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

Percentage Accurate: 99.4% → 99.8%

Time: 16.9s

Alternatives: 19

Speedup: 1.0×

• 44.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-*l* 99.8%

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

   2. 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. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. pow2 99.8%

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

   2. exp-prod 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt(((2.0 * z) * exp(pow(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(((2.0d0 * z) * exp((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(((2.0 * z) * Math.exp(Math.pow(t, 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-*l* 99.8%

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

   2. 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. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

8. Simplified 99.8%

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

Alternative 3: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 3.15:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;t\_1 \cdot {\left({z}^{2} \cdot 4\right)}^{0.25}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+154} \lor \neg \left(t \leq 1.9 \cdot 10^{+258}\right):\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 3.15)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= t 1.85e+50)
       (* t_1 (pow (* (pow z 2.0) 4.0) 0.25))
       (if (or (<= t 8.5e+154) (not (<= t 1.9e+258)))
         (* (* t (* t_1 (sqrt 2.0))) (sqrt z))
         (* (sqrt (* (* 2.0 z) (fma t t 1.0))) (* x 0.5)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 3.15) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if (t <= 1.85e+50) {
		tmp = t_1 * pow((pow(z, 2.0) * 4.0), 0.25);
	} else if ((t <= 8.5e+154) || !(t <= 1.9e+258)) {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	} else {
		tmp = sqrt(((2.0 * z) * fma(t, t, 1.0))) * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 3.15)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (t <= 1.85e+50)
		tmp = Float64(t_1 * (Float64((z ^ 2.0) * 4.0) ^ 0.25));
	elseif ((t <= 8.5e+154) || !(t <= 1.9e+258))
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 3.15], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.85e+50], N[(t$95$1 * N[Power[N[(N[Power[z, 2.0], $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 8.5e+154], N[Not[LessEqual[t, 1.9e+258]], $MachinePrecision]], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.14999999999999991

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 78.4%

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

   if 3.14999999999999991 < t < 1.85e50

   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. associate-*l* 99.8%

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

      2. 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. exp-prod 99.9%

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

   3. Simplified 99.9%

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

      1. pow1 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. associate-*l* 99.9%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 19.9%

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

       1. sqrt-prod 19.9%

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

       2. pow1/2 19.9%

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

       3. metadata-eval 19.9%

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

       4. pow-prod-up 19.9%

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

       5. pow-prod-down 71.3%

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

       6. swap-sqr 71.3%

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

       7. pow2 71.3%

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

       8. metadata-eval 71.3%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({z}^{2} \cdot \color{blue}{4}\right)}^{0.25} \]

   11. Applied egg-rr 71.3%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left({z}^{2} \cdot 4\right)}^{0.25}} \]

   if 1.85e50 < t < 8.5000000000000002e154 or 1.90000000000000004e258 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 59.5%

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

       1. +-commutative 59.5%

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

       2. unpow2 59.5%

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

       3. fma-define 59.5%

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

   11. Simplified 59.5%

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

   12. Taylor expanded in t around inf 52.3%

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

   if 8.5000000000000002e154 < t < 1.90000000000000004e258

   1. Initial program 94.7%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 89.5%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right)} \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.15:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot {\left({z}^{2} \cdot 4\right)}^{0.25}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+154} \lor \neg \left(t \leq 1.9 \cdot 10^{+258}\right):\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 89:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 86000000000:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot t\_2\right)\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+153} \lor \neg \left(t \leq 9.5 \cdot 10^{+257}\right):\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* 2.0 z))))
   (if (<= t 89.0)
     (* t_1 t_2)
     (if (<= t 86000000000.0)
       (* (exp (/ (* t t) 2.0)) (* y t_2))
       (if (or (<= t 5.9e+153) (not (<= t 9.5e+257)))
         (* (* t (* t_1 (sqrt 2.0))) (sqrt z))
         (* (sqrt (* (* 2.0 z) (fma t t 1.0))) (* x 0.5)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((2.0 * z));
	double tmp;
	if (t <= 89.0) {
		tmp = t_1 * t_2;
	} else if (t <= 86000000000.0) {
		tmp = exp(((t * t) / 2.0)) * (y * t_2);
	} else if ((t <= 5.9e+153) || !(t <= 9.5e+257)) {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	} else {
		tmp = sqrt(((2.0 * z) * fma(t, t, 1.0))) * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 89.0)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 86000000000.0)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * t_2));
	elseif ((t <= 5.9e+153) || !(t <= 9.5e+257))
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 89.0], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 86000000000.0], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * t$95$2), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 5.9e+153], N[Not[LessEqual[t, 9.5e+257]], $MachinePrecision]], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 89

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 78.1%

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

   if 89 < t < 8.6e10

   1. Initial program 100.0%

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

      1. add-cbrt-cube 50.0%

         \[\leadsto \left(\left(\color{blue}{\sqrt[3]{\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot 0.5\right)}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. pow3 50.0%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{\left(x \cdot 0.5\right)}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. unpow-prod-down 50.0%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {0.5}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. metadata-eval 50.0%

         \[\leadsto \left(\left(\sqrt[3]{{x}^{3} \cdot \color{blue}{0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 50.0%

      \[\leadsto \left(\left(\color{blue}{\sqrt[3]{{x}^{3} \cdot 0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Taylor expanded in x around 0 0.0%

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

      1. neg-mul-1 0.0%

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

   7. Simplified 0.0%

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

      1. neg-sub0 0.0%

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

      2. sub-neg 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. sqr-neg 0.0%

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

      6. sqrt-unprod 50.0%

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

      7. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. +-lft-identity 100.0%

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

   11. Simplified 100.0%

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

   if 8.6e10 < t < 5.9000000000000002e153 or 9.50000000000000022e257 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 55.5%

