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

Percentage Accurate: 99.5% → 99.8%

Time: 17.1s

Alternatives: 15

Speedup: 1.0×

• 45.1% 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: 99.5% 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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. sqr-neg 98.7%

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

   2. associate-/l* 98.7%

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

   3. distribute-frac-neg 98.7%

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

   4. exp-neg 98.7%

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

   5. associate-*r/ 98.7%

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

   6. *-rgt-identity 98.7%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. associate-*r/ 99.8%

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

   10. exp-neg 99.8%

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

   11. distribute-frac-neg 99.8%

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

   12. associate-/l* 99.8%

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

   13. sqr-neg 99.8%

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

   14. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 78.4%

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

   3. sqrt-unprod 78.4%

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

   4. associate-*l* 78.4%

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

   5. pow2 78.4%

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

5. Applied egg-rr 78.4%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 64.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z} \cdot \left(t_1 \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{if}\;t \leq 175000:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 3.42 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.04 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{+144} \lor \neg \left(t \leq 9.6 \cdot 10^{+169}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (* (sqrt z) (* t_1 (* t (sqrt 2.0))))))
   (if (<= t 175000.0)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= t 3.42e+40)
       (sqrt (* 2.0 (* z (pow (* x 0.5) 2.0))))
       (if (<= t 2.15e+104)
         t_2
         (if (<= t 1.04e+116)
           (sqrt (* (* 2.0 z) (pow y 2.0)))
           (if (or (<= t 9.4e+144) (not (<= t 9.6e+169)))
             t_2
             (* t_1 (cbrt (pow (* 2.0 z) 1.5))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	double tmp;
	if (t <= 175000.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if (t <= 3.42e+40) {
		tmp = sqrt((2.0 * (z * pow((x * 0.5), 2.0))));
	} else if (t <= 2.15e+104) {
		tmp = t_2;
	} else if (t <= 1.04e+116) {
		tmp = sqrt(((2.0 * z) * pow(y, 2.0)));
	} else if ((t <= 9.4e+144) || !(t <= 9.6e+169)) {
		tmp = t_2;
	} else {
		tmp = t_1 * cbrt(pow((2.0 * z), 1.5));
	}
	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(z) * (t_1 * (t * Math.sqrt(2.0)));
	double tmp;
	if (t <= 175000.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else if (t <= 3.42e+40) {
		tmp = Math.sqrt((2.0 * (z * Math.pow((x * 0.5), 2.0))));
	} else if (t <= 2.15e+104) {
		tmp = t_2;
	} else if (t <= 1.04e+116) {
		tmp = Math.sqrt(((2.0 * z) * Math.pow(y, 2.0)));
	} else if ((t <= 9.4e+144) || !(t <= 9.6e+169)) {
		tmp = t_2;
	} else {
		tmp = t_1 * Math.cbrt(Math.pow((2.0 * z), 1.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = Float64(sqrt(z) * Float64(t_1 * Float64(t * sqrt(2.0))))
	tmp = 0.0
	if (t <= 175000.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (t <= 3.42e+40)
		tmp = sqrt(Float64(2.0 * Float64(z * (Float64(x * 0.5) ^ 2.0))));
	elseif (t <= 2.15e+104)
		tmp = t_2;
	elseif (t <= 1.04e+116)
		tmp = sqrt(Float64(Float64(2.0 * z) * (y ^ 2.0)));
	elseif ((t <= 9.4e+144) || !(t <= 9.6e+169))
		tmp = t_2;
	else
		tmp = Float64(t_1 * cbrt((Float64(2.0 * z) ^ 1.5)));
	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[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 175000.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.42e+40], N[Sqrt[N[(2.0 * N[(z * N[Power[N[(x * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 2.15e+104], t$95$2, If[LessEqual[t, 1.04e+116], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[t, 9.4e+144], N[Not[LessEqual[t, 9.6e+169]], $MachinePrecision]], t$95$2, N[(t$95$1 * N[Power[N[Power[N[(2.0 * z), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < 175000

   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. sqr-neg 98.8%

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

      2. associate-/l* 98.8%

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

      3. distribute-frac-neg 98.8%

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

      4. exp-neg 98.8%

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

      5. associate-*r/ 98.8%

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

      6. *-rgt-identity 98.8%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 72.2%

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

      3. sqrt-unprod 72.2%

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

      4. associate-*l* 72.2%

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

      5. pow2 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 64.4%

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

   if 175000 < t < 3.42000000000000018e40

   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. Taylor expanded in t around 0 14.1%

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

      1. associate-*r* 14.1%

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

      2. *-commutative 14.1%

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

      3. *-commutative 14.1%

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

      4. fma-neg 14.1%

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

      5. associate-*l* 14.1%

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

      6. fma-neg 14.1%

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

      7. *-commutative 14.1%

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

   4. Simplified 14.1%

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

   5. Taylor expanded in x around inf 3.5%

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

      1. add-sqr-sqrt 2.5%

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

      2. sqrt-unprod 30.7%

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

      3. swap-sqr 30.7%

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

      4. rem-square-sqrt 30.7%

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

      5. associate-*r* 30.7%

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

      6. *-commutative 30.7%

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

      7. associate-*r* 30.7%

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

      8. *-commutative 30.7%

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

      9. swap-sqr 30.7%

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

      10. pow2 30.7%

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

      11. add-sqr-sqrt 30.7%

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

   7. Applied egg-rr 30.7%

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

   if 3.42000000000000018e40 < t < 2.1500000000000001e104 or 1.04e116 < t < 9.4000000000000004e144 or 9.5999999999999994e169 < t

   1. Initial program 97.4%

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

      1. sqr-neg 97.4%

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

      2. associate-/l* 97.4%

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

      3. distribute-frac-neg 97.4%

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

      4. exp-neg 97.4%

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

      5. associate-*r/ 97.4%

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

      6. *-rgt-identity 97.4%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 82.5%

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

      1. distribute-lft-out 82.5%

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

      2. *-commutative 82.5%

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

   10. Simplified 82.5%

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

   11. Taylor expanded in z around 0 82.5%

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

       1. +-commutative 82.5%

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

       2. unpow2 82.5%

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

       3. fma-def 82.5%

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

   13. Simplified 82.5%

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

   14. Taylor expanded in t around inf 57.6%

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

       1. *-commutative 57.6%

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

       2. associate-*r* 57.6%

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

       3. *-commutative 57.6%

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

   16. Simplified 57.6%

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

   if 2.1500000000000001e104 < t < 1.04e116

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 4.7%

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

   9. Taylor expanded in x around 0 4.4%

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

       1. mul-1-neg 4.4%

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

       2. *-commutative 4.4%

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

       3. distribute-rgt-neg-in 4.4%

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

       4. *-commutative 4.4%

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

   11. Simplified 4.4%

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

       1. add-sqr-sqrt 4.4%

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

       2. sqrt-unprod 4.4%

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

       3. swap-sqr 100.0%

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

       4. add-sqr-sqrt 100.0%

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

       5. sqr-neg 100.0%

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

       6. swap-sqr 100.0%

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

       7. rem-square-sqrt 100.0%

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

       8. pow2 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. associate-*r* 100.0%

