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

Percentage Accurate: 99.3% → 99.8%

Time: 17.1s

Alternatives: 10

Speedup: 1.0×

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l* 99.9%

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

   3. exp-sqrt 99.9%

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

3. Simplified 99.9%

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

   1. exp-sqrt 99.9%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. expm1-log1p-u 52.3%

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

   5. expm1-udef 42.5%

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

6. Applied egg-rr 42.5%

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

   1. expm1-def 52.3%

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

   2. expm1-log1p 99.9%

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

   3. fma-neg 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*l* 99.9%

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

8. Simplified 99.9%

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

   1. pow2 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 43.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot \left(y - x\right)\right)\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-20} \lor \neg \left(y \leq 9.5 \cdot 10^{+28}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \sqrt{0.5}\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.5e+56)
   (sqrt (* 2.0 (* z (* y (- y x)))))
   (if (<= y 3.8e-98)
     (* x (sqrt (* 0.5 z)))
     (if (or (<= y 2.8e-20) (not (<= y 9.5e+28)))
       (* y (- (sqrt (* z 2.0))))
       (* (* x (sqrt 0.5)) (sqrt z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e+56) {
		tmp = sqrt((2.0 * (z * (y * (y - x)))));
	} else if (y <= 3.8e-98) {
		tmp = x * sqrt((0.5 * z));
	} else if ((y <= 2.8e-20) || !(y <= 9.5e+28)) {
		tmp = y * -sqrt((z * 2.0));
	} else {
		tmp = (x * sqrt(0.5)) * sqrt(z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5.5d+56)) then
        tmp = sqrt((2.0d0 * (z * (y * (y - x)))))
    else if (y <= 3.8d-98) then
        tmp = x * sqrt((0.5d0 * z))
    else if ((y <= 2.8d-20) .or. (.not. (y <= 9.5d+28))) then
        tmp = y * -sqrt((z * 2.0d0))
    else
        tmp = (x * sqrt(0.5d0)) * sqrt(z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e+56) {
		tmp = Math.sqrt((2.0 * (z * (y * (y - x)))));
	} else if (y <= 3.8e-98) {
		tmp = x * Math.sqrt((0.5 * z));
	} else if ((y <= 2.8e-20) || !(y <= 9.5e+28)) {
		tmp = y * -Math.sqrt((z * 2.0));
	} else {
		tmp = (x * Math.sqrt(0.5)) * Math.sqrt(z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -5.5e+56:
		tmp = math.sqrt((2.0 * (z * (y * (y - x)))))
	elif y <= 3.8e-98:
		tmp = x * math.sqrt((0.5 * z))
	elif (y <= 2.8e-20) or not (y <= 9.5e+28):
		tmp = y * -math.sqrt((z * 2.0))
	else:
		tmp = (x * math.sqrt(0.5)) * math.sqrt(z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.5e+56)
		tmp = sqrt(Float64(2.0 * Float64(z * Float64(y * Float64(y - x)))));
	elseif (y <= 3.8e-98)
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	elseif ((y <= 2.8e-20) || !(y <= 9.5e+28))
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	else
		tmp = Float64(Float64(x * sqrt(0.5)) * sqrt(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -5.5e+56)
		tmp = sqrt((2.0 * (z * (y * (y - x)))));
	elseif (y <= 3.8e-98)
		tmp = x * sqrt((0.5 * z));
	elseif ((y <= 2.8e-20) || ~((y <= 9.5e+28)))
		tmp = y * -sqrt((z * 2.0));
	else
		tmp = (x * sqrt(0.5)) * sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.5e+56], N[Sqrt[N[(2.0 * N[(z * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 3.8e-98], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 2.8e-20], N[Not[LessEqual[y, 9.5e+28]], $MachinePrecision]], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(x * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.5000000000000002e56

   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. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 51.0%

