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

Percentage Accurate: 99.4% → 99.8%

Time: 15.0s

Alternatives: 10

Speedup: 1.0×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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) * ((z * (2.0d0 * (exp(t) ** t))) ** 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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/ 99.0%

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

   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. 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. pow1/2 99.8%

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

   2. pow1/2 99.8%

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

   3. pow-prod-down 99.8%

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

   4. associate-*l* 99.8%

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

   5. exp-prod 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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/ 99.0%

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

   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. 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 74.4%

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

   3. sqrt-unprod 74.4%

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

   4. associate-*l* 74.4%

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

   5. exp-prod 74.4%

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

5. Applied egg-rr 74.4%

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

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

   5. exp-prod 99.8%

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

   6. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 87.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;z \cdot 2 \leq 10^{+59}:\\ \;\;\;\;t_1 \cdot \left(\left(1 + 0.5 \cdot \left(t \cdot t\right)\right) \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot {\left(z \cdot \left(2 \cdot \left(t \cdot t + 1\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* z 2.0) 1e+59)
     (* t_1 (* (+ 1.0 (* 0.5 (* t t))) (sqrt (* z 2.0))))
     (* t_1 (pow (* z (* 2.0 (+ (* t t) 1.0))) 0.5)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((z * 2.0) <= 1e+59) {
		tmp = t_1 * ((1.0 + (0.5 * (t * t))) * sqrt((z * 2.0)));
	} else {
		tmp = t_1 * pow((z * (2.0 * ((t * t) + 1.0))), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if ((z * 2.0d0) <= 1d+59) then
        tmp = t_1 * ((1.0d0 + (0.5d0 * (t * t))) * sqrt((z * 2.0d0)))
    else
        tmp = t_1 * ((z * (2.0d0 * ((t * t) + 1.0d0))) ** 0.5d0)
    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 ((z * 2.0) <= 1e+59) {
		tmp = t_1 * ((1.0 + (0.5 * (t * t))) * Math.sqrt((z * 2.0)));
	} else {
		tmp = t_1 * Math.pow((z * (2.0 * ((t * t) + 1.0))), 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (z * 2.0) <= 1e+59:
		tmp = t_1 * ((1.0 + (0.5 * (t * t))) * math.sqrt((z * 2.0)))
	else:
		tmp = t_1 * math.pow((z * (2.0 * ((t * t) + 1.0))), 0.5)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(z * 2.0) <= 1e+59)
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(0.5 * Float64(t * t))) * sqrt(Float64(z * 2.0))));
	else
		tmp = Float64(t_1 * (Float64(z * Float64(2.0 * Float64(Float64(t * t) + 1.0))) ^ 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((z * 2.0) <= 1e+59)
		tmp = t_1 * ((1.0 + (0.5 * (t * t))) * sqrt((z * 2.0)));
	else
		tmp = t_1 * ((z * (2.0 * ((t * t) + 1.0))) ^ 0.5);
	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[(z * 2.0), $MachinePrecision], 1e+59], N[(t$95$1 * N[(N[(1.0 + N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[(z * N[(2.0 * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 2) < 9.99999999999999972e58

   1. Initial program 98.6%

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

      1. sqr-neg 98.6%

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

      2. associate-/l* 98.6%

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

      3. distribute-frac-neg 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-*r/ 98.5%

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

      6. *-rgt-identity 98.5%

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

      7. associate-*r/ 99.9%

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

      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. Taylor expanded in t around 0 89.6%

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

      1. unpow2 89.6%

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

   6. Simplified 89.6%

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

   if 9.99999999999999972e58 < (*.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. pow1/2 99.8%

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

      2. pow1/2 99.8%

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

      3. pow-prod-down 99.8%

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

      4. associate-*l* 99.8%

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

      5. exp-prod 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in t around 0 95.2%