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

   15. Simplified 100.0%

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

   if 9.4000000000000004e144 < t < 9.5999999999999994e169

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 16.6%

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

      1. sqrt-prod 16.6%

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

      2. add-cbrt-cube 76.4%

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

      3. add-sqr-sqrt 76.4%

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

      4. pow1/3 76.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\left(z \cdot 2\right) \cdot \sqrt{z \cdot 2}\right)}^{0.3333333333333333}} \]

      5. pow1 76.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\color{blue}{{\left(z \cdot 2\right)}^{1}} \cdot \sqrt{z \cdot 2}\right)}^{0.3333333333333333} \]

      6. pow1/2 76.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\left(z \cdot 2\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 76.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left({\left(z \cdot 2\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      8. *-commutative 76.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\color{blue}{\left(2 \cdot z\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 76.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\left(2 \cdot z\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 76.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left({\left(2 \cdot z\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   7. Step-by-step derivation

      1. unpow1/3 76.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}} \]

   8. Simplified 76.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 175000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 3.42 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t \leq 1.04 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{+144} \lor \neg \left(t \leq 9.6 \cdot 10^{+169}\right):\\ \;\;\;\;\sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \end{array} \]

Alternative 3: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 240000:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 3.42 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{z} \cdot \left(t_1 \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 240000.0)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= t 3.42e+40)
       (sqrt (* 2.0 (* z (pow (* x 0.5) 2.0))))
       (if (<= t 6.5e+105)
         (* (sqrt z) (* t_1 (* t (sqrt 2.0))))
         (* t_1 (sqrt (* 2.0 (* z (pow t 2.0))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 240000.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if (t <= 3.42e+40) {
		tmp = sqrt((2.0 * (z * pow((x * 0.5), 2.0))));
	} else if (t <= 6.5e+105) {
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	} else {
		tmp = t_1 * sqrt((2.0 * (z * pow(t, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 240000.0d0) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else if (t <= 3.42d+40) then
        tmp = sqrt((2.0d0 * (z * ((x * 0.5d0) ** 2.0d0))))
    else if (t <= 6.5d+105) then
        tmp = sqrt(z) * (t_1 * (t * sqrt(2.0d0)))
    else
        tmp = t_1 * sqrt((2.0d0 * (z * (t ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 240000.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else if (t <= 3.42e+40) {
		tmp = Math.sqrt((2.0 * (z * Math.pow((x * 0.5), 2.0))));
	} else if (t <= 6.5e+105) {
		tmp = Math.sqrt(z) * (t_1 * (t * Math.sqrt(2.0)));
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z * Math.pow(t, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 240000.0:
		tmp = t_1 * math.sqrt((2.0 * z))
	elif t <= 3.42e+40:
		tmp = math.sqrt((2.0 * (z * math.pow((x * 0.5), 2.0))))
	elif t <= 6.5e+105:
		tmp = math.sqrt(z) * (t_1 * (t * math.sqrt(2.0)))
	else:
		tmp = t_1 * math.sqrt((2.0 * (z * math.pow(t, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 240000.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (t <= 3.42e+40)
		tmp = sqrt(Float64(2.0 * Float64(z * (Float64(x * 0.5) ^ 2.0))));
	elseif (t <= 6.5e+105)
		tmp = Float64(sqrt(z) * Float64(t_1 * Float64(t * sqrt(2.0))));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * (t ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 240000.0)
		tmp = t_1 * sqrt((2.0 * z));
	elseif (t <= 3.42e+40)
		tmp = sqrt((2.0 * (z * ((x * 0.5) ^ 2.0))));
	elseif (t <= 6.5e+105)
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	else
		tmp = t_1 * sqrt((2.0 * (z * (t ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 240000.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.42e+40], N[Sqrt[N[(2.0 * N[(z * N[Power[N[(x * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 6.5e+105], N[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.4e5

   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. sqr-neg 98.8%

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

      2. associate-/l* 98.8%

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

      3. distribute-frac-neg 98.8%

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

      4. exp-neg 98.8%

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

      5. associate-*r/ 98.8%

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

      6. *-rgt-identity 98.8%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 72.2%

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

      3. sqrt-unprod 72.2%

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

      4. associate-*l* 72.2%

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

      5. pow2 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 64.4%

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

   if 2.4e5 < t < 3.42000000000000018e40

   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. Taylor expanded in t around 0 14.1%

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

      1. associate-*r* 14.1%

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

      2. *-commutative 14.1%

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

      3. *-commutative 14.1%

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

      4. fma-neg 14.1%

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

      5. associate-*l* 14.1%

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

      6. fma-neg 14.1%

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

      7. *-commutative 14.1%

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

   4. Simplified 14.1%

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

   5. Taylor expanded in x around inf 3.5%

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

      1. add-sqr-sqrt 2.5%

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

      2. sqrt-unprod 30.7%

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

      3. swap-sqr 30.7%

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

      4. rem-square-sqrt 30.7%

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

      5. associate-*r* 30.7%

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

      6. *-commutative 30.7%

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

      7. associate-*r* 30.7%

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

      8. *-commutative 30.7%

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

      9. swap-sqr 30.7%

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

      10. pow2 30.7%

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

      11. add-sqr-sqrt 30.7%

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

   7. Applied egg-rr 30.7%

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

   if 3.42000000000000018e40 < t < 6.50000000000000049e105

   1. Initial program 83.3%

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

      1. sqr-neg 83.3%

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

      2. associate-/l* 83.3%

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

      3. distribute-frac-neg 83.3%

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

      4. exp-neg 83.3%

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

      5. associate-*r/ 83.3%

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

      6. *-rgt-identity 83.3%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 20.8%

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

      1. distribute-lft-out 20.8%

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

      2. *-commutative 20.8%

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

   10. Simplified 20.8%

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

   11. Taylor expanded in z around 0 20.8%

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

       1. +-commutative 20.8%

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

       2. unpow2 20.8%

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

       3. fma-def 20.8%

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

   13. Simplified 20.8%

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

   14. Taylor expanded in t around inf 20.8%

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

       1. *-commutative 20.8%

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

       2. associate-*r* 20.8%

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

       3. *-commutative 20.8%

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

   16. Simplified 20.8%

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

   if 6.50000000000000049e105 < t

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 93.1%

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

      1. distribute-lft-out 93.1%

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

      2. *-commutative 93.1%

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

   10. Simplified 93.1%

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

   11. Taylor expanded in z around 0 93.1%

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

       1. +-commutative 93.1%

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

       2. unpow2 93.1%

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

       3. fma-def 93.1%

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

   13. Simplified 93.1%

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

   14. Taylor expanded in t around inf 93.1%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 240000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 3.42 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]

