z) <= 5e+73) {
		tmp = Math.sqrt((2.0 * z)) * ((0.5 * (x + (t * (t * t_1)))) - y);
	} else {
		tmp = t_1 * Math.sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (2.0 * z) <= 5e+73:
		tmp = math.sqrt((2.0 * z)) * ((0.5 * (x + (t * (t * t_1)))) - y)
	else:
		tmp = t_1 * math.sqrt(((2.0 * z) * ((t * t) + 1.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(2.0 * z) <= 5e+73)
		tmp = Float64(sqrt(Float64(2.0 * z)) * Float64(Float64(0.5 * Float64(x + Float64(t * Float64(t * t_1)))) - y));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * Float64(Float64(t * t) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((2.0 * z) <= 5e+73)
		tmp = sqrt((2.0 * z)) * ((0.5 * (x + (t * (t * t_1)))) - y);
	else
		tmp = t_1 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(2.0 * z), $MachinePrecision], 5e+73], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * N[(x + N[(t * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 2) < 4.99999999999999976e73

   1. Initial program 99.1%

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

      1. sqr-neg 99.1%

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

      2. associate-/l* 99.1%

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

      3. distribute-frac-neg 99.1%

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

      4. exp-neg 99.1%

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

      5. associate-*r/ 99.1%

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

      6. *-rgt-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. associate-*r/ 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. associate-*r/ 99.8%

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

      11. exp-neg 99.8%

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

      12. distribute-frac-neg 99.8%

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

      13. associate-/l* 99.8%

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

      14. sqr-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 83.2%

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

      1. distribute-lft-out 83.2%

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

      2. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in t around 0 83.2%

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

      1. unpow2 83.2%

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

      2. *-commutative 83.2%

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

      3. associate-*r* 83.2%

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

      4. *-commutative 83.2%

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

   9. Simplified 83.2%

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

   if 4.99999999999999976e73 < (*.f64 z 2)

   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. Step-by-step derivation

      1. sqr-neg 99.9%

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

      2. associate-/l* 99.9%

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

      3. distribute-frac-neg 99.9%

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

      4. exp-neg 99.9%

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

      5. associate-*r/ 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. associate-*r/ 99.9%

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

      8. *-rgt-identity 99.9%

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

      9. associate-*r/ 99.9%

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

      10. exp-neg 99.9%

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

      11. distribute-frac-neg 99.9%

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

      12. associate-/l* 99.9%

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

      13. sqr-neg 99.9%

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

      14. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 96.7%

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

      2. expm1-udef 96.7%

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

      3. pow-exp 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. associate-*l* 96.7%

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

   5. Applied egg-rr 96.7%

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

      1. expm1-def 96.7%

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

      2. expm1-log1p 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t around 0 92.4%

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

      1. +-commutative 92.4%

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

      2. unpow2 92.4%

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

   10. Simplified 92.4%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \color{blue}{\left(t \cdot t + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot z \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(x + t \cdot \left(t \cdot \left(x \cdot 0.5 - y\right)\right)\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \]

Alternative 5: 88.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;2 \cdot z \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(x + \left(t \cdot t\right) \cdot t_1\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* 2.0 z) 2e+99)
     (* (sqrt (* 2.0 z)) (- (* 0.5 (+ x (* (* t t) t_1))) y))
     (* t_1 (sqrt (* (* 2.0 z) (+ (* t t) 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((2.0 * z) <= 2e+99) {
		tmp = sqrt((2.0 * z)) * ((0.5 * (x + ((t * t) * t_1))) - y);
	} else {
		tmp = t_1 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if ((2.0d0 * z) <= 2d+99) then
        tmp = sqrt((2.0d0 * z)) * ((0.5d0 * (x + ((t * t) * t_1))) - y)
    else
        tmp = t_1 * sqrt(((2.0d0 * z) * ((t * t) + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((2.0 * z) <= 2e+99) {
		tmp = Math.sqrt((2.0 * z)) * ((0.5 * (x + ((t * t) * t_1))) - y);
	} else {
		tmp = t_1 * Math.sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (2.0 * z) <= 2e+99:
		tmp = math.sqrt((2.0 * z)) * ((0.5 * (x + ((t * t) * t_1))) - y)
	else:
		tmp = t_1 * math.sqrt(((2.0 * z) * ((t * t) + 1.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(2.0 * z) <= 2e+99)
		tmp = Float64(sqrt(Float64(2.0 * z)) * Float64(Float64(0.5 * Float64(x + Float64(Float64(t * t) * t_1))) - y));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * Float64(Float64(t * t) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((2.0 * z) <= 2e+99)
		tmp = sqrt((2.0 * z)) * ((0.5 * (x + ((t * t) * t_1))) - y);
	else
		tmp = t_1 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(2.0 * z), $MachinePrecision], 2e+99], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * N[(x + N[(N[(t * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 2) < 1.9999999999999999e99