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

       1. +-commutative 55.5%

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

       2. unpow2 55.5%

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

       3. fma-define 55.5%

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

   11. Simplified 55.5%

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

   12. Taylor expanded in t around inf 47.5%

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

   if 5.9000000000000002e153 < t < 9.50000000000000022e257

   1. Initial program 94.7%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 89.5%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right)} \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 89:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 86000000000:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+153} \lor \neg \left(t \leq 9.5 \cdot 10^{+257}\right):\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 89:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {t\_1}^{2}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+152} \lor \neg \left(t \leq 6.2 \cdot 10^{+258}\right):\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 89.0)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= t 2.7e+35)
       (sqrt (* (* 2.0 z) (pow t_1 2.0)))
       (if (or (<= t 5.2e+152) (not (<= t 6.2e+258)))
         (* (* t (* t_1 (sqrt 2.0))) (sqrt z))
         (* (sqrt (* (* 2.0 z) (fma t t 1.0))) (* x 0.5)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 89.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if (t <= 2.7e+35) {
		tmp = sqrt(((2.0 * z) * pow(t_1, 2.0)));
	} else if ((t <= 5.2e+152) || !(t <= 6.2e+258)) {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	} else {
		tmp = sqrt(((2.0 * z) * fma(t, t, 1.0))) * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 89.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (t <= 2.7e+35)
		tmp = sqrt(Float64(Float64(2.0 * z) * (t_1 ^ 2.0)));
	elseif ((t <= 5.2e+152) || !(t <= 6.2e+258))
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 89.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e+35], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[t, 5.2e+152], N[Not[LessEqual[t, 6.2e+258]], $MachinePrecision]], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 89

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 78.1%

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

   if 89 < t < 2.70000000000000003e35

   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. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 4.3%

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

       1. add-cbrt-cube 24.9%

          \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot 0.5\right)}} - y\right) \cdot \sqrt{2 \cdot z} \]

       2. pow3 24.9%

          \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(x \cdot 0.5\right)}^{3}}} - y\right) \cdot \sqrt{2 \cdot z} \]

       3. unpow-prod-down 24.9%

          \[\leadsto \left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {0.5}^{3}}} - y\right) \cdot \sqrt{2 \cdot z} \]

       4. metadata-eval 24.9%

          \[\leadsto \left(\sqrt[3]{{x}^{3} \cdot \color{blue}{0.125}} - y\right) \cdot \sqrt{2 \cdot z} \]

       5. *-commutative 24.9%

          \[\leadsto \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

       6. add-sqr-sqrt 13.6%

          \[\leadsto \color{blue}{\sqrt{\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}} \cdot \sqrt{\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}}} \]

       7. sqrt-unprod 24.2%

          \[\leadsto \color{blue}{\sqrt{\left(\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}\right)}} \]

       8. *-commutative 24.2%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right)} \cdot \left(\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}\right)} \]

       9. *-commutative 24.2%

          \[\leadsto \sqrt{\left(\sqrt{\color{blue}{2 \cdot z}} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right) \cdot \left(\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}\right)} \]

       10. *-commutative 24.2%

           \[\leadsto \sqrt{\left(\sqrt{2 \cdot z} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right)}} \]

       11. *-commutative 24.2%

           \[\leadsto \sqrt{\left(\sqrt{2 \cdot z} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right) \cdot \left(\sqrt{\color{blue}{2 \cdot z}} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right)} \]

   11. Applied egg-rr 24.4%

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

       1. fma-neg 24.4%

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

       2. *-commutative 24.4%

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

   13. Simplified 24.4%

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

   if 2.70000000000000003e35 < t < 5.2000000000000001e152 or 6.1999999999999996e258 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 59.2%

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

       1. +-commutative 59.2%

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

       2. unpow2 59.2%

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

       3. fma-define 59.2%

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

   11. Simplified 59.2%

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

   12. Taylor expanded in t around inf 52.4%

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

   if 5.2000000000000001e152 < t < 6.1999999999999996e258

   1. Initial program 94.7%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 89.5%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right)} \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 89:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+152} \lor \neg \left(t \leq 6.2 \cdot 10^{+258}\right):\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 10^{-9}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+293}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))) (t_2 (- (* x 0.5) y)))
   (if (<= (* t t) 1e-9)
     (* t_2 (* t_1 (hypot 1.0 t)))
     (if (<= (* t t) 2e+293)
       (* (exp (/ (* t t) 2.0)) (* 0.5 (* x t_1)))
       (* t_2 (sqrt (* (* 2.0 z) (fma t t 1.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 1e-9) {
		tmp = t_2 * (t_1 * hypot(1.0, t));
	} else if ((t * t) <= 2e+293) {
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * t_1));
	} else {
		tmp = t_2 * sqrt(((2.0 * z) * fma(t, t, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 1e-9)
		tmp = Float64(t_2 * Float64(t_1 * hypot(1.0, t)));
	elseif (Float64(t * t) <= 2e+293)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(0.5 * Float64(x * t_1)));
	else
		tmp = Float64(t_2 * sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1e-9], N[(t$95$2 * N[(t$95$1 * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 2e+293], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 1.00000000000000006e-9

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. exp-sqrt 99.6%

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

      3. exp-prod 99.6%

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

   3. Simplified 99.6%

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

      1. pow1 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. associate-*l* 99.6%

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

      4. pow-exp 99.6%

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

      5. pow2 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow1 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in t around 0 99.6%

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

       1. +-commutative 99.6%

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

       2. unpow2 99.6%

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

       3. fma-define 99.6%

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

   11. Simplified 99.6%

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

       1. sqrt-prod 99.6%

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

   13. Applied egg-rr 99.6%

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

       1. *-commutative 99.6%

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

       2. fma-undefine 99.6%

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

       3. unpow2 99.6%

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

       4. +-commutative 99.6%

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

       5. unpow2 99.6%

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

       6. hypot-1-def 99.6%

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

       7. *-commutative 99.6%

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

   15. Simplified 99.6%

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

   if 1.00000000000000006e-9 < (*.f64 t t) < 1.9999999999999998e293

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 100.0%

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

   4. Taylor expanded in x around inf 70.9%

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

      1. associate-*l* 70.9%

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

      2. *-commutative 70.9%

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

      3. unpow1/2 70.9%

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

      4. metadata-eval 70.9%

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

      5. pow-sqr 70.9%

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

      6. unpow1/2 70.9%

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

      7. metadata-eval 70.9%

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

      8. pow-sqr 70.9%

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

      9. unswap-sqr 70.9%

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

      10. exp-to-pow 70.9%

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

      11. exp-to-pow 70.9%

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

      12. exp-sum 70.9%

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

      13. distribute-rgt-in 70.9%

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

      14. +-commutative 70.9%

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

      15. *-commutative 70.9%

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

      16. exp-prod 70.9%

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

      17. exp-sum 70.9%

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

      18. rem-exp-log 70.9%

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

      19. rem-exp-log 70.9%

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

      20. exp-to-pow 70.9%

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

      21. exp-to-pow 70.9%

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

   6. Simplified 70.9%

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

   if 1.9999999999999998e293 < (*.f64 t 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. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 98.6%