Alternative 4: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 240000:\\ \;\;\;\;t_1 \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 3.42 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{z} \cdot \left(t_1 \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 240000.0)
     (* t_1 (* (sqrt (* 2.0 z)) (hypot 1.0 t)))
     (if (<= t 3.42e+40)
       (sqrt (* 2.0 (* z (pow (* x 0.5) 2.0))))
       (if (<= t 1.1e+116)
         (* (sqrt z) (* t_1 (* t (sqrt 2.0))))
         (* t_1 (sqrt (* 2.0 (* z (pow t 2.0))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 240000.0) {
		tmp = t_1 * (sqrt((2.0 * z)) * hypot(1.0, t));
	} else if (t <= 3.42e+40) {
		tmp = sqrt((2.0 * (z * pow((x * 0.5), 2.0))));
	} else if (t <= 1.1e+116) {
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	} else {
		tmp = t_1 * sqrt((2.0 * (z * pow(t, 2.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 240000.0) {
		tmp = t_1 * (Math.sqrt((2.0 * z)) * Math.hypot(1.0, t));
	} else if (t <= 3.42e+40) {
		tmp = Math.sqrt((2.0 * (z * Math.pow((x * 0.5), 2.0))));
	} else if (t <= 1.1e+116) {
		tmp = Math.sqrt(z) * (t_1 * (t * Math.sqrt(2.0)));
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z * Math.pow(t, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 240000.0:
		tmp = t_1 * (math.sqrt((2.0 * z)) * math.hypot(1.0, t))
	elif t <= 3.42e+40:
		tmp = math.sqrt((2.0 * (z * math.pow((x * 0.5), 2.0))))
	elif t <= 1.1e+116:
		tmp = math.sqrt(z) * (t_1 * (t * math.sqrt(2.0)))
	else:
		tmp = t_1 * math.sqrt((2.0 * (z * math.pow(t, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 240000.0)
		tmp = Float64(t_1 * Float64(sqrt(Float64(2.0 * z)) * hypot(1.0, t)));
	elseif (t <= 3.42e+40)
		tmp = sqrt(Float64(2.0 * Float64(z * (Float64(x * 0.5) ^ 2.0))));
	elseif (t <= 1.1e+116)
		tmp = Float64(sqrt(z) * Float64(t_1 * Float64(t * sqrt(2.0))));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * (t ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 240000.0)
		tmp = t_1 * (sqrt((2.0 * z)) * hypot(1.0, t));
	elseif (t <= 3.42e+40)
		tmp = sqrt((2.0 * (z * ((x * 0.5) ^ 2.0))));
	elseif (t <= 1.1e+116)
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	else
		tmp = t_1 * sqrt((2.0 * (z * (t ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 240000.0], N[(t$95$1 * N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.42e+40], N[Sqrt[N[(2.0 * N[(z * N[Power[N[(x * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.1e+116], N[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.4e5

   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. sqr-neg 98.8%

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

      2. associate-/l* 98.8%

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

      3. distribute-frac-neg 98.8%

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

      4. exp-neg 98.8%

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

      5. associate-*r/ 98.8%

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

      6. *-rgt-identity 98.8%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 72.2%

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

      3. sqrt-unprod 72.2%

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

      4. associate-*l* 72.2%

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

      5. pow2 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 86.8%

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

      1. distribute-lft-out 86.8%

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

      2. *-commutative 86.8%

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

   10. Simplified 86.8%

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

   11. Taylor expanded in z around 0 86.8%

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

       1. +-commutative 86.8%

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

       2. unpow2 86.8%

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

       3. fma-def 86.8%

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

   13. Simplified 86.8%

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

       1. associate-*r* 86.8%

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

       2. *-commutative 86.8%

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

       3. sqrt-prod 85.8%

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

       4. *-commutative 85.8%

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

   15. Applied egg-rr 85.8%

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

       1. *-commutative 85.8%

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

       2. fma-udef 85.8%

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

       3. unpow2 85.8%

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

       4. +-commutative 85.8%

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

       5. unpow2 85.8%

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

       6. hypot-1-def 77.8%

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

   17. Simplified 77.8%

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

   if 2.4e5 < t < 3.42000000000000018e40

   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. Taylor expanded in t around 0 14.1%

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

      1. associate-*r* 14.1%

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

      2. *-commutative 14.1%

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

      3. *-commutative 14.1%

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

      4. fma-neg 14.1%

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

      5. associate-*l* 14.1%

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

      6. fma-neg 14.1%

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

      7. *-commutative 14.1%

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

   4. Simplified 14.1%

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

   5. Taylor expanded in x around inf 3.5%

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

      1. add-sqr-sqrt 2.5%

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

      2. sqrt-unprod 30.7%

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

      3. swap-sqr 30.7%

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

      4. rem-square-sqrt 30.7%

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

      5. associate-*r* 30.7%

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

      6. *-commutative 30.7%

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

      7. associate-*r* 30.7%

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

      8. *-commutative 30.7%

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

      9. swap-sqr 30.7%

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

      10. pow2 30.7%

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

      11. add-sqr-sqrt 30.7%

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

   7. Applied egg-rr 30.7%

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

   if 3.42000000000000018e40 < t < 1.1e116

   1. Initial program 85.7%

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

      1. sqr-neg 85.7%

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

      2. associate-/l* 85.7%

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

      3. distribute-frac-neg 85.7%

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

      4. exp-neg 85.7%

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

      5. associate-*r/ 85.7%

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

      6. *-rgt-identity 85.7%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 18.9%

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

      1. distribute-lft-out 18.9%

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

      2. *-commutative 18.9%

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

   10. Simplified 18.9%

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

   11. Taylor expanded in z around 0 18.9%

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

       1. +-commutative 18.9%

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

       2. unpow2 18.9%

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

       3. fma-def 18.9%

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

   13. Simplified 18.9%

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

   14. Taylor expanded in t around inf 18.9%

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

       1. *-commutative 18.9%

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

       2. associate-*r* 18.9%

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

       3. *-commutative 18.9%

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

   16. Simplified 18.9%

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

   if 1.1e116 < t

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 95.3%

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

      1. distribute-lft-out 95.3%

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

      2. *-commutative 95.3%

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

   10. Simplified 95.3%

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

   11. Taylor expanded in z around 0 95.3%

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

       1. +-commutative 95.3%

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

       2. unpow2 95.3%

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

       3. fma-def 95.3%

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

   13. Simplified 95.3%

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

   14. Taylor expanded in t around inf 95.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 240000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 3.42 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]

Alternative 5: 99.5% 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 98.7%

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

2. Final simplification 98.7%

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

Alternative 6: 56.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 240000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+132} \lor \neg \left(t \leq 7.5 \cdot 10^{+183}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 240000.0)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (if (or (<= t 9.2e+132) (not (<= t 7.5e+183)))
     (sqrt (* 2.0 (* z (pow (* x 0.5) 2.0))))
     (* 0.5 (* x (cbrt (pow (* 2.0 z) 1.5)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 240000.0) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else if ((t <= 9.2e+132) || !(t <= 7.5e+183)) {
		tmp = sqrt((2.0 * (z * pow((x * 0.5), 2.0))));
	} else {
		tmp = 0.5 * (x * cbrt(pow((2.0 * z), 1.5)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 240000.0) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else if ((t <= 9.2e+132) || !(t <= 7.5e+183)) {
		tmp = Math.sqrt((2.0 * (z * Math.pow((x * 0.5), 2.0))));
	} else {
		tmp = 0.5 * (x * Math.cbrt(Math.pow((2.0 * z), 1.5)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 240000.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	elseif ((t <= 9.2e+132) || !(t <= 7.5e+183))
		tmp = sqrt(Float64(2.0 * Float64(z * (Float64(x * 0.5) ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(x * cbrt((Float64(2.0 * z) ^ 1.5))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 240000.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 9.2e+132], N[Not[LessEqual[t, 7.5e+183]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(z * N[Power[N[(x * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(x * N[Power[N[Power[N[(2.0 * z), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.4e5