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

      1. +-commutative 95.2%

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

      2. unpow2 95.2%

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

   8. Simplified 95.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 2 \leq 10^{+59}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\left(1 + 0.5 \cdot \left(t \cdot t\right)\right) \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot {\left(z \cdot \left(2 \cdot \left(t \cdot t + 1\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 4: 84.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 0.02:\\ \;\;\;\;t_1 \cdot {\left(z \cdot 2\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot {\left(2 \cdot \left(z \cdot \left(t \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 0.02)
     (* t_1 (pow (* z 2.0) 0.5))
     (* t_1 (pow (* 2.0 (* z (* t t))) 0.5)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 0.02) {
		tmp = t_1 * pow((z * 2.0), 0.5);
	} else {
		tmp = t_1 * pow((2.0 * (z * (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 = (x * 0.5d0) - y
    if ((t * t) <= 0.02d0) then
        tmp = t_1 * ((z * 2.0d0) ** 0.5d0)
    else
        tmp = t_1 * ((2.0d0 * (z * (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 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 0.02) {
		tmp = t_1 * Math.pow((z * 2.0), 0.5);
	} else {
		tmp = t_1 * Math.pow((2.0 * (z * (t * t))), 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (t * t) <= 0.02:
		tmp = t_1 * math.pow((z * 2.0), 0.5)
	else:
		tmp = t_1 * math.pow((2.0 * (z * (t * t))), 0.5)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 0.02)
		tmp = Float64(t_1 * (Float64(z * 2.0) ^ 0.5));
	else
		tmp = Float64(t_1 * (Float64(2.0 * Float64(z * Float64(t * t))) ^ 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((t * t) <= 0.02)
		tmp = t_1 * ((z * 2.0) ^ 0.5);
	else
		tmp = t_1 * ((2.0 * (z * (t * t))) ^ 0.5);
	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[(t * t), $MachinePrecision], 0.02], N[(t$95$1 * N[Power[N[(z * 2.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[(2.0 * N[(z * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 0.0200000000000000004

   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. associate-*r/ 99.7%

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

      8. *-rgt-identity 99.7%

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

      9. associate-*r/ 99.7%

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

      10. exp-neg 99.7%

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

      11. distribute-frac-neg 99.7%

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

      12. associate-/l* 99.7%

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

      13. sqr-neg 99.7%

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

      14. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 98.6%

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

      1. *-commutative 98.6%

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

      2. sqrt-prod 99.0%

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

      3. pow1/2 99.0%

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

   6. Applied egg-rr 99.0%

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

   if 0.0200000000000000004 < (*.f64 t t)

   1. Initial program 98.3%

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

      1. sqr-neg 98.3%

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

      2. associate-/l* 98.3%

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

      3. distribute-frac-neg 98.3%

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

      4. exp-neg 98.3%

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

      5. associate-*r/ 98.3%

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

      6. *-rgt-identity 98.3%

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

      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. pow1/2 100.0%

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

      2. pow1/2 100.0%

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

      3. pow-prod-down 100.0%

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

      4. associate-*l* 100.0%

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

      5. exp-prod 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in t around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. unpow2 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in t around inf 79.0%

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

       1. unpow2 79.0%

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

       2. *-commutative 79.0%

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

   11. Simplified 79.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.02:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot {\left(z \cdot 2\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot {\left(2 \cdot \left(z \cdot \left(t \cdot t\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 5: 84.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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) * ((z * (2.0d0 * ((t * t) + 1.0d0))) ** 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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/ 99.0%

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

   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. 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. pow1/2 99.8%

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

   2. pow1/2 99.8%

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

   3. pow-prod-down 99.8%

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

   4. associate-*l* 99.8%

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

   5. exp-prod 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in t around 0 90.0%