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

       1. +-commutative 98.6%

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

       2. unpow2 98.6%

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

       3. fma-define 98.6%

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

   11. Simplified 98.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-9}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+293}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 5:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-t\_2\right)\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+258}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* 2.0 z))))
   (if (<= t 5.0)
     (* t_1 (* t_2 (hypot 1.0 t)))
     (if (<= t 1.32e+154)
       (* (exp (/ (* t t) 2.0)) (* y (- t_2)))
       (if (<= t 6.5e+258)
         (* (sqrt (* (* 2.0 z) (fma t t 1.0))) (* x 0.5))
         (* (* t (* t_1 (sqrt 2.0))) (sqrt z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((2.0 * z));
	double tmp;
	if (t <= 5.0) {
		tmp = t_1 * (t_2 * hypot(1.0, t));
	} else if (t <= 1.32e+154) {
		tmp = exp(((t * t) / 2.0)) * (y * -t_2);
	} else if (t <= 6.5e+258) {
		tmp = sqrt(((2.0 * z) * fma(t, t, 1.0))) * (x * 0.5);
	} else {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 5.0)
		tmp = Float64(t_1 * Float64(t_2 * hypot(1.0, t)));
	elseif (t <= 1.32e+154)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * Float64(-t_2)));
	elseif (t <= 6.5e+258)
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * Float64(x * 0.5));
	else
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 5.0], N[(t$95$1 * N[(t$95$2 * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.32e+154], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * (-t$95$2)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e+258], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 5

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 92.5%

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

       1. +-commutative 92.5%

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

       2. unpow2 92.5%

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

       3. fma-define 92.5%

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

   11. Simplified 92.5%

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

       1. sqrt-prod 92.0%

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

   13. Applied egg-rr 92.0%

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

       1. *-commutative 92.0%

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

       2. fma-undefine 92.0%

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

       3. unpow2 92.0%

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

       4. +-commutative 92.0%

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

       5. unpow2 92.0%

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

       6. hypot-1-def 87.4%

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

       7. *-commutative 87.4%

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

   15. Simplified 87.4%

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

   if 5 < t < 1.31999999999999998e154

   1. Initial program 100.0%

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

      1. add-cbrt-cube 91.4%

         \[\leadsto \left(\left(\color{blue}{\sqrt[3]{\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot 0.5\right)}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. pow3 91.4%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{\left(x \cdot 0.5\right)}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. unpow-prod-down 91.4%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {0.5}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. metadata-eval 91.4%

         \[\leadsto \left(\left(\sqrt[3]{{x}^{3} \cdot \color{blue}{0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 91.4%

      \[\leadsto \left(\left(\color{blue}{\sqrt[3]{{x}^{3} \cdot 0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Taylor expanded in x around 0 71.4%

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

      1. neg-mul-1 71.4%

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

   7. Simplified 71.4%

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

   if 1.31999999999999998e154 < t < 6.50000000000000005e258

   1. Initial program 94.7%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 89.5%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right)} \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \]

   if 6.50000000000000005e258 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 84.2%

       \[\leadsto \color{blue}{\left(t \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot \sqrt{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{2 \cdot z}\right)\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+258}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 4.8:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+151}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-t\_2\right)\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* 2.0 z))))
   (if (<= t 4.8)
     (* t_1 t_2)
     (if (<= t 5e+151)
       (* (exp (/ (* t t) 2.0)) (* y (- t_2)))
       (if (<= t 8e+257)
         (* (sqrt (* (* 2.0 z) (fma t t 1.0))) (* x 0.5))
         (* (* t (* t_1 (sqrt 2.0))) (sqrt z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((2.0 * z));
	double tmp;
	if (t <= 4.8) {
		tmp = t_1 * t_2;
	} else if (t <= 5e+151) {
		tmp = exp(((t * t) / 2.0)) * (y * -t_2);
	} else if (t <= 8e+257) {
		tmp = sqrt(((2.0 * z) * fma(t, t, 1.0))) * (x * 0.5);
	} else {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 4.8)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 5e+151)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * Float64(-t_2)));
	elseif (t <= 8e+257)
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * Float64(x * 0.5));
	else
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 4.8], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 5e+151], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * (-t$95$2)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e+257], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 4.79999999999999982

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 78.1%

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

   if 4.79999999999999982 < t < 5.0000000000000002e151

   1. Initial program 100.0%

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

      1. add-cbrt-cube 91.4%

         \[\leadsto \left(\left(\color{blue}{\sqrt[3]{\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot 0.5\right)}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. pow3 91.4%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{\left(x \cdot 0.5\right)}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. unpow-prod-down 91.4%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {0.5}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. metadata-eval 91.4%

         \[\leadsto \left(\left(\sqrt[3]{{x}^{3} \cdot \color{blue}{0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied egg-rr 91.4%

      \[\leadsto \left(\left(\color{blue}{\sqrt[3]{{x}^{3} \cdot 0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Taylor expanded in x around 0 71.4%

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

      1. neg-mul-1 71.4%

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

   7. Simplified 71.4%

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

   if 5.0000000000000002e151 < t < 8.00000000000000024e257

   1. Initial program 94.7%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 89.5%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right)} \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \]

   if 8.00000000000000024e257 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 84.2%

       \[\leadsto \color{blue}{\left(t \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot \sqrt{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+151}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{2 \cdot z}\right)\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 0.37:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+258}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* 2.0 z))))
   (if (<= t 0.37)
     (* t_1 (* t_2 (hypot 1.0 t)))
     (if (<= t 4.6e+258)
       (* (exp (/ (* t t) 2.0)) (* 0.5 (* x t_2)))
       (* (* t (* t_1 (sqrt 2.0))) (sqrt z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((2.0 * z));
	double tmp;
	if (t <= 0.37) {
		tmp = t_1 * (t_2 * hypot(1.0, t));
	} else if (t <= 4.6e+258) {
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * t_2));
	} else {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt((2.0 * z));
	double tmp;
	if (t <= 0.37) {
		tmp = t_1 * (t_2 * Math.hypot(1.0, t));
	} else if (t <= 4.6e+258) {
		tmp = Math.exp(((t * t) / 2.0)) * (0.5 * (x * t_2));
	} else {
		tmp = (t * (t_1 * Math.sqrt(2.0))) * Math.sqrt(z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((2.0 * z))
	tmp = 0
	if t <= 0.37:
		tmp = t_1 * (t_2 * math.hypot(1.0, t))
	elif t <= 4.6e+258:
		tmp = math.exp(((t * t) / 2.0)) * (0.5 * (x * t_2))
	else:
		tmp = (t * (t_1 * math.sqrt(2.0))) * math.sqrt(z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 0.37)
		tmp = Float64(t_1 * Float64(t_2 * hypot(1.0, t)));
	elseif (t <= 4.6e+258)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(0.5 * Float64(x * t_2)));
	else
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((2.0 * z));
	tmp = 0.0;
	if (t <= 0.37)
		tmp = t_1 * (t_2 * hypot(1.0, t));
	elseif (t <= 4.6e+258)
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * t_2));
	else
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.37], N[(t$95$1 * N[(t$95$2 * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e+258], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.37