   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. sqr-neg 98.8%

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

      2. associate-/l* 98.8%

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

      3. distribute-frac-neg 98.8%

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

      4. exp-neg 98.8%

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

      5. associate-*r/ 98.8%

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

      6. *-rgt-identity 98.8%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 72.2%

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

      3. sqrt-unprod 72.2%

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

      4. associate-*l* 72.2%

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

      5. pow2 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 64.4%

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

   if 2.4e5 < t < 9.2000000000000006e132 or 7.49999999999999966e183 < t

   1. Initial program 97.7%

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

   2. Taylor expanded in t around 0 13.7%

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

      1. associate-*r* 13.7%

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

      2. *-commutative 13.7%

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

      3. *-commutative 13.7%

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

      4. fma-neg 13.7%

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

      5. associate-*l* 13.7%

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

      6. fma-neg 13.7%

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

      7. *-commutative 13.7%

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

   4. Simplified 13.7%

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

   5. Taylor expanded in x around inf 8.3%

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

      1. add-sqr-sqrt 4.6%

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

      2. sqrt-unprod 26.6%

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

      3. swap-sqr 26.6%

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

      4. rem-square-sqrt 26.6%

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

      5. associate-*r* 26.6%

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

      6. *-commutative 26.6%

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

      7. associate-*r* 26.6%

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

      8. *-commutative 26.6%

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

      9. swap-sqr 28.9%

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

      10. pow2 28.9%

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

      11. add-sqr-sqrt 28.9%

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

   7. Applied egg-rr 28.9%

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

   if 9.2000000000000006e132 < t < 7.49999999999999966e183

   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. Taylor expanded in t around 0 17.9%

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

      1. associate-*r* 17.9%

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

      2. *-commutative 17.9%

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

      3. *-commutative 17.9%

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

      4. fma-neg 17.9%

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

      5. associate-*l* 17.9%

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

      6. fma-neg 17.9%

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

      7. *-commutative 17.9%

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

   4. Simplified 17.9%

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

   5. Taylor expanded in x around inf 9.6%

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

      1. expm1-log1p-u 1.7%

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

      2. expm1-udef 1.5%

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

      3. *-commutative 1.5%

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

      4. associate-*r* 1.5%

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

      5. *-commutative 1.5%

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

      6. associate-*l* 1.5%

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

      7. sqrt-prod 1.5%

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

      8. *-commutative 1.5%

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

      9. *-commutative 1.5%

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

   7. Applied egg-rr 1.5%

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

      1. expm1-def 1.7%

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

      2. expm1-log1p 9.6%

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

      3. *-commutative 9.6%

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

      4. associate-*r* 9.6%

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

      5. *-commutative 9.6%

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

      6. *-commutative 9.6%

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

   9. Simplified 9.6%

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

       1. add-cbrt-cube 37.1%

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

       2. add-sqr-sqrt 37.1%

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

       3. pow1 37.1%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{\color{blue}{{\left(z \cdot 2\right)}^{1}} \cdot \sqrt{z \cdot 2}}\right) \]

       4. pow1/2 37.1%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}}\right) \]

       5. pow-prod-up 37.1%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{\color{blue}{{\left(z \cdot 2\right)}^{\left(1 + 0.5\right)}}}\right) \]

       6. metadata-eval 37.1%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{\color{blue}{1.5}}}\right) \]

   11. Applied egg-rr 37.1%

       \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 240000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+132} \lor \neg \left(t \leq 7.5 \cdot 10^{+183}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\right)\\ \end{array} \]

Alternative 7: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 210000:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 210000.0)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= t 9.2e+124)
       (sqrt (* 2.0 (* z (pow (* x 0.5) 2.0))))
       (* t_1 (cbrt (pow (* 2.0 z) 1.5)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 210000.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if (t <= 9.2e+124) {
		tmp = sqrt((2.0 * (z * pow((x * 0.5), 2.0))));
	} else {
		tmp = t_1 * cbrt(pow((2.0 * z), 1.5));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 210000.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else if (t <= 9.2e+124) {
		tmp = Math.sqrt((2.0 * (z * Math.pow((x * 0.5), 2.0))));
	} else {
		tmp = t_1 * Math.cbrt(Math.pow((2.0 * z), 1.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 210000.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (t <= 9.2e+124)
		tmp = sqrt(Float64(2.0 * Float64(z * (Float64(x * 0.5) ^ 2.0))));
	else
		tmp = Float64(t_1 * cbrt((Float64(2.0 * z) ^ 1.5)));
	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, 210000.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e+124], N[Sqrt[N[(2.0 * N[(z * N[Power[N[(x * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Power[N[Power[N[(2.0 * z), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.1e5

   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. sqr-neg 98.8%

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

      2. associate-/l* 98.8%

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

      3. distribute-frac-neg 98.8%

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

      4. exp-neg 98.8%

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

      5. associate-*r/ 98.8%

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

      6. *-rgt-identity 98.8%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 72.2%

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

      3. sqrt-unprod 72.2%

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

      4. associate-*l* 72.2%

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

      5. pow2 72.2%

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

   5. Applied egg-rr 72.2%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 64.4%

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

   if 2.1e5 < t < 9.19999999999999938e124

   1. Initial program 95.0%

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

   2. Taylor expanded in t around 0 14.5%

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

      1. associate-*r* 14.5%

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

      2. *-commutative 14.5%

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

      3. *-commutative 14.5%

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

      4. fma-neg 14.5%

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

      5. associate-*l* 14.5%

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

      6. fma-neg 14.5%

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

      7. *-commutative 14.5%

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

   4. Simplified 14.5%

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

   5. Taylor expanded in x around inf 9.0%

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

      1. add-sqr-sqrt 2.8%

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

      2. sqrt-unprod 26.0%

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

      3. swap-sqr 26.0%

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

      4. rem-square-sqrt 26.0%

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

      5. associate-*r* 26.0%

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

      6. *-commutative 26.0%

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

      7. associate-*r* 26.0%

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

      8. *-commutative 26.0%

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

      9. swap-sqr 30.8%

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

      10. pow2 30.8%

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

      11. add-sqr-sqrt 30.8%

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

   7. Applied egg-rr 30.8%

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

   if 9.19999999999999938e124 < t

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 14.9%

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

      1. sqrt-prod 14.9%

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

      2. add-cbrt-cube 32.6%

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

      3. add-sqr-sqrt 32.6%

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

      4. pow1/3 32.6%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(\left(z \cdot 2\right) \cdot \sqrt{z \cdot 2}\right)}^{0.3333333333333333}} \]