   1. Initial program 99.2%

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

      1. sqr-neg 99.2%

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

      2. associate-/l* 99.2%

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

      3. distribute-frac-neg 99.2%

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

      4. exp-neg 99.2%

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

      5. associate-*r/ 99.2%

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

      6. *-rgt-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. associate-*r/ 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. associate-*r/ 99.8%

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

      11. exp-neg 99.8%

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

      12. distribute-frac-neg 99.8%

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

      13. associate-/l* 99.8%

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

      14. sqr-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 83.4%

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

      1. distribute-lft-out 83.4%

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

      2. unpow2 83.4%

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

   6. Simplified 83.4%

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

   if 1.9999999999999999e99 < (*.f64 z 2)

   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. Step-by-step derivation

      1. sqr-neg 99.8%

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

      2. associate-/l* 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. exp-neg 99.8%

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

      5. associate-*r/ 99.8%

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

      6. *-rgt-identity 99.8%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 96.7%

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

      2. expm1-udef 96.7%

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

      3. pow-exp 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. associate-*l* 96.7%

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

   5. Applied egg-rr 96.7%

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

      1. expm1-def 96.7%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 93.3%

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

      1. +-commutative 93.3%

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

      2. unpow2 93.3%

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

   10. Simplified 93.3%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \color{blue}{\left(t \cdot t + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot z \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(x + \left(t \cdot t\right) \cdot \left(x \cdot 0.5 - y\right)\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \]

Alternative 6: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;t_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(y \cdot \left(\left(t \cdot t\right) \cdot -0.5\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= (* t t) 2e+121)
     (* t_1 (- (* x 0.5) y))
     (* t_1 (- (* y (* (* t t) -0.5)) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 2e+121) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * ((y * ((t * t) * -0.5)) - 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) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if ((t * t) <= 2d+121) then
        tmp = t_1 * ((x * 0.5d0) - y)
    else
        tmp = t_1 * ((y * ((t * t) * (-0.5d0))) - 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((2.0 * z));
	double tmp;
	if ((t * t) <= 2e+121) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * ((y * ((t * t) * -0.5)) - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (t * t) <= 2e+121:
		tmp = t_1 * ((x * 0.5) - y)
	else:
		tmp = t_1 * ((y * ((t * t) * -0.5)) - y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 2e+121)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(t_1 * Float64(Float64(y * Float64(Float64(t * t) * -0.5)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((t * t) <= 2e+121)
		tmp = t_1 * ((x * 0.5) - y);
	else
		tmp = t_1 * ((y * ((t * t) * -0.5)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2e+121], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(y * N[(N[(t * t), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.00000000000000007e121

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-frac-neg 99.7%

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

      4. exp-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. *-rgt-identity 99.7%

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

      10. associate-*r/ 99.7%

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

      11. exp-neg 99.7%

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

      12. distribute-frac-neg 99.7%

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

      13. associate-/l* 99.7%

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

      14. sqr-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 83.8%

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

   if 2.00000000000000007e121 < (*.f64 t t)

   1. Initial program 99.0%

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

      1. sqr-neg 99.0%

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

      2. associate-/l* 99.0%

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

      3. distribute-frac-neg 99.0%

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

      4. exp-neg 99.0%

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

      5. associate-*r/ 99.0%

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

      6. *-rgt-identity 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. exp-neg 100.0%

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

      12. distribute-frac-neg 100.0%

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

      13. associate-/l* 100.0%

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

      14. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 77.9%

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

      1. distribute-lft-out 77.9%

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

      2. unpow2 77.9%

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

   6. Simplified 77.9%

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

   7. Taylor expanded in t around 0 77.9%

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

      1. unpow2 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-*r* 74.3%

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

      4. *-commutative 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in x around 0 52.1%

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

       1. mul-1-neg 52.1%

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

       2. *-commutative 52.1%

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

       3. distribute-rgt-neg-in 52.1%

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

   12. Simplified 52.1%

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

   13. Taylor expanded in x around 0 57.4%

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

       1. associate-*r* 57.4%

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

       2. *-commutative 57.4%

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

       3. unpow2 57.4%

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

   15. Simplified 57.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(y \cdot \left(\left(t \cdot t\right) \cdot -0.5\right) - y\right)\\ \end{array} \]