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

   1. +-commutative 90.0%

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

   2. unpow2 90.0%

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

8. Simplified 90.0%

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

9. Final simplification 90.0%

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

Alternative 6: 56.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ t_2 := \left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 760000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 0.5 (* z (* x x)))))
        (t_2 (* (- (* x 0.5) y) (sqrt (* z 2.0)))))
   (if (<= t 760000000000.0)
     t_2
     (if (<= t 1.95e+66)
       t_1
       (if (<= t 1.1e+94)
         t_2
         (if (<= t 6.8e+191) (sqrt (* (* z 2.0) (* y y))) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((0.5 * (z * (x * x))));
	double t_2 = ((x * 0.5) - y) * sqrt((z * 2.0));
	double tmp;
	if (t <= 760000000000.0) {
		tmp = t_2;
	} else if (t <= 1.95e+66) {
		tmp = t_1;
	} else if (t <= 1.1e+94) {
		tmp = t_2;
	} else if (t <= 6.8e+191) {
		tmp = sqrt(((z * 2.0) * (y * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((0.5d0 * (z * (x * x))))
    t_2 = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    if (t <= 760000000000.0d0) then
        tmp = t_2
    else if (t <= 1.95d+66) then
        tmp = t_1
    else if (t <= 1.1d+94) then
        tmp = t_2
    else if (t <= 6.8d+191) then
        tmp = sqrt(((z * 2.0d0) * (y * y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((0.5 * (z * (x * x))));
	double t_2 = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 760000000000.0) {
		tmp = t_2;
	} else if (t <= 1.95e+66) {
		tmp = t_1;
	} else if (t <= 1.1e+94) {
		tmp = t_2;
	} else if (t <= 6.8e+191) {
		tmp = Math.sqrt(((z * 2.0) * (y * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((0.5 * (z * (x * x))))
	t_2 = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	tmp = 0
	if t <= 760000000000.0:
		tmp = t_2
	elif t <= 1.95e+66:
		tmp = t_1
	elif t <= 1.1e+94:
		tmp = t_2
	elif t <= 6.8e+191:
		tmp = math.sqrt(((z * 2.0) * (y * y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(0.5 * Float64(z * Float64(x * x))))
	t_2 = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)))
	tmp = 0.0
	if (t <= 760000000000.0)
		tmp = t_2;
	elseif (t <= 1.95e+66)
		tmp = t_1;
	elseif (t <= 1.1e+94)
		tmp = t_2;
	elseif (t <= 6.8e+191)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((0.5 * (z * (x * x))));
	t_2 = ((x * 0.5) - y) * sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 760000000000.0)
		tmp = t_2;
	elseif (t <= 1.95e+66)
		tmp = t_1;
	elseif (t <= 1.1e+94)
		tmp = t_2;
	elseif (t <= 6.8e+191)
		tmp = sqrt(((z * 2.0) * (y * y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(0.5 * N[(z * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 760000000000.0], t$95$2, If[LessEqual[t, 1.95e+66], t$95$1, If[LessEqual[t, 1.1e+94], t$95$2, If[LessEqual[t, 6.8e+191], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.6e11 or 1.9500000000000002e66 < t < 1.10000000000000006e94

   1. Initial program 99.3%

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

      1. sqr-neg 99.3%

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

      2. associate-/l* 99.3%

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

      3. distribute-frac-neg 99.3%

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

      4. exp-neg 99.3%

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

      5. associate-*r/ 99.3%

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

      6. *-rgt-identity 99.3%

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

      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. Taylor expanded in t around 0 71.6%

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

      1. *-commutative 71.6%

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

      2. sqrt-prod 71.9%

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

   6. Applied egg-rr 71.9%

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

   if 7.6e11 < t < 1.9500000000000002e66 or 6.80000000000000018e191 < t

   1. Initial program 97.1%

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

      1. sqr-neg 97.1%

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

      2. associate-/l* 97.1%

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

      3. distribute-frac-neg 97.1%

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

      4. exp-neg 97.1%

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

      5. associate-*r/ 97.1%

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

      6. *-rgt-identity 97.1%

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

      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. Taylor expanded in t around 0 12.9%