      5. pow1 32.6%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left(\color{blue}{{\left(z \cdot 2\right)}^{1}} \cdot \sqrt{z \cdot 2}\right)}^{0.3333333333333333} \]

      6. pow1/2 32.6%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\left(z \cdot 2\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      7. pow-prod-up 32.6%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\color{blue}{\left({\left(z \cdot 2\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      8. *-commutative 32.6%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\color{blue}{\left(2 \cdot z\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 32.6%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot {\left({\left(2 \cdot z\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 32.6%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left({\left(2 \cdot z\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   7. Step-by-step derivation

      1. unpow1/3 32.6%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}} \]

   8. Simplified 32.6%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 210000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}\\ \end{array} \]

Alternative 8: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. sqr-neg 98.7%

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

   2. associate-/l* 98.7%

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

   3. distribute-frac-neg 98.7%

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

   4. exp-neg 98.7%

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

   5. associate-*r/ 98.7%

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

   6. *-rgt-identity 98.7%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. associate-*r/ 99.8%

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

   10. exp-neg 99.8%

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

   11. distribute-frac-neg 99.8%

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

   12. associate-/l* 99.8%

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

   13. sqr-neg 99.8%

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

   14. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 78.4%

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

   3. sqrt-unprod 78.4%

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

   4. associate-*l* 78.4%

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

   5. pow2 78.4%

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

5. Applied egg-rr 78.4%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 83.4%

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

   1. distribute-lft-out 83.4%

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

   2. *-commutative 83.4%

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

10. Simplified 83.4%

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

11. Final simplification 83.4%

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

Alternative 9: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. sqr-neg 98.7%

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

   2. associate-/l* 98.7%

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

   3. distribute-frac-neg 98.7%

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

   4. exp-neg 98.7%

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

   5. associate-*r/ 98.7%

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

   6. *-rgt-identity 98.7%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. associate-*r/ 99.8%

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

   10. exp-neg 99.8%

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

   11. distribute-frac-neg 99.8%

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

   12. associate-/l* 99.8%

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

   13. sqr-neg 99.8%

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

   14. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 78.4%

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

   3. sqrt-unprod 78.4%

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

   4. associate-*l* 78.4%

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

   5. pow2 78.4%

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

5. Applied egg-rr 78.4%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 83.4%

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

   1. distribute-lft-out 83.4%

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

   2. *-commutative 83.4%

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

10. Simplified 83.4%

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

11. Taylor expanded in z around 0 83.4%

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

    1. +-commutative 83.4%

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

    2. unpow2 83.4%

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

    3. fma-def 83.4%

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

13. Simplified 83.4%

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

14. Final simplification 83.4%

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

Alternative 10: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq 10^{+146}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x 0.5) 1e+146)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (sqrt (* 2.0 (* z (pow (* x 0.5) 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * 0.5) <= 1e+146) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt((2.0 * (z * pow((x * 0.5), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * 0.5d0) <= 1d+146) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt((2.0d0 * (z * ((x * 0.5d0) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * 0.5) <= 1e+146) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt((2.0 * (z * Math.pow((x * 0.5), 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * 0.5) <= 1e+146:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt((2.0 * (z * math.pow((x * 0.5), 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * 0.5) <= 1e+146)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(2.0 * Float64(z * (Float64(x * 0.5) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * 0.5) <= 1e+146)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt((2.0 * (z * ((x * 0.5) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], 1e+146], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(z * N[Power[N[(x * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < 9.99999999999999934e145

   1. Initial program 98.5%

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

      1. sqr-neg 98.5%

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

      2. associate-/l* 98.5%

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

      3. distribute-frac-neg 98.5%

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

      4. exp-neg 98.5%

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

      5. associate-*r/ 98.5%

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

      6. *-rgt-identity 98.5%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.3%

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

      2. expm1-udef 77.3%

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

      3. sqrt-unprod 77.3%

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

      4. associate-*l* 77.3%

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

      5. pow2 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. expm1-def 98.3%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 55.0%

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

   if 9.99999999999999934e145 < (*.f64 x 1/2)

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 40.6%

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

      1. associate-*r* 40.5%

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

      2. *-commutative 40.5%

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

      3. *-commutative 40.5%

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

      4. fma-neg 40.5%

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

      5. associate-*l* 40.5%

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

      6. fma-neg 40.5%

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

      7. *-commutative 40.5%

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

   4. Simplified 40.5%

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

   5. Taylor expanded in x around inf 35.6%

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

      1. add-sqr-sqrt 35.5%

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

      2. sqrt-unprod 68.3%

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

      3. swap-sqr 68.3%

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

      4. rem-square-sqrt 68.3%

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

      5. associate-*r* 68.3%

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

      6. *-commutative 68.3%

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

      7. associate-*r* 68.3%

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

      8. *-commutative 68.3%

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

      9. swap-sqr 77.9%

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

      10. pow2 77.9%

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

      11. add-sqr-sqrt 77.9%

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

   7. Applied egg-rr 77.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq 10^{+146}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot {\left(x \cdot 0.5\right)}^{2}\right)}\\ \end{array} \]

Alternative 11: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 72000000000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 72000000000000.0)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (sqrt (* (* 2.0 z) (pow y 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 72000000000000.0) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt(((2.0 * z) * pow(y, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 72000000000000.0d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt(((2.0d0 * z) * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 72000000000000.0) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(((2.0 * z) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 72000000000000.0:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(((2.0 * z) * math.pow(y, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 72000000000000.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(Float64(2.0 * z) * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 72000000000000.0)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt(((2.0 * z) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 72000000000000.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e13

   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. sqr-neg 98.8%

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

      2. associate-/l* 98.8%

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

      3. distribute-frac-neg 98.8%

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

      4. exp-neg 98.8%

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

      5. associate-*r/ 98.8%

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

      6. *-rgt-identity 98.8%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 72.6%

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

      3. sqrt-unprod 72.6%

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

      4. associate-*l* 72.6%

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

      5. pow2 72.6%

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

   5. Applied egg-rr 72.6%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t around 0 63.5%

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

   if 7.2e13 < t

   1. Initial program 98.1%

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

      1. sqr-neg 98.1%

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

      2. associate-/l* 98.1%

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

      3. distribute-frac-neg 98.1%

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

      4. exp-neg 98.1%

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

      5. associate-*r/ 98.1%

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

      6. *-rgt-identity 98.1%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-*r/ 100.0%

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

      10. exp-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. associate-/l* 100.0%

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

      13. sqr-neg 100.0%

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

      14. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot e^{{t}^{2}}} \]

   7. Simplified 100.0%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]

   8. Taylor expanded in t around 0 15.3%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]

   9. Taylor expanded in x around 0 8.3%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 8.3%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. *-commutative 8.3%

          \[\leadsto -\color{blue}{\sqrt{z} \cdot \left(y \cdot \sqrt{2}\right)} \]