Alternative 7: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;t_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(y \cdot \left(t \cdot \left(t \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= (* t t) 2e+121)
     (* t_1 (- (* x 0.5) y))
     (* t_1 (* y (* t (* t -0.5)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 2e+121) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * (y * (t * (t * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if ((t * t) <= 2d+121) then
        tmp = t_1 * ((x * 0.5d0) - y)
    else
        tmp = t_1 * (y * (t * (t * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 2e+121) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * (y * (t * (t * -0.5)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (t * t) <= 2e+121:
		tmp = t_1 * ((x * 0.5) - y)
	else:
		tmp = t_1 * (y * (t * (t * -0.5)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 2e+121)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(t_1 * Float64(y * Float64(t * Float64(t * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((t * t) <= 2e+121)
		tmp = t_1 * ((x * 0.5) - y);
	else
		tmp = t_1 * (y * (t * (t * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2e+121], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(y * N[(t * N[(t * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.00000000000000007e121

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-frac-neg 99.7%

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

      4. exp-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. *-rgt-identity 99.7%

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

      10. associate-*r/ 99.7%

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

      11. exp-neg 99.7%

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

      12. distribute-frac-neg 99.7%

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

      13. associate-/l* 99.7%

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

      14. sqr-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 83.8%

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

   if 2.00000000000000007e121 < (*.f64 t t)

   1. Initial program 99.0%

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

      1. sqr-neg 99.0%

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

      2. associate-/l* 99.0%

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

      3. distribute-frac-neg 99.0%

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

      4. exp-neg 99.0%

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

      5. associate-*r/ 99.0%

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

      6. *-rgt-identity 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. exp-neg 100.0%

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

      12. distribute-frac-neg 100.0%

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

      13. associate-/l* 100.0%

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

      14. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 77.9%

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

      1. distribute-lft-out 77.9%

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

      2. unpow2 77.9%

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

   6. Simplified 77.9%

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

   7. Taylor expanded in y around -inf 57.4%

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

      1. associate-*r* 57.4%

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

      2. neg-mul-1 57.4%

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

      3. *-commutative 57.4%

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

      4. unpow2 57.4%

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

   9. Simplified 57.4%

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

   10. Taylor expanded in t around inf 57.4%

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

       1. associate-*r* 57.4%

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

       2. *-commutative 57.4%

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

       3. unpow2 57.4%

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

       4. associate-*r* 57.4%

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

   12. Simplified 57.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(y \cdot \left(t \cdot \left(t \cdot -0.5\right)\right)\right)\\ \end{array} \]

Alternative 8: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (* 2.0 z) (+ (* t t) 1.0)))))

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt(((2.0 * z) * ((t * t) + 1.0)));
}

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(((2.0d0 * z) * ((t * t) + 1.0d0)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * Math.sqrt(((2.0 * z) * ((t * t) + 1.0)));
}

Python

def code(x, y, z, t):
	return ((x * 0.5) - y) * math.sqrt(((2.0 * z) * ((t * t) + 1.0)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(Float64(2.0 * z) * Float64(Float64(t * t) + 1.0))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * 0.5) - y) * sqrt(((2.0 * z) * ((t * t) + 1.0)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. sqr-neg 99.4%

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

   2. associate-/l* 99.4%

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

   3. distribute-frac-neg 99.4%

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

   4. exp-neg 99.5%

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

   5. associate-*r/ 99.5%

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

   6. *-rgt-identity 99.5%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. associate-*r/ 99.8%

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

   10. exp-neg 99.8%

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

   11. distribute-frac-neg 99.8%

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

   12. associate-/l* 99.8%

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

   13. sqr-neg 99.8%

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

   14. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.3%

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

   2. expm1-udef 80.4%

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

   3. pow-exp 80.4%

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

   4. sqrt-unprod 80.4%

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

   5. associate-*l* 80.4%

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

5. Applied egg-rr 80.4%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 83.9%

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

   1. +-commutative 83.9%

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

   2. unpow2 83.9%

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

10. Simplified 83.9%

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

11. Final simplification 83.9%

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

Alternative 9: 43.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+30} \lor \neg \left(y \leq 6.8 \cdot 10^{-29}\right):\\ \;\;\;\;t_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (or (<= y -1.25e+30) (not (<= y 6.8e-29)))
     (* t_1 (- y))
     (* t_1 (* x 0.5)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((y <= -1.25e+30) || !(y <= 6.8e-29)) {
		tmp = t_1 * -y;
	} else {
		tmp = t_1 * (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if ((y <= (-1.25d+30)) .or. (.not. (y <= 6.8d-29))) then
        tmp = t_1 * -y
    else
        tmp = t_1 * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if ((y <= -1.25e+30) || !(y <= 6.8e-29)) {
		tmp = t_1 * -y;
	} else {
		tmp = t_1 * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (y <= -1.25e+30) or not (y <= 6.8e-29):
		tmp = t_1 * -y
	else:
		tmp = t_1 * (x * 0.5)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if ((y <= -1.25e+30) || !(y <= 6.8e-29))
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = Float64(t_1 * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((y <= -1.25e+30) || ~((y <= 6.8e-29)))
		tmp = t_1 * -y;
	else
		tmp = t_1 * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -1.25e+30], N[Not[LessEqual[y, 6.8e-29]], $MachinePrecision]], N[(t$95$1 * (-y)), $MachinePrecision], N[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e30 or 6.79999999999999945e-29 < 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. Step-by-step derivation