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

      1. add-sqr-sqrt 11.0%

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

      2. sqrt-prod 27.8%

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

      3. unpow2 27.8%

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

      4. pow1/2 27.8%

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

      5. *-rgt-identity 27.8%

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

      6. pow1/2 27.8%

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

      7. unpow-prod-down 30.6%

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

      8. *-commutative 30.6%

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

      9. *-commutative 30.6%

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

      10. associate-*l* 30.6%

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

   6. Applied egg-rr 30.6%

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

      1. unpow1/2 30.6%

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

      2. associate-*r* 30.6%

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

      3. *-commutative 30.6%

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

      4. *-commutative 30.6%

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

      5. *-commutative 30.6%

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

      6. *-commutative 30.6%

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

   8. Simplified 30.6%

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

   9. Taylor expanded in x around inf 30.4%

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

       1. unpow2 30.4%

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

   11. Simplified 30.4%

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

   if 1.10000000000000006e94 < t < 6.80000000000000018e191

   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. Taylor expanded in t around 0 15.6%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-prod 17.6%

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

      3. unpow2 17.6%

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

      4. pow1/2 17.6%

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

      5. *-rgt-identity 17.6%

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

      6. pow1/2 17.6%

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

      7. unpow-prod-down 28.2%

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

      8. *-commutative 28.2%

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

      9. *-commutative 28.2%

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

      10. associate-*l* 28.2%

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

   6. Applied egg-rr 28.2%

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

      1. unpow1/2 28.2%

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

      2. associate-*r* 28.2%

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

      3. *-commutative 28.2%

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

      4. *-commutative 28.2%

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

      5. *-commutative 28.2%

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

      6. *-commutative 28.2%

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

   8. Simplified 28.2%

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

   9. Taylor expanded in x around 0 28.3%

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

       1. unpow2 28.3%

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

   11. Simplified 28.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 760000000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+94}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternative 7: 56.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ t_2 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1250000000000:\\ \;\;\;\;t_2 \cdot {\left(z \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+91}:\\ \;\;\;\;t_2 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 0.5 (* z (* x x))))) (t_2 (- (* x 0.5) y)))
   (if (<= t 1250000000000.0)
     (* t_2 (pow (* z 2.0) 0.5))
     (if (<= t 2.7e+66)
       t_1
       (if (<= t 7.5e+91)
         (* t_2 (sqrt (* z 2.0)))
         (if (<= t 6.8e+191) (sqrt (* (* z 2.0) (* y y))) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((0.5 * (z * (x * x))));
	double t_2 = (x * 0.5) - y;
	double tmp;
	if (t <= 1250000000000.0) {
		tmp = t_2 * pow((z * 2.0), 0.5);
	} else if (t <= 2.7e+66) {
		tmp = t_1;
	} else if (t <= 7.5e+91) {
		tmp = t_2 * sqrt((z * 2.0));
	} else if (t <= 6.8e+191) {
		tmp = sqrt(((z * 2.0) * (y * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((0.5d0 * (z * (x * x))))
    t_2 = (x * 0.5d0) - y
    if (t <= 1250000000000.0d0) then
        tmp = t_2 * ((z * 2.0d0) ** 0.5d0)
    else if (t <= 2.7d+66) then
        tmp = t_1
    else if (t <= 7.5d+91) then
        tmp = t_2 * sqrt((z * 2.0d0))
    else if (t <= 6.8d+191) then
        tmp = sqrt(((z * 2.0d0) * (y * y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((0.5 * (z * (x * x))));
	double t_2 = (x * 0.5) - y;
	double tmp;
	if (t <= 1250000000000.0) {
		tmp = t_2 * Math.pow((z * 2.0), 0.5);
	} else if (t <= 2.7e+66) {
		tmp = t_1;
	} else if (t <= 7.5e+91) {
		tmp = t_2 * Math.sqrt((z * 2.0));
	} else if (t <= 6.8e+191) {
		tmp = Math.sqrt(((z * 2.0) * (y * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((0.5 * (z * (x * x))))
	t_2 = (x * 0.5) - y
	tmp = 0
	if t <= 1250000000000.0:
		tmp = t_2 * math.pow((z * 2.0), 0.5)
	elif t <= 2.7e+66:
		tmp = t_1
	elif t <= 7.5e+91:
		tmp = t_2 * math.sqrt((z * 2.0))
	elif t <= 6.8e+191:
		tmp = math.sqrt(((z * 2.0) * (y * y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(0.5 * Float64(z * Float64(x * x))))
	t_2 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1250000000000.0)
		tmp = Float64(t_2 * (Float64(z * 2.0) ^ 0.5));
	elseif (t <= 2.7e+66)
		tmp = t_1;
	elseif (t <= 7.5e+91)
		tmp = Float64(t_2 * sqrt(Float64(z * 2.0)));
	elseif (t <= 6.8e+191)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((0.5 * (z * (x * x))));
	t_2 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 1250000000000.0)
		tmp = t_2 * ((z * 2.0) ^ 0.5);
	elseif (t <= 2.7e+66)
		tmp = t_1;
	elseif (t <= 7.5e+91)
		tmp = t_2 * sqrt((z * 2.0));
	elseif (t <= 6.8e+191)
		tmp = sqrt(((z * 2.0) * (y * y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(0.5 * N[(z * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1250000000000.0], N[(t$95$2 * N[Power[N[(z * 2.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e+66], t$95$1, If[LessEqual[t, 7.5e+91], N[(t$95$2 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e+191], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.25e12