       3. distribute-rgt-neg-in 8.3%

          \[\leadsto \color{blue}{\sqrt{z} \cdot \left(-y \cdot \sqrt{2}\right)} \]

       4. *-commutative 8.3%

          \[\leadsto \sqrt{z} \cdot \left(-\color{blue}{\sqrt{2} \cdot y}\right) \]

   11. Simplified 8.3%

       \[\leadsto \color{blue}{\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 1.4%

          \[\leadsto \color{blue}{\sqrt{\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)} \cdot \sqrt{\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)}} \]

       2. sqrt-unprod 7.2%

          \[\leadsto \color{blue}{\sqrt{\left(\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)\right) \cdot \left(\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)\right)}} \]

       3. swap-sqr 10.7%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\left(-\sqrt{2} \cdot y\right) \cdot \left(-\sqrt{2} \cdot y\right)\right)}} \]

       4. add-sqr-sqrt 10.7%

          \[\leadsto \sqrt{\color{blue}{z} \cdot \left(\left(-\sqrt{2} \cdot y\right) \cdot \left(-\sqrt{2} \cdot y\right)\right)} \]

       5. sqr-neg 10.7%

          \[\leadsto \sqrt{z \cdot \color{blue}{\left(\left(\sqrt{2} \cdot y\right) \cdot \left(\sqrt{2} \cdot y\right)\right)}} \]

       6. swap-sqr 10.7%

          \[\leadsto \sqrt{z \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(y \cdot y\right)\right)}} \]

       7. rem-square-sqrt 10.7%

          \[\leadsto \sqrt{z \cdot \left(\color{blue}{2} \cdot \left(y \cdot y\right)\right)} \]

       8. pow2 10.7%

          \[\leadsto \sqrt{z \cdot \left(2 \cdot \color{blue}{{y}^{2}}\right)} \]

   13. Applied egg-rr 10.7%

       \[\leadsto \color{blue}{\sqrt{z \cdot \left(2 \cdot {y}^{2}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 10.7%

          \[\leadsto \sqrt{\color{blue}{\left(z \cdot 2\right) \cdot {y}^{2}}} \]

   15. Simplified 10.7%

       \[\leadsto \color{blue}{\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 72000000000000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {y}^{2}}\\ \end{array} \]

Alternative 12: 43.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+39} \lor \neg \left(y \leq 6.2 \cdot 10^{+41}\right):\\ \;\;\;\;y \cdot \left(-t_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (or (<= y -9e+39) (not (<= y 6.2e+41)))
     (* 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 <= -9e+39) || !(y <= 6.2e+41)) {
		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 <= (-9d+39)) .or. (.not. (y <= 6.2d+41))) 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 <= -9e+39) || !(y <= 6.2e+41)) {
		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 <= -9e+39) or not (y <= 6.2e+41):
		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 <= -9e+39) || !(y <= 6.2e+41))
		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 <= -9e+39) || ~((y <= 6.2e+41)))
		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, -9e+39], N[Not[LessEqual[y, 6.2e+41]], $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 < -8.99999999999999991e39 or 6.2e41 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Step-by-step derivation

      1. sqr-neg 99.9%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}{2}} \]

      2. associate-/l* 99.9%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{-t}{\frac{2}{-t}}}} \]

      3. distribute-frac-neg 99.9%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{-\frac{t}{\frac{2}{-t}}}} \]

      4. exp-neg 99.9%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\frac{1}{e^{\frac{t}{\frac{2}{-t}}}}} \]

      5. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot 1}{e^{\frac{t}{\frac{2}{-t}}}}} \]

      6. *-rgt-identity 99.9%

         \[\leadsto \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}}{e^{\frac{t}{\frac{2}{-t}}}} \]

      7. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{\sqrt{z \cdot 2}}{e^{\frac{t}{\frac{2}{-t}}}}} \]

      8. *-rgt-identity 99.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \frac{\color{blue}{\sqrt{z \cdot 2} \cdot 1}}{e^{\frac{t}{\frac{2}{-t}}}} \]

      9. associate-*r/ 99.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \frac{1}{e^{\frac{t}{\frac{2}{-t}}}}\right)} \]

      10. exp-neg 99.9%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{e^{-\frac{t}{\frac{2}{-t}}}}\right) \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{-t}{\frac{2}{-t}}}}\right) \]

      12. associate-/l* 99.9%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{\left(-t\right) \cdot \left(-t\right)}{2}}}\right) \]

      13. sqr-neg 99.9%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{\color{blue}{t \cdot t}}{2}}\right) \]

      14. exp-sqrt 99.9%

          \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 98.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)\right)} \]

      2. expm1-udef 71.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} - 1\right)} \]

      3. sqrt-unprod 71.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1\right) \]

      4. associate-*l* 71.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot e^{t \cdot t}\right)}}\right)} - 1\right) \]

      5. pow2 71.4%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)}\right)} - 1\right) \]

   5. Applied egg-rr 71.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)} - 1\right)} \]
   6. Step-by-step derivation

      1. expm1-def 98.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)\right)} \]

      2. expm1-log1p 99.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}} \]

      3. associate-*r* 99.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}} \]

      4. *-commutative 99.9%

         \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot e^{{t}^{2}}} \]

   7. Simplified 99.9%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]

   8. Taylor expanded in t around 0 58.2%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]

   9. Taylor expanded in x around 0 51.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg 51.6%

          \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

       2. *-commutative 51.6%

          \[\leadsto -\color{blue}{\sqrt{z} \cdot \left(y \cdot \sqrt{2}\right)} \]

       3. distribute-rgt-neg-in 51.6%

          \[\leadsto \color{blue}{\sqrt{z} \cdot \left(-y \cdot \sqrt{2}\right)} \]

       4. *-commutative 51.6%

          \[\leadsto \sqrt{z} \cdot \left(-\color{blue}{\sqrt{2} \cdot y}\right) \]

   11. Simplified 51.6%

       \[\leadsto \color{blue}{\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)} \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-out 51.6%

          \[\leadsto \color{blue}{-\sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)} \]

       2. neg-sub0 51.6%

          \[\leadsto \color{blue}{0 - \sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)} \]

       3. associate-*r* 51.6%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot y} \]

       4. sqrt-prod 51.8%

          \[\leadsto 0 - \color{blue}{\sqrt{z \cdot 2}} \cdot y \]

       5. *-commutative 51.8%

          \[\leadsto 0 - \sqrt{\color{blue}{2 \cdot z}} \cdot y \]

   13. Applied egg-rr 51.8%

       \[\leadsto \color{blue}{0 - \sqrt{2 \cdot z} \cdot y} \]
   14. Step-by-step derivation

       1. neg-sub0 51.8%

          \[\leadsto \color{blue}{-\sqrt{2 \cdot z} \cdot y} \]

       2. distribute-rgt-neg-in 51.8%

          \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \left(-y\right)} \]

       3. *-commutative 51.8%

          \[\leadsto \sqrt{\color{blue}{z \cdot 2}} \cdot \left(-y\right) \]