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 92.9%

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

       1. +-commutative 92.9%

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

       2. unpow2 92.9%

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

       3. fma-define 92.9%

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

   11. Simplified 92.9%

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

       1. sqrt-prod 92.4%

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

   13. Applied egg-rr 92.4%

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

       1. *-commutative 92.4%

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

       2. fma-undefine 92.4%

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

       3. unpow2 92.4%

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

       4. +-commutative 92.4%

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

       5. unpow2 92.4%

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

       6. hypot-1-def 87.7%

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

       7. *-commutative 87.7%

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

   15. Simplified 87.7%

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

   if 0.37 < t < 4.6000000000000002e258

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0 98.2%

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

   4. Taylor expanded in x around inf 71.4%

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

      1. associate-*l* 71.4%

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

      2. *-commutative 71.4%

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

      3. unpow1/2 71.4%

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

      4. metadata-eval 71.4%

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

      5. pow-sqr 71.4%

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

      6. unpow1/2 71.4%

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

      7. metadata-eval 71.4%

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

      8. pow-sqr 71.4%

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

      9. unswap-sqr 71.4%

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

      10. exp-to-pow 71.4%

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

      11. exp-to-pow 71.4%

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

      12. exp-sum 71.4%

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

      13. distribute-rgt-in 71.4%

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

      14. +-commutative 71.4%

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

      15. *-commutative 71.4%

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

      16. exp-prod 71.4%

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

      17. exp-sum 71.4%

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

      18. rem-exp-log 71.4%

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

      19. rem-exp-log 71.4%

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

      20. exp-to-pow 71.4%

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

      21. exp-to-pow 71.4%

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

   6. Simplified 71.4%

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

   if 4.6000000000000002e258 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 84.2%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.37:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+258}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 89:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {t\_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 89.0)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= t 2.8e+39)
       (sqrt (* (* 2.0 z) (pow t_1 2.0)))
       (* (sqrt (* (* 2.0 z) (fma t t 1.0))) (* x 0.5))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 89.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if (t <= 2.8e+39) {
		tmp = sqrt(((2.0 * z) * pow(t_1, 2.0)));
	} else {
		tmp = sqrt(((2.0 * z) * fma(t, t, 1.0))) * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 89.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (t <= 2.8e+39)
		tmp = sqrt(Float64(Float64(2.0 * z) * (t_1 ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 89.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+39], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 89

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 78.1%

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

   if 89 < t < 2.80000000000000001e39

   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. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 4.3%

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

       1. add-cbrt-cube 24.9%

          \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot 0.5\right)}} - y\right) \cdot \sqrt{2 \cdot z} \]

       2. pow3 24.9%

          \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(x \cdot 0.5\right)}^{3}}} - y\right) \cdot \sqrt{2 \cdot z} \]

       3. unpow-prod-down 24.9%

          \[\leadsto \left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {0.5}^{3}}} - y\right) \cdot \sqrt{2 \cdot z} \]

       4. metadata-eval 24.9%

          \[\leadsto \left(\sqrt[3]{{x}^{3} \cdot \color{blue}{0.125}} - y\right) \cdot \sqrt{2 \cdot z} \]

       5. *-commutative 24.9%

          \[\leadsto \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]

       6. add-sqr-sqrt 13.6%

          \[\leadsto \color{blue}{\sqrt{\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}} \cdot \sqrt{\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}}} \]

       7. sqrt-unprod 24.2%

          \[\leadsto \color{blue}{\sqrt{\left(\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}\right)}} \]

       8. *-commutative 24.2%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right)} \cdot \left(\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}\right)} \]

       9. *-commutative 24.2%

          \[\leadsto \sqrt{\left(\sqrt{\color{blue}{2 \cdot z}} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right) \cdot \left(\left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right) \cdot \sqrt{z \cdot 2}\right)} \]

       10. *-commutative 24.2%

           \[\leadsto \sqrt{\left(\sqrt{2 \cdot z} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right)}} \]

       11. *-commutative 24.2%

           \[\leadsto \sqrt{\left(\sqrt{2 \cdot z} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right) \cdot \left(\sqrt{\color{blue}{2 \cdot z}} \cdot \left(\sqrt[3]{{x}^{3} \cdot 0.125} - y\right)\right)} \]

   11. Applied egg-rr 24.4%

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

       1. fma-neg 24.4%

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

       2. *-commutative 24.4%

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

   13. Simplified 24.4%

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

   if 2.80000000000000001e39 < t

   1. Initial program 98.4%

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

      1. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 71.5%

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

       1. +-commutative 71.5%

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

       2. unpow2 71.5%

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

       3. fma-define 71.5%

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

   11. Simplified 71.5%

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

   12. Taylor expanded in x around inf 55.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 89:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+54}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8e+54)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* y (- (sqrt (* (* 2.0 z) (fma t t 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8e+54) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = y * -sqrt(((2.0 * z) * fma(t, t, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8e+54)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(y * Float64(-sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8e+54], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * (-N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.0000000000000006e54

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 74.1%

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

   if 8.0000000000000006e54 < 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. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 73.2%