      1. sqr-neg 99.9%

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

      2. associate-/l* 99.9%

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

      3. distribute-frac-neg 99.9%

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

      4. exp-neg 99.9%

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

      5. associate-*r/ 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*r/ 99.9%

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

      9. *-rgt-identity 99.9%

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

      10. associate-*r/ 99.9%

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

      11. exp-neg 99.9%

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

      12. distribute-frac-neg 99.9%

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

      13. associate-/l* 99.9%

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

      14. sqr-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 87.4%

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

      1. distribute-lft-out 87.4%

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

      2. unpow2 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in y around -inf 71.7%

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

      1. associate-*r* 71.7%

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

      2. neg-mul-1 71.7%

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

      3. *-commutative 71.7%

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

      4. unpow2 71.7%

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

   9. Simplified 71.7%

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

   10. Taylor expanded in t around 0 45.2%

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

       1. neg-mul-1 45.2%

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

   12. Simplified 45.2%

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

   if -1.25e30 < y < 6.79999999999999945e-29

   1. Initial program 99.0%

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

      1. sqr-neg 99.0%

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

      2. associate-/l* 99.0%

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

      3. distribute-frac-neg 99.0%

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

      4. exp-neg 99.0%

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

      5. associate-*r/ 99.0%

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

      6. *-rgt-identity 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-*r/ 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. associate-*r/ 99.8%

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

      11. exp-neg 99.8%

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

      12. distribute-frac-neg 99.8%

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

      13. associate-/l* 99.8%

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

      14. sqr-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 55.6%

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

   5. Taylor expanded in x around inf 42.9%

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

      1. *-commutative 42.9%

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

      2. *-commutative 42.9%

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

      3. associate-*l* 42.9%

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

      4. *-commutative 42.9%

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

      5. associate-*l* 42.9%

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

      6. *-commutative 42.9%

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

   7. Simplified 42.9%

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

      1. pow1 42.9%

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

      2. associate-*r* 42.9%

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

      3. sqrt-prod 43.0%

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

      4. *-commutative 43.0%

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

      5. *-commutative 43.0%

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

   9. Applied egg-rr 43.0%

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

       1. unpow1 43.0%

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

       2. *-commutative 43.0%

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

       3. *-commutative 43.0%

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

   11. Simplified 43.0%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+30} \lor \neg \left(y \leq 6.8 \cdot 10^{-29}\right):\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 56.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt((2.0 * z)) * ((x * 0.5) - 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((2.0d0 * z)) * ((x * 0.5d0) - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sqr-neg 99.4%

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

   2. associate-/l* 99.4%

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

   3. distribute-frac-neg 99.4%

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

   4. exp-neg 99.5%

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

   5. associate-*r/ 99.5%

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

   6. *-rgt-identity 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*r/ 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. associate-*r/ 99.8%

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

   11. exp-neg 99.8%

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

   12. distribute-frac-neg 99.8%

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

   13. associate-/l* 99.8%

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

   14. sqr-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 55.5%

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

5. Final simplification 55.5%

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

Alternative 11: 29.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sqr-neg 99.4%

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

   2. associate-/l* 99.4%

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

   3. distribute-frac-neg 99.4%

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

   4. exp-neg 99.5%

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

   5. associate-*r/ 99.5%

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

   6. *-rgt-identity 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*r/ 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. associate-*r/ 99.8%

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

   11. exp-neg 99.8%

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

   12. distribute-frac-neg 99.8%

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

   13. associate-/l* 99.8%

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

   14. sqr-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 83.3%

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

   1. distribute-lft-out 83.3%

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

   2. unpow2 83.3%

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

6. Simplified 83.3%

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

7. Taylor expanded in y around -inf 47.9%

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

   1. associate-*r* 47.9%

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

   2. neg-mul-1 47.9%

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

   3. *-commutative 47.9%

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

   4. unpow2 47.9%

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

9. Simplified 47.9%

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

10. Taylor expanded in t around 0 28.4%

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

    1. neg-mul-1 28.4%

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

12. Simplified 28.4%

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

13. Final simplification 28.4%

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

Developer target: 99.3% 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 2023275 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* 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))))