   1. Initial program 99.3%

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

      1. sqr-neg 99.3%

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

      2. associate-/l* 99.3%

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

      3. distribute-frac-neg 99.3%

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

      4. exp-neg 99.3%

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

      5. associate-*r/ 99.3%

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

      6. *-rgt-identity 99.3%

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

      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. Taylor expanded in t around 0 72.9%

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

      1. *-commutative 72.9%

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

      2. sqrt-prod 73.2%

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

      3. pow1/2 73.2%

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

   6. Applied egg-rr 73.2%

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

   if 1.25e12 < t < 2.7e66 or 6.80000000000000018e191 < t

   1. Initial program 97.1%

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

      1. sqr-neg 97.1%

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

      2. associate-/l* 97.1%

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

      3. distribute-frac-neg 97.1%

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

      4. exp-neg 97.1%

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

      5. associate-*r/ 97.1%

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

      6. *-rgt-identity 97.1%

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

      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. Taylor expanded in t around 0 12.9%

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

      1. add-sqr-sqrt 11.0%

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

      2. sqrt-prod 27.8%

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

      3. unpow2 27.8%

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

      4. pow1/2 27.8%

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

      5. *-rgt-identity 27.8%

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

      6. pow1/2 27.8%

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

      7. unpow-prod-down 30.6%

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

      8. *-commutative 30.6%

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

      9. *-commutative 30.6%

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

      10. associate-*l* 30.6%

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

   6. Applied egg-rr 30.6%

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

      1. unpow1/2 30.6%

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

      2. associate-*r* 30.6%

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

      3. *-commutative 30.6%

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

      4. *-commutative 30.6%

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

      5. *-commutative 30.6%

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

      6. *-commutative 30.6%

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

   8. Simplified 30.6%

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

   9. Taylor expanded in x around inf 30.4%

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

       1. unpow2 30.4%

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

   11. Simplified 30.4%

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

   if 2.7e66 < t < 7.50000000000000033e91

   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. Taylor expanded in t around 0 40.4%

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

      1. *-commutative 40.4%

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

      2. sqrt-prod 40.4%

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

   6. Applied egg-rr 40.4%

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

   if 7.50000000000000033e91 < t < 6.80000000000000018e191

   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. Taylor expanded in t around 0 15.6%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-prod 17.6%