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

       1. +-commutative 73.2%

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

       2. unpow2 73.2%

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

       3. fma-define 73.2%

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

   11. Simplified 73.2%

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

   12. Taylor expanded in x around 0 54.1%

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

       1. mul-1-neg 54.1%

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

   14. Simplified 54.1%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+54}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 13: 59.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 4.5:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(y \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= t 4.5)
     (* (- (* x 0.5) y) t_1)
     (* t_1 (* y (+ (* 0.5 (/ x y)) -1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if (t <= 4.5) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if (t <= 4.5d0) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = t_1 * (y * ((0.5d0 * (x / y)) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if (t <= 4.5) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if t <= 4.5:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 4.5)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(t_1 * Float64(y * Float64(Float64(0.5 * Float64(x / y)) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if (t <= 4.5)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 4.5], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(y * N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 4.5

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 78.0%

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

   if 4.5 < t

   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. Add Preprocessing

   3. Taylor expanded in t around 0 15.8%

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

      1. add-cbrt-cube 90.4%

         \[\leadsto \left(\left(\color{blue}{\sqrt[3]{\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot 0.5\right)}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. pow3 90.4%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{\left(x \cdot 0.5\right)}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. unpow-prod-down 90.4%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {0.5}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. metadata-eval 90.4%

         \[\leadsto \left(\left(\sqrt[3]{{x}^{3} \cdot \color{blue}{0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Applied egg-rr 36.4%

      \[\leadsto \left(\left(\color{blue}{\sqrt[3]{{x}^{3} \cdot 0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

   6. Taylor expanded in y around inf 30.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(y \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+110} \lor \neg \left(y \leq 1.5 \cdot 10^{+110}\right):\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (or (<= y -1.25e+110) (not (<= y 1.5e+110)))
     (* y (- t_1))
     (* 0.5 (* x t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((y <= -1.25e+110) || !(y <= 1.5e+110)) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * (x * 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) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if ((y <= (-1.25d+110)) .or. (.not. (y <= 1.5d+110))) then
        tmp = y * -t_1
    else
        tmp = 0.5d0 * (x * 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((2.0 * z));
	double tmp;
	if ((y <= -1.25e+110) || !(y <= 1.5e+110)) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (y <= -1.25e+110) or not (y <= 1.5e+110):
		tmp = y * -t_1
	else:
		tmp = 0.5 * (x * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if ((y <= -1.25e+110) || !(y <= 1.5e+110))
		tmp = Float64(y * Float64(-t_1));
	else
		tmp = Float64(0.5 * Float64(x * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((y <= -1.25e+110) || ~((y <= 1.5e+110)))
		tmp = y * -t_1;
	else
		tmp = 0.5 * (x * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -1.25e+110], N[Not[LessEqual[y, 1.5e+110]], $MachinePrecision]], N[(y * (-t$95$1)), $MachinePrecision], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.24999999999999995e110 or 1.50000000000000004e110 < y

   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. associate-*l* 99.8%

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

      2. 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. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 73.6%

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

   10. Taylor expanded in x around 0 64.7%

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

       1. mul-1-neg 64.7%

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

       2. associate-*l* 64.7%

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

       3. *-commutative 64.7%

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

       4. unpow1/2 64.7%

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

       5. metadata-eval 64.7%

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

       6. pow-sqr 64.7%

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

       7. unpow1/2 64.7%

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

       8. metadata-eval 64.7%

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

       9. pow-sqr 64.7%

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

       10. unswap-sqr 64.7%

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

       11. exp-to-pow 63.2%

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

       12. exp-to-pow 63.2%

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

       13. exp-sum 63.1%

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

       14. distribute-rgt-in 63.1%

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

       15. +-commutative 63.1%

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

       16. *-commutative 63.1%

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

       17. exp-prod 63.1%

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

       18. exp-sum 63.2%

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

       19. rem-exp-log 63.2%

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

       20. rem-exp-log 64.8%

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

       21. exp-to-pow 63.2%

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

       22. exp-to-pow 63.2%

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

   12. Simplified 64.9%

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

   if -1.24999999999999995e110 < y < 1.50000000000000004e110

   1. Initial program 98.8%

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

      1. associate-*l* 99.8%

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

      2. 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. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 55.5%

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

   10. Taylor expanded in x around inf 44.6%

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

       1. associate-*l* 75.3%

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

       2. *-commutative 75.3%

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

       3. unpow1/2 75.3%

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

       4. metadata-eval 75.3%

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

       5. pow-sqr 75.4%

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

       6. unpow1/2 75.4%

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

       7. metadata-eval 75.4%

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

       8. pow-sqr 75.3%

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

       9. unswap-sqr 75.3%

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

       10. exp-to-pow 73.7%

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

       11. exp-to-pow 73.7%

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

       12. exp-sum 73.4%

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

       13. distribute-rgt-in 73.4%

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

       14. +-commutative 73.4%

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

       15. *-commutative 73.4%

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

       16. exp-prod 73.4%

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

       17. exp-sum 73.7%

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

       18. rem-exp-log 73.7%

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

       19. rem-exp-log 75.4%

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

       20. exp-to-pow 73.7%

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

       21. exp-to-pow 73.7%

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

   12. Simplified 44.8%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+110} \lor \neg \left(y \leq 1.5 \cdot 10^{+110}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 1100000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= t 1100000.0)
     (* (- (* x 0.5) y) t_1)
     (* t_1 (* x (- 0.5 (/ y x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if (t <= 1100000.0) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (x * (0.5 - (y / 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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if (t <= 1100000.0d0) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = t_1 * (x * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if (t <= 1100000.0) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if t <= 1100000.0:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = t_1 * (x * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 1100000.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(t_1 * Float64(x * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if (t <= 1100000.0)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = t_1 * (x * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1100000.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e6

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 77.3%

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

   if 1.1e6 < t

   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. Add Preprocessing

   3. Taylor expanded in t around 0 15.0%

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

      1. add-cbrt-cube 92.9%

         \[\leadsto \left(\left(\color{blue}{\sqrt[3]{\left(\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot 0.5\right)}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. pow3 92.9%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{\left(x \cdot 0.5\right)}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. unpow-prod-down 92.9%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {0.5}^{3}}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. metadata-eval 92.9%

         \[\leadsto \left(\left(\sqrt[3]{{x}^{3} \cdot \color{blue}{0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Applied egg-rr 36.5%

      \[\leadsto \left(\left(\color{blue}{\sqrt[3]{{x}^{3} \cdot 0.125}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