   15. Simplified 51.8%

       \[\leadsto \color{blue}{\sqrt{z \cdot 2} \cdot \left(-y\right)} \]

   if -8.99999999999999991e39 < y < 6.2e41

   1. Initial program 97.7%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. Taylor expanded in t around 0 49.4%

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 49.3%

         \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \left(0.5 \cdot x - y\right)} \]

      2. *-commutative 49.3%

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{z}\right)} \cdot \left(0.5 \cdot x - y\right) \]

      3. *-commutative 49.3%

         \[\leadsto \left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \left(\color{blue}{x \cdot 0.5} - y\right) \]

      4. fma-neg 49.3%

         \[\leadsto \left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)} \]

      5. associate-*l* 49.2%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{z} \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right)} \]

      6. fma-neg 49.2%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot 0.5 - y\right)}\right) \]

      7. *-commutative 49.2%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{z} \cdot \left(\color{blue}{0.5 \cdot x} - y\right)\right) \]

   4. Simplified 49.2%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{z} \cdot \left(0.5 \cdot x - y\right)\right)} \]

   5. Taylor expanded in x around inf 40.8%

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 21.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{z}\right)\right)\right)\right)} \]

      2. expm1-udef 12.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{z}\right)\right)\right)} - 1} \]

      3. *-commutative 12.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot \left(x \cdot \sqrt{z}\right)\right) \cdot \sqrt{2}}\right)} - 1 \]

      4. associate-*r* 12.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(0.5 \cdot x\right) \cdot \sqrt{z}\right)} \cdot \sqrt{2}\right)} - 1 \]

      5. *-commutative 12.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\left(x \cdot 0.5\right)} \cdot \sqrt{z}\right) \cdot \sqrt{2}\right)} - 1 \]

      6. associate-*l* 12.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 0.5\right) \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)}\right)} - 1 \]

      7. sqrt-prod 12.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right)} - 1 \]

      8. *-commutative 12.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5\right) \cdot \sqrt{\color{blue}{2 \cdot z}}\right)} - 1 \]

      9. *-commutative 12.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(x \cdot 0.5\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right)} - 1 \]

   7. Applied egg-rr 12.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 21.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\right)\right)} \]

      2. expm1-log1p 40.9%

         \[\leadsto \color{blue}{\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}} \]

      3. *-commutative 40.9%

         \[\leadsto \color{blue}{\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)} \]

      4. associate-*r* 40.9%

         \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot x\right) \cdot 0.5} \]

      5. *-commutative 40.9%

         \[\leadsto \color{blue}{0.5 \cdot \left(\sqrt{z \cdot 2} \cdot x\right)} \]

      6. *-commutative 40.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \sqrt{z \cdot 2}\right)} \]

   9. Simplified 40.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+39} \lor \neg \left(y \leq 6.2 \cdot 10^{+41}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\\ \end{array} \]

Alternative 13: 55.9% 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 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. sqr-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}{2}} \]

   2. associate-/l* 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{-t}{\frac{2}{-t}}}} \]

   3. distribute-frac-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{-\frac{t}{\frac{2}{-t}}}} \]

   4. exp-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\frac{1}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   5. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot 1}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   6. *-rgt-identity 98.7%

      \[\leadsto \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}}{e^{\frac{t}{\frac{2}{-t}}}} \]

   7. associate-*r/ 99.8%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{\sqrt{z \cdot 2}}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   8. *-rgt-identity 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \frac{\color{blue}{\sqrt{z \cdot 2} \cdot 1}}{e^{\frac{t}{\frac{2}{-t}}}} \]

   9. associate-*r/ 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \frac{1}{e^{\frac{t}{\frac{2}{-t}}}}\right)} \]

   10. exp-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{e^{-\frac{t}{\frac{2}{-t}}}}\right) \]

   11. distribute-frac-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{-t}{\frac{2}{-t}}}}\right) \]

   12. associate-/l* 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{\left(-t\right) \cdot \left(-t\right)}{2}}}\right) \]

   13. sqr-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{\color{blue}{t \cdot t}}{2}}\right) \]

   14. exp-sqrt 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 98.5%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)\right)} \]

   2. expm1-udef 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} - 1\right)} \]

   3. sqrt-unprod 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1\right) \]

   4. associate-*l* 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot e^{t \cdot t}\right)}}\right)} - 1\right) \]

   5. pow2 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)}\right)} - 1\right) \]

5. Applied egg-rr 78.4%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 98.5%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}} \]

   3. associate-*r* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}} \]

   4. *-commutative 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot e^{{t}^{2}}} \]

7. Simplified 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]

8. Taylor expanded in t around 0 53.4%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]

9. Final simplification 53.4%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} \]

Alternative 14: 29.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(-\sqrt{2 \cdot z}\right) \end{array} \]

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 * Float64(-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 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. sqr-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}{2}} \]

   2. associate-/l* 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{-t}{\frac{2}{-t}}}} \]

   3. distribute-frac-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{-\frac{t}{\frac{2}{-t}}}} \]

   4. exp-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\frac{1}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   5. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot 1}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   6. *-rgt-identity 98.7%

      \[\leadsto \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}}{e^{\frac{t}{\frac{2}{-t}}}} \]

   7. associate-*r/ 99.8%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{\sqrt{z \cdot 2}}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   8. *-rgt-identity 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \frac{\color{blue}{\sqrt{z \cdot 2} \cdot 1}}{e^{\frac{t}{\frac{2}{-t}}}} \]

   9. associate-*r/ 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \frac{1}{e^{\frac{t}{\frac{2}{-t}}}}\right)} \]

   10. exp-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{e^{-\frac{t}{\frac{2}{-t}}}}\right) \]

   11. distribute-frac-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{-t}{\frac{2}{-t}}}}\right) \]

   12. associate-/l* 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{\left(-t\right) \cdot \left(-t\right)}{2}}}\right) \]

   13. sqr-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{\color{blue}{t \cdot t}}{2}}\right) \]

   14. exp-sqrt 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 98.5%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)\right)} \]

   2. expm1-udef 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} - 1\right)} \]

   3. sqrt-unprod 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1\right) \]

   4. associate-*l* 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot e^{t \cdot t}\right)}}\right)} - 1\right) \]

   5. pow2 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)}\right)} - 1\right) \]

5. Applied egg-rr 78.4%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 98.5%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}} \]

   3. associate-*r* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}} \]

   4. *-commutative 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot e^{{t}^{2}}} \]

7. Simplified 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]

8. Taylor expanded in t around 0 53.4%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]

9. Taylor expanded in x around 0 29.5%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
10. Step-by-step derivation

    1. mul-1-neg 29.5%

       \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

    2. *-commutative 29.5%

       \[\leadsto -\color{blue}{\sqrt{z} \cdot \left(y \cdot \sqrt{2}\right)} \]

    3. distribute-rgt-neg-in 29.5%

       \[\leadsto \color{blue}{\sqrt{z} \cdot \left(-y \cdot \sqrt{2}\right)} \]

    4. *-commutative 29.5%

       \[\leadsto \sqrt{z} \cdot \left(-\color{blue}{\sqrt{2} \cdot y}\right) \]