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

      3. unpow2 17.6%

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

      4. pow1/2 17.6%

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

      5. *-rgt-identity 17.6%

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

      6. pow1/2 17.6%

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

      7. unpow-prod-down 28.2%

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

      8. *-commutative 28.2%

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

      9. *-commutative 28.2%

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

      10. associate-*l* 28.2%

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

   6. Applied egg-rr 28.2%

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

      1. unpow1/2 28.2%

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

      2. associate-*r* 28.2%

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

      3. *-commutative 28.2%

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

      4. *-commutative 28.2%

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

      5. *-commutative 28.2%

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

      6. *-commutative 28.2%

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

   8. Simplified 28.2%

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

   9. Taylor expanded in x around 0 28.3%

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

       1. unpow2 28.3%

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

   11. Simplified 28.3%

       \[\leadsto \sqrt{\color{blue}{\left(y \cdot y\right)} \cdot \left(2 \cdot z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1250000000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot {\left(z \cdot 2\right)}^{0.5}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+91}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternative 8: 36.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-41}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 5.6e-41) (* (- y) (sqrt (* z 2.0))) (sqrt (* 0.5 (* z (* x x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 5.6e-41) {
		tmp = -y * sqrt((z * 2.0));
	} else {
		tmp = sqrt((0.5 * (z * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 5.6d-41) then
        tmp = -y * sqrt((z * 2.0d0))
    else
        tmp = sqrt((0.5d0 * (z * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 5.6e-41) {
		tmp = -y * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt((0.5 * (z * (x * x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 5.6e-41:
		tmp = -y * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt((0.5 * (z * (x * x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 5.6e-41)
		tmp = Float64(Float64(-y) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(0.5 * Float64(z * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 5.6e-41)
		tmp = -y * sqrt((z * 2.0));
	else
		tmp = sqrt((0.5 * (z * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 5.6e-41], N[((-y) * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(z * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.6000000000000003e-41

   1. Initial program 98.7%

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

      1. sqr-neg 98.7%

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

      2. associate-/l* 98.7%

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

      3. distribute-frac-neg 98.7%

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

      4. exp-neg 98.7%

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

      5. associate-*r/ 98.7%

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

      6. *-rgt-identity 98.7%

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

      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. Taylor expanded in t around 0 62.2%

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

   5. Taylor expanded in x around 0 35.1%

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

      1. mul-1-neg 35.1%

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

      2. associate-*l* 35.1%

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

      3. distribute-rgt-neg-in 35.1%

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

      4. distribute-lft-neg-out 35.1%

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

      5. *-commutative 35.1%

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

   7. Simplified 35.1%

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

      1. *-commutative 35.1%

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

      2. *-commutative 35.1%

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

      3. associate-*l* 35.1%

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

      4. sqrt-prod 35.2%

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

      5. add-cube-cbrt 34.8%

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

      6. unpow3 34.8%

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

      7. *-commutative 34.8%

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

      8. distribute-rgt-neg-out 34.8%

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

      9. add-sqr-sqrt 20.0%

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

      10. sqrt-unprod 19.8%

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

      11. sqr-neg 19.8%

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

      12. sqrt-unprod 2.2%

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

      13. add-sqr-sqrt 3.1%

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

      14. *-commutative 3.1%

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

      15. add-sqr-sqrt 2.2%

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

      16. sqrt-unprod 19.8%

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

      17. sqr-neg 19.8%

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

      18. sqrt-unprod 20.0%

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

      19. add-sqr-sqrt 34.8%

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

      20. unpow3 34.8%

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

      21. add-cube-cbrt 35.2%

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

      22. *-commutative 35.2%

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

   9. Applied egg-rr 35.2%

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

       1. distribute-lft-neg-in 35.2%

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

   11. Simplified 35.2%

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

   if 5.6000000000000003e-41 < x

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. 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. Taylor expanded in t around 0 55.5%