   6. Taylor expanded in x around inf 20.4%

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

      1. mul-1-neg 20.4%

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

      2. unsub-neg 20.4%

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

   8. Simplified 20.4%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1100000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{+57}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{{\left(2 \cdot z\right)}^{1.5}}{2 \cdot \left(-z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.45e+57)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* y (/ (pow (* 2.0 z) 1.5) (* 2.0 (- z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.45e+57) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = y * (pow((2.0 * z), 1.5) / (2.0 * -z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.45d+57) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = y * (((2.0d0 * z) ** 1.5d0) / (2.0d0 * -z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.45e+57) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = y * (Math.pow((2.0 * z), 1.5) / (2.0 * -z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.45e+57:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = y * (math.pow((2.0 * z), 1.5) / (2.0 * -z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.45e+57)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(y * Float64((Float64(2.0 * z) ^ 1.5) / Float64(2.0 * Float64(-z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.45e+57)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = y * (((2.0 * z) ^ 1.5) / (2.0 * -z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.45e+57], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Power[N[(2.0 * z), $MachinePrecision], 1.5], $MachinePrecision] / N[(2.0 * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4500000000000001e57

   1. Initial program 99.2%

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

      1. associate-*l* 99.7%

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

      2. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 73.4%

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

   if 1.4500000000000001e57 < 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. associate-*l* 100.0%

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

      2. 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 15.5%

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

   10. Taylor expanded in x around 0 11.0%

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

       1. mul-1-neg 11.0%

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

       2. associate-*l* 11.0%

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

       3. *-commutative 11.0%

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

       4. unpow1/2 11.0%

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

       5. metadata-eval 11.0%

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

       6. pow-sqr 11.0%

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

       7. unpow1/2 11.0%

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

       8. metadata-eval 11.0%

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

       9. pow-sqr 11.0%

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

       10. unswap-sqr 11.0%

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

       11. exp-to-pow 11.0%

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

       12. exp-to-pow 11.0%

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

       13. exp-sum 11.0%

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

       14. distribute-rgt-in 11.0%

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

       15. +-commutative 11.0%

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

       16. *-commutative 11.0%

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

       17. exp-prod 11.0%

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

       18. exp-sum 11.0%

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

       19. rem-exp-log 11.0%

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

       20. rem-exp-log 11.0%

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

       21. exp-to-pow 11.0%

           \[\leadsto -y \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(\color{blue}{e^{\log z \cdot 0.25}} \cdot {2}^{0.25}\right)\right) \]

       22. exp-to-pow 11.0%

           \[\leadsto -y \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(e^{\log z \cdot 0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25}}\right)\right) \]

   12. Simplified 11.0%

       \[\leadsto \color{blue}{y \cdot \left(-\sqrt{2 \cdot z}\right)} \]
   13. Step-by-step derivation

       1. neg-sub0 11.0%

          \[\leadsto y \cdot \color{blue}{\left(0 - \sqrt{2 \cdot z}\right)} \]

       2. *-commutative 11.0%

          \[\leadsto y \cdot \left(0 - \sqrt{\color{blue}{z \cdot 2}}\right) \]

       3. flip3-- 24.1%

          \[\leadsto y \cdot \color{blue}{\frac{{0}^{3} - {\left(\sqrt{z \cdot 2}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)}} \]

       4. metadata-eval 24.1%

          \[\leadsto y \cdot \frac{\color{blue}{0} - {\left(\sqrt{z \cdot 2}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       5. pow3 24.1%

          \[\leadsto y \cdot \frac{0 - \color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{z \cdot 2}}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       6. add-sqr-sqrt 24.1%

          \[\leadsto y \cdot \frac{0 - \color{blue}{\left(z \cdot 2\right)} \cdot \sqrt{z \cdot 2}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       7. *-commutative 24.1%

          \[\leadsto y \cdot \frac{0 - \color{blue}{\left(2 \cdot z\right)} \cdot \sqrt{z \cdot 2}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       8. pow1 24.1%

          \[\leadsto y \cdot \frac{0 - \color{blue}{{\left(2 \cdot z\right)}^{1}} \cdot \sqrt{z \cdot 2}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       9. pow1/2 24.1%

          \[\leadsto y \cdot \frac{0 - {\left(2 \cdot z\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       10. *-commutative 24.1%

           \[\leadsto y \cdot \frac{0 - {\left(2 \cdot z\right)}^{1} \cdot {\color{blue}{\left(2 \cdot z\right)}}^{0.5}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       11. pow-prod-up 24.1%

           \[\leadsto y \cdot \frac{0 - \color{blue}{{\left(2 \cdot z\right)}^{\left(1 + 0.5\right)}}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       12. metadata-eval 24.1%

           \[\leadsto y \cdot \frac{0 - {\left(2 \cdot z\right)}^{\color{blue}{1.5}}}{0 \cdot 0 + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       13. metadata-eval 24.1%

           \[\leadsto y \cdot \frac{0 - {\left(2 \cdot z\right)}^{1.5}}{\color{blue}{0} + \left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       14. add-sqr-sqrt 24.1%

           \[\leadsto y \cdot \frac{0 - {\left(2 \cdot z\right)}^{1.5}}{0 + \left(\color{blue}{z \cdot 2} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       15. *-commutative 24.1%

           \[\leadsto y \cdot \frac{0 - {\left(2 \cdot z\right)}^{1.5}}{0 + \left(\color{blue}{2 \cdot z} + 0 \cdot \sqrt{z \cdot 2}\right)} \]

       16. *-commutative 24.1%

           \[\leadsto y \cdot \frac{0 - {\left(2 \cdot z\right)}^{1.5}}{0 + \left(2 \cdot z + 0 \cdot \sqrt{\color{blue}{2 \cdot z}}\right)} \]

   14. Applied egg-rr 24.1%

       \[\leadsto y \cdot \color{blue}{\frac{0 - {\left(2 \cdot z\right)}^{1.5}}{0 + \left(2 \cdot z + 0 \cdot \sqrt{2 \cdot z}\right)}} \]
   15. Step-by-step derivation

       1. sub0-neg 24.1%

          \[\leadsto y \cdot \frac{\color{blue}{-{\left(2 \cdot z\right)}^{1.5}}}{0 + \left(2 \cdot z + 0 \cdot \sqrt{2 \cdot z}\right)} \]

       2. *-commutative 24.1%

          \[\leadsto y \cdot \frac{-{\color{blue}{\left(z \cdot 2\right)}}^{1.5}}{0 + \left(2 \cdot z + 0 \cdot \sqrt{2 \cdot z}\right)} \]