11. Simplified 29.5%

    \[\leadsto \color{blue}{\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)} \]
12. Step-by-step derivation

    1. distribute-rgt-neg-out 29.5%

       \[\leadsto \color{blue}{-\sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)} \]

    2. neg-sub0 29.5%

       \[\leadsto \color{blue}{0 - \sqrt{z} \cdot \left(\sqrt{2} \cdot y\right)} \]

    3. associate-*r* 29.5%

       \[\leadsto 0 - \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot y} \]

    4. sqrt-prod 29.6%

       \[\leadsto 0 - \color{blue}{\sqrt{z \cdot 2}} \cdot y \]

    5. *-commutative 29.6%

       \[\leadsto 0 - \sqrt{\color{blue}{2 \cdot z}} \cdot y \]

13. Applied egg-rr 29.6%

    \[\leadsto \color{blue}{0 - \sqrt{2 \cdot z} \cdot y} \]
14. Step-by-step derivation

    1. neg-sub0 29.6%

       \[\leadsto \color{blue}{-\sqrt{2 \cdot z} \cdot y} \]

    2. distribute-rgt-neg-in 29.6%

       \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \left(-y\right)} \]

    3. *-commutative 29.6%

       \[\leadsto \sqrt{\color{blue}{z \cdot 2}} \cdot \left(-y\right) \]

15. Simplified 29.6%

    \[\leadsto \color{blue}{\sqrt{z \cdot 2} \cdot \left(-y\right)} \]

16. Final simplification 29.6%

    \[\leadsto y \cdot \left(-\sqrt{2 \cdot z}\right) \]

Alternative 15: 2.5% 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 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. sqr-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}{2}} \]

   2. associate-/l* 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{-t}{\frac{2}{-t}}}} \]

   3. distribute-frac-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{-\frac{t}{\frac{2}{-t}}}} \]

   4. exp-neg 98.7%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\frac{1}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   5. associate-*r/ 98.7%

      \[\leadsto \color{blue}{\frac{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot 1}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   6. *-rgt-identity 98.7%

      \[\leadsto \frac{\color{blue}{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}}{e^{\frac{t}{\frac{2}{-t}}}} \]

   7. associate-*r/ 99.8%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \frac{\sqrt{z \cdot 2}}{e^{\frac{t}{\frac{2}{-t}}}}} \]

   8. *-rgt-identity 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \frac{\color{blue}{\sqrt{z \cdot 2} \cdot 1}}{e^{\frac{t}{\frac{2}{-t}}}} \]

   9. associate-*r/ 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \frac{1}{e^{\frac{t}{\frac{2}{-t}}}}\right)} \]

   10. exp-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{e^{-\frac{t}{\frac{2}{-t}}}}\right) \]

   11. distribute-frac-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{-t}{\frac{2}{-t}}}}\right) \]

   12. associate-/l* 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\color{blue}{\frac{\left(-t\right) \cdot \left(-t\right)}{2}}}\right) \]

   13. sqr-neg 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot e^{\frac{\color{blue}{t \cdot t}}{2}}\right) \]

   14. exp-sqrt 99.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 98.5%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)\right)} \]

   2. expm1-udef 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} - 1\right)} \]

   3. sqrt-unprod 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}}\right)} - 1\right) \]

   4. associate-*l* 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{z \cdot \left(2 \cdot e^{t \cdot t}\right)}}\right)} - 1\right) \]

   5. pow2 78.4%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{\color{blue}{{t}^{2}}}\right)}\right)} - 1\right) \]

5. Applied egg-rr 78.4%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 98.5%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{z \cdot \left(2 \cdot e^{{t}^{2}}\right)}} \]

   3. associate-*r* 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(z \cdot 2\right) \cdot e^{{t}^{2}}}} \]

   4. *-commutative 99.8%

      \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(2 \cdot z\right)} \cdot e^{{t}^{2}}} \]

7. Simplified 99.8%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot e^{{t}^{2}}}} \]

8. Taylor expanded in t around 0 53.4%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}} \]

9. Taylor expanded in x around 0 29.5%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
10. Step-by-step derivation

    1. mul-1-neg 29.5%

       \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

    2. *-commutative 29.5%

       \[\leadsto -\color{blue}{\sqrt{z} \cdot \left(y \cdot \sqrt{2}\right)} \]

    3. distribute-rgt-neg-in 29.5%

       \[\leadsto \color{blue}{\sqrt{z} \cdot \left(-y \cdot \sqrt{2}\right)} \]

    4. *-commutative 29.5%

       \[\leadsto \sqrt{z} \cdot \left(-\color{blue}{\sqrt{2} \cdot y}\right) \]

11. Simplified 29.5%

    \[\leadsto \color{blue}{\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)} \]
12. Step-by-step derivation

    1. expm1-log1p-u 18.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)\right)\right)} \]

    2. expm1-udef 12.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{z} \cdot \left(-\sqrt{2} \cdot y\right)\right)} - 1} \]

    3. add-sqr-sqrt 11.2%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{z} \cdot \color{blue}{\left(\sqrt{-\sqrt{2} \cdot y} \cdot \sqrt{-\sqrt{2} \cdot y}\right)}\right)} - 1 \]

    4. sqrt-unprod 13.6%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{z} \cdot \color{blue}{\sqrt{\left(-\sqrt{2} \cdot y\right) \cdot \left(-\sqrt{2} \cdot y\right)}}\right)} - 1 \]

    5. sqr-neg 13.6%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{z} \cdot \sqrt{\color{blue}{\left(\sqrt{2} \cdot y\right) \cdot \left(\sqrt{2} \cdot y\right)}}\right)} - 1 \]

    6. sqrt-unprod 1.3%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{z} \cdot \color{blue}{\left(\sqrt{\sqrt{2} \cdot y} \cdot \sqrt{\sqrt{2} \cdot y}\right)}\right)} - 1 \]

    7. add-sqr-sqrt 1.8%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{z} \cdot \color{blue}{\left(\sqrt{2} \cdot y\right)}\right)} - 1 \]

    8. associate-*r* 1.8%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot y}\right)} - 1 \]

    9. sqrt-prod 1.8%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{z \cdot 2}} \cdot y\right)} - 1 \]

    10. *-commutative 1.8%

        \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{2 \cdot z}} \cdot y\right)} - 1 \]

13. Applied egg-rr 1.8%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot z} \cdot y\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 1.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot z} \cdot y\right)\right)} \]

    2. expm1-log1p 2.1%

       \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot y} \]

    3. *-commutative 2.1%

       \[\leadsto \color{blue}{y \cdot \sqrt{2 \cdot z}} \]

    4. *-commutative 2.1%

       \[\leadsto y \cdot \sqrt{\color{blue}{z \cdot 2}} \]

15. Simplified 2.1%

    \[\leadsto \color{blue}{y \cdot \sqrt{z \cdot 2}} \]

16. Final simplification 2.1%

    \[\leadsto y \cdot \sqrt{2 \cdot z} \]

Developer target: 99.5% 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 2023336 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))