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

      1. add-sqr-sqrt 51.1%

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

      2. sqrt-prod 51.5%

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

      3. unpow2 51.5%

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

      4. pow1/2 51.5%

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

      5. *-rgt-identity 51.5%

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

      6. pow1/2 51.5%

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

      7. unpow-prod-down 51.5%

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

      8. *-commutative 51.5%

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

      9. *-commutative 51.5%

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

      10. associate-*l* 51.5%

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

   6. Applied egg-rr 51.5%

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

      1. unpow1/2 51.5%

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

      2. associate-*r* 51.5%

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

      3. *-commutative 51.5%

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

      4. *-commutative 51.5%

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

      5. *-commutative 51.5%

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

      6. *-commutative 51.5%

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

   8. Simplified 51.5%

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

   9. Taylor expanded in x around inf 50.3%

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

       1. unpow2 50.3%

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

   11. Simplified 50.3%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-41}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternative 9: 29.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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/ 99.0%

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

   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. 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. Taylor expanded in t around 0 60.1%

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

5. Taylor expanded in x around 0 29.3%

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

   1. mul-1-neg 29.3%

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

   2. associate-*l* 29.3%

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

   3. distribute-rgt-neg-in 29.3%

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

   4. distribute-lft-neg-out 29.3%

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

   5. *-commutative 29.3%

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

7. Simplified 29.3%

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

   1. *-commutative 29.3%

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

   2. *-commutative 29.3%

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

   3. associate-*l* 29.2%

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

   4. sqrt-prod 29.4%

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

   5. add-cube-cbrt 29.0%

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

   6. unpow3 29.0%

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

   7. *-commutative 29.0%

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

   8. distribute-rgt-neg-out 29.0%

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

   9. add-sqr-sqrt 15.2%

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

   10. sqrt-unprod 16.4%

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

   11. sqr-neg 16.4%

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

   12. sqrt-unprod 1.7%

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

   13. add-sqr-sqrt 2.8%

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

   14. *-commutative 2.8%

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

   15. add-sqr-sqrt 1.7%

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

   16. sqrt-unprod 16.4%

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

   17. sqr-neg 16.4%

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

   18. sqrt-unprod 15.2%

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

   19. add-sqr-sqrt 29.0%

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

   20. unpow3 29.0%

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

   21. add-cube-cbrt 29.4%

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

   22. *-commutative 29.4%

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

9. Applied egg-rr 29.4%

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

    1. distribute-lft-neg-in 29.4%

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

11. Simplified 29.4%

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

12. Final simplification 29.4%

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

Alternative 10: 2.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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/ 99.0%

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

   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. 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. Taylor expanded in t around 0 60.1%

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

5. Taylor expanded in x around 0 29.3%

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

   1. mul-1-neg 29.3%

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

   2. associate-*l* 29.3%

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

   3. distribute-rgt-neg-in 29.3%

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

   4. distribute-lft-neg-out 29.3%

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

   5. *-commutative 29.3%

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

7. Simplified 29.3%

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

   1. expm1-log1p-u 16.5%

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

   2. expm1-udef 10.5%

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

   3. *-commutative 10.5%

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

   4. *-commutative 10.5%

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

   5. associate-*l* 10.5%

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

   6. add-sqr-sqrt 9.8%

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

   7. sqrt-unprod 12.6%

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

   8. sqr-neg 12.6%

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

   9. sqrt-unprod 1.1%

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

   10. add-sqr-sqrt 2.6%

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

   11. sqrt-prod 2.6%

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

   12. *-commutative 2.6%

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

9. Applied egg-rr 2.6%

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

    1. expm1-def 2.5%

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

    2. expm1-log1p 2.8%

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

11. Simplified 2.8%

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

12. Final simplification 2.8%

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

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (exp(1.0d0) ** ((t * t) / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(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))))