       3. +-lft-identity 24.1%

          \[\leadsto y \cdot \frac{-{\left(z \cdot 2\right)}^{1.5}}{\color{blue}{2 \cdot z + 0 \cdot \sqrt{2 \cdot z}}} \]

       4. mul0-lft 24.1%

          \[\leadsto y \cdot \frac{-{\left(z \cdot 2\right)}^{1.5}}{2 \cdot z + \color{blue}{0}} \]

       5. +-rgt-identity 24.1%

          \[\leadsto y \cdot \frac{-{\left(z \cdot 2\right)}^{1.5}}{\color{blue}{2 \cdot z}} \]

       6. *-commutative 24.1%

          \[\leadsto y \cdot \frac{-{\left(z \cdot 2\right)}^{1.5}}{\color{blue}{z \cdot 2}} \]

   16. Simplified 24.1%

       \[\leadsto y \cdot \color{blue}{\frac{-{\left(z \cdot 2\right)}^{1.5}}{z \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{+57}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{{\left(2 \cdot z\right)}^{1.5}}{2 \cdot \left(-z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (- (* x 0.5) y) (sqrt (* 2.0 z))))

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * sqrt((2.0 * z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * Math.sqrt((2.0 * z));
}

Python

def code(x, y, z, t):
	return ((x * 0.5) - y) * math.sqrt((2.0 * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   2. 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. exp-prod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right)}^{1}} \]

   2. sqrt-unprod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left(\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}\right)}}^{1} \]

   3. associate-*l* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}}\right)}^{1} \]

   4. pow-exp 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)}\right)}^{1} \]

   5. pow2 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)}\right)}^{1} \]

6. Applied egg-rr 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}} \]

   2. associate-*r* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}} \]

   3. *-commutative 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot e^{{t}^{2}}} \]

8. Simplified 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]

9. Taylor expanded in t around 0 60.3%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]
10. Add Preprocessing

Alternative 18: 30.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot z} \cdot \left(-y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (sqrt (* 2.0 z)) (- y)))

C

double code(double x, double y, double z, double t) {
	return sqrt((2.0 * z)) * -y;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((2.0d0 * z)) * -y
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.sqrt((2.0 * z)) * -y;
}

Python

def code(x, y, z, t):
	return math.sqrt((2.0 * z)) * -y

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(2.0 * z)) * Float64(-y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = sqrt((2.0 * z)) * -y;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   2. 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. exp-prod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right)}^{1}} \]

   2. sqrt-unprod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left(\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}\right)}}^{1} \]

   3. associate-*l* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}}\right)}^{1} \]

   4. pow-exp 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)}\right)}^{1} \]

   5. pow2 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)}\right)}^{1} \]

6. Applied egg-rr 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}} \]

   2. associate-*r* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}} \]

   3. *-commutative 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot e^{{t}^{2}}} \]

8. Simplified 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]

9. Taylor expanded in t around 0 60.1%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \]

10. Taylor expanded in x around 0 25.4%

    \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
11. Step-by-step derivation

    1. mul-1-neg 25.4%

       \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

    2. associate-*l* 25.4%

       \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

    3. *-commutative 25.4%

       \[\leadsto -y \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \]

    4. unpow1/2 25.4%

       \[\leadsto -y \cdot \left(\sqrt{z} \cdot \color{blue}{{2}^{0.5}}\right) \]

    5. metadata-eval 25.4%

       \[\leadsto -y \cdot \left(\sqrt{z} \cdot {2}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \]

    6. pow-sqr 25.4%

       \[\leadsto -y \cdot \left(\sqrt{z} \cdot \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)}\right) \]

    7. unpow1/2 25.4%

       \[\leadsto -y \cdot \left(\color{blue}{{z}^{0.5}} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    8. metadata-eval 25.4%

       \[\leadsto -y \cdot \left({z}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    9. pow-sqr 25.4%

       \[\leadsto -y \cdot \left(\color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    10. unswap-sqr 25.4%

        \[\leadsto -y \cdot \color{blue}{\left(\left({z}^{0.25} \cdot {2}^{0.25}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)} \]

    11. exp-to-pow 24.6%

        \[\leadsto -y \cdot \left(\left(\color{blue}{e^{\log z \cdot 0.25}} \cdot {2}^{0.25}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    12. exp-to-pow 24.6%

        \[\leadsto -y \cdot \left(\left(e^{\log z \cdot 0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25}}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    13. exp-sum 24.6%

        \[\leadsto -y \cdot \left(\color{blue}{e^{\log z \cdot 0.25 + \log 2 \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    14. distribute-rgt-in 24.6%

        \[\leadsto -y \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log z + \log 2\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    15. +-commutative 24.6%

        \[\leadsto -y \cdot \left(e^{0.25 \cdot \color{blue}{\left(\log 2 + \log z\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    16. *-commutative 24.6%

        \[\leadsto -y \cdot \left(e^{\color{blue}{\left(\log 2 + \log z\right) \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    17. exp-prod 24.6%

        \[\leadsto -y \cdot \left(\color{blue}{{\left(e^{\log 2 + \log z}\right)}^{0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    18. exp-sum 24.6%

        \[\leadsto -y \cdot \left({\color{blue}{\left(e^{\log 2} \cdot e^{\log z}\right)}}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    19. rem-exp-log 24.6%

        \[\leadsto -y \cdot \left({\left(\color{blue}{2} \cdot e^{\log z}\right)}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    20. rem-exp-log 25.4%

        \[\leadsto -y \cdot \left({\left(2 \cdot \color{blue}{z}\right)}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    21. exp-to-pow 24.6%

        \[\leadsto -y \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(\color{blue}{e^{\log z \cdot 0.25}} \cdot {2}^{0.25}\right)\right) \]

    22. exp-to-pow 24.6%

        \[\leadsto -y \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(e^{\log z \cdot 0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25}}\right)\right) \]

12. Simplified 25.5%

    \[\leadsto \color{blue}{y \cdot \left(-\sqrt{2 \cdot z}\right)} \]

13. Final simplification 25.5%

    \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
14. Add Preprocessing

Alternative 19: 2.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \sqrt{2 \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y (sqrt (* 2.0 z))))

C

double code(double x, double y, double z, double t) {
	return y * sqrt((2.0 * z));
}

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((2.0d0 * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * Math.sqrt((2.0 * z));
}

Python

def code(x, y, z, t):
	return y * math.sqrt((2.0 * z))

Julia

function code(x, y, z, t)
	return Float64(y * sqrt(Float64(2.0 * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y * sqrt((2.0 * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   2. 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. exp-prod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\sqrt{z \cdot 2} \cdot \sqrt{{\left(e^{t}\right)}^{t}}\right)}^{1}} \]

   2. sqrt-unprod 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left(\sqrt{\left(z \cdot 2\right) \cdot {\left(e^{t}\right)}^{t}}\right)}}^{1} \]

   3. associate-*l* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}}\right)}^{1} \]

   4. pow-exp 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{z \cdot \left(2 \cdot \color{blue}{e^{t \cdot t}}\right)}\right)}^{1} \]

   5. pow2 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)}\right)}^{1} \]

6. Applied egg-rr 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}} \]

   2. associate-*r* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}} \]

   3. *-commutative 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot e^{{t}^{2}}} \]

8. Simplified 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]

9. Taylor expanded in t around 0 60.1%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \]

10. Taylor expanded in x around 0 25.4%

    \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
11. Step-by-step derivation

    1. mul-1-neg 25.4%

       \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

    2. associate-*l* 25.4%

       \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

    3. *-commutative 25.4%

       \[\leadsto -y \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \]

    4. unpow1/2 25.4%

       \[\leadsto -y \cdot \left(\sqrt{z} \cdot \color{blue}{{2}^{0.5}}\right) \]

    5. metadata-eval 25.4%

       \[\leadsto -y \cdot \left(\sqrt{z} \cdot {2}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \]

    6. pow-sqr 25.4%

       \[\leadsto -y \cdot \left(\sqrt{z} \cdot \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)}\right) \]

    7. unpow1/2 25.4%

       \[\leadsto -y \cdot \left(\color{blue}{{z}^{0.5}} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    8. metadata-eval 25.4%

       \[\leadsto -y \cdot \left({z}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    9. pow-sqr 25.4%

       \[\leadsto -y \cdot \left(\color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    10. unswap-sqr 25.4%

        \[\leadsto -y \cdot \color{blue}{\left(\left({z}^{0.25} \cdot {2}^{0.25}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right)} \]

    11. exp-to-pow 24.6%

        \[\leadsto -y \cdot \left(\left(\color{blue}{e^{\log z \cdot 0.25}} \cdot {2}^{0.25}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    12. exp-to-pow 24.6%

        \[\leadsto -y \cdot \left(\left(e^{\log z \cdot 0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25}}\right) \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    13. exp-sum 24.6%

        \[\leadsto -y \cdot \left(\color{blue}{e^{\log z \cdot 0.25 + \log 2 \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    14. distribute-rgt-in 24.6%

        \[\leadsto -y \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log z + \log 2\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    15. +-commutative 24.6%

        \[\leadsto -y \cdot \left(e^{0.25 \cdot \color{blue}{\left(\log 2 + \log z\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    16. *-commutative 24.6%

        \[\leadsto -y \cdot \left(e^{\color{blue}{\left(\log 2 + \log z\right) \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    17. exp-prod 24.6%

        \[\leadsto -y \cdot \left(\color{blue}{{\left(e^{\log 2 + \log z}\right)}^{0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    18. exp-sum 24.6%

        \[\leadsto -y \cdot \left({\color{blue}{\left(e^{\log 2} \cdot e^{\log z}\right)}}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    19. rem-exp-log 24.6%

        \[\leadsto -y \cdot \left({\left(\color{blue}{2} \cdot e^{\log z}\right)}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    20. rem-exp-log 25.4%

        \[\leadsto -y \cdot \left({\left(2 \cdot \color{blue}{z}\right)}^{0.25} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

    21. exp-to-pow 24.6%

        \[\leadsto -y \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(\color{blue}{e^{\log z \cdot 0.25}} \cdot {2}^{0.25}\right)\right) \]

    22. exp-to-pow 24.6%

        \[\leadsto -y \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(e^{\log z \cdot 0.25} \cdot \color{blue}{e^{\log 2 \cdot 0.25}}\right)\right) \]

12. Simplified 25.5%

    \[\leadsto \color{blue}{y \cdot \left(-\sqrt{2 \cdot z}\right)} \]
13. Step-by-step derivation

    1. *-commutative 25.5%

       \[\leadsto y \cdot \left(-\sqrt{\color{blue}{z \cdot 2}}\right) \]

    2. distribute-rgt-neg-in 25.5%

       \[\leadsto \color{blue}{-y \cdot \sqrt{z \cdot 2}} \]

    3. distribute-lft-neg-out 25.5%

       \[\leadsto \color{blue}{\left(-y\right) \cdot \sqrt{z \cdot 2}} \]

    4. add-sqr-sqrt 11.4%

       \[\leadsto \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)} \cdot \sqrt{z \cdot 2} \]

    5. sqrt-unprod 11.6%

       \[\leadsto \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}} \cdot \sqrt{z \cdot 2} \]

    6. sqr-neg 11.6%

       \[\leadsto \sqrt{\color{blue}{y \cdot y}} \cdot \sqrt{z \cdot 2} \]

    7. sqrt-unprod 0.9%

       \[\leadsto \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{z \cdot 2} \]

    8. add-sqr-sqrt 1.9%

       \[\leadsto \color{blue}{y} \cdot \sqrt{z \cdot 2} \]

    9. pow1 1.9%

       \[\leadsto \color{blue}{{\left(y \cdot \sqrt{z \cdot 2}\right)}^{1}} \]

    10. *-commutative 1.9%

        \[\leadsto {\left(y \cdot \sqrt{\color{blue}{2 \cdot z}}\right)}^{1} \]

14. Applied egg-rr 1.9%

    \[\leadsto \color{blue}{{\left(y \cdot \sqrt{2 \cdot z}\right)}^{1}} \]
15. Step-by-step derivation

    1. unpow1 1.9%

       \[\leadsto \color{blue}{y \cdot \sqrt{2 \cdot z}} \]

    2. *-commutative 1.9%

       \[\leadsto y \cdot \sqrt{\color{blue}{z \cdot 2}} \]

16. Simplified 1.9%

    \[\leadsto \color{blue}{y \cdot \sqrt{z \cdot 2}} \]

17. Final simplification 1.9%

    \[\leadsto y \cdot \sqrt{2 \cdot z} \]
18. Add Preprocessing

Developer Target 1: 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 2024116 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))