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

      1. *-commutative 51.0%

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

      2. *-commutative 51.0%

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

      3. fma-neg 51.0%

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

      4. associate-*l* 50.9%

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

      5. fma-neg 50.9%

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

      6. *-commutative 50.9%

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

   7. Simplified 50.9%

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

      1. associate-*r* 51.0%

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

      2. *-commutative 51.0%

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

      3. add-sqr-sqrt 47.0%

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

      4. sqrt-unprod 57.6%

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

      5. *-commutative 57.6%

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

      6. associate-*r* 57.6%

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

      7. *-commutative 57.6%

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

      8. associate-*r* 57.6%

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

      9. swap-sqr 57.6%

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

      10. rem-square-sqrt 57.7%

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

      11. *-commutative 57.7%

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

      12. *-commutative 57.7%

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

      13. swap-sqr 63.0%

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

      14. add-sqr-sqrt 63.0%

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

      15. unpow2 63.0%

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

   9. Applied egg-rr 63.0%

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

   10. Taylor expanded in x around 0 59.4%

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

       1. +-commutative 59.4%

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

       2. unpow2 59.4%

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

       3. associate-*r* 59.4%

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

       4. distribute-rgt-out 61.2%

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

       5. mul-1-neg 61.2%

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

   12. Simplified 61.2%

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

   if -5.5000000000000002e56 < y < 3.8000000000000003e-98

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 48.7%

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

      1. *-commutative 48.7%

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

      2. *-commutative 48.7%

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

      3. fma-neg 48.7%

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

      4. associate-*l* 48.8%

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

      5. fma-neg 48.8%

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

      6. *-commutative 48.8%

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

   7. Simplified 48.8%

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

      1. associate-*r* 48.7%

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

      2. *-commutative 48.7%

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

      3. add-sqr-sqrt 23.9%

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

      4. sqrt-unprod 24.4%

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

      5. *-commutative 24.4%

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

      6. associate-*r* 24.5%

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

      7. *-commutative 24.5%

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

      8. associate-*r* 24.4%

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

      9. swap-sqr 24.4%

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

      10. rem-square-sqrt 24.5%

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

      11. *-commutative 24.5%

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

      12. *-commutative 24.5%

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

      13. swap-sqr 22.7%

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

      14. add-sqr-sqrt 22.7%

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

      15. unpow2 22.7%

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

   9. Applied egg-rr 22.7%

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

   10. Taylor expanded in x around inf 20.4%

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

       1. associate-*r* 20.4%

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

       2. unpow2 20.4%

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

       3. associate-*l* 20.4%

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

       4. *-commutative 20.4%

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

       5. associate-*l* 20.4%

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

       6. unpow2 20.4%

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

   12. Simplified 20.4%

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

       1. associate-*r* 20.4%

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

       2. sqrt-prod 18.7%

          \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{{x}^{2}}} \]

       3. *-commutative 18.7%

          \[\leadsto \sqrt{\color{blue}{0.5 \cdot z}} \cdot \sqrt{{x}^{2}} \]

       4. unpow2 18.7%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \sqrt{\color{blue}{x \cdot x}} \]

       5. sqrt-prod 20.6%

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

       6. add-sqr-sqrt 43.1%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \color{blue}{x} \]

   14. Applied egg-rr 43.1%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot x} \]

   if 3.8000000000000003e-98 < y < 2.8000000000000003e-20 or 9.49999999999999927e28 < y

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 58.7%

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

      1. *-commutative 58.7%

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

      2. *-commutative 58.7%

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

      3. fma-neg 58.7%

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

      4. associate-*l* 58.6%

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

      5. fma-neg 58.6%

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

      6. *-commutative 58.6%

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

   7. Simplified 58.6%

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

   8. Taylor expanded in x around 0 48.1%

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

      1. mul-1-neg 48.1%

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

      2. distribute-lft-neg-out 48.1%

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

      3. *-commutative 48.1%

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

   10. Simplified 48.1%

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

       1. associate-*r* 48.1%

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

       2. distribute-rgt-neg-out 48.1%

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

       3. add-sqr-sqrt 48.1%

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

       4. sqrt-unprod 54.6%

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

       5. sqr-neg 54.6%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 1.1%

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

       8. *-commutative 1.1%

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

       9. add-sqr-sqrt 0.0%

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

       10. sqrt-unprod 54.6%

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

       11. sqr-neg 54.6%

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

       12. sqrt-unprod 48.1%

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

       13. add-sqr-sqrt 48.1%

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

       14. sqrt-unprod 48.2%

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

       15. *-commutative 48.2%

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

   12. Applied egg-rr 48.2%

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

       1. distribute-rgt-neg-in 48.2%

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

       2. *-commutative 48.2%

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

   14. Simplified 48.2%

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

   if 2.8000000000000003e-20 < y < 9.49999999999999927e28

   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. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 49.6%

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

      1. *-commutative 49.6%

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

      2. *-commutative 49.6%

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

      3. fma-neg 49.6%

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

      4. associate-*l* 49.4%

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

      5. fma-neg 49.4%

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

      6. *-commutative 49.4%

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

   7. Simplified 49.4%

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

      1. associate-*r* 49.6%

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

      2. *-commutative 49.6%

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

      3. add-sqr-sqrt 16.7%

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

      4. sqrt-unprod 23.9%

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

      5. *-commutative 23.9%

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

      6. associate-*r* 24.0%

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

      7. *-commutative 24.0%

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

      8. associate-*r* 24.0%

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

      9. swap-sqr 24.0%

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

      10. rem-square-sqrt 24.1%

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

      11. *-commutative 24.1%

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

      12. *-commutative 24.1%

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

      13. swap-sqr 23.9%

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

      14. add-sqr-sqrt 23.9%

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

      15. unpow2 23.9%

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

   9. Applied egg-rr 23.9%

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

   10. Taylor expanded in x around inf 19.5%

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

       1. associate-*r* 19.5%

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

       2. unpow2 19.5%

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

       3. associate-*l* 19.5%

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

       4. *-commutative 19.5%

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

       5. associate-*l* 19.5%

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

       6. unpow2 19.5%

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

   12. Simplified 19.5%

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

   13. Taylor expanded in x around 0 45.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot \left(y - x\right)\right)\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-20} \lor \neg \left(y \leq 9.5 \cdot 10^{+28}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \sqrt{0.5}\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{elif}\;x \leq 10^{-62}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.25 \cdot \left(z \cdot {x}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.6e+14)
   (* x (sqrt (* 0.5 z)))
   (if (<= x 1e-62)
     (* y (- (sqrt (* z 2.0))))
     (sqrt (* 2.0 (* 0.25 (* z (pow x 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.6e+14) {
		tmp = x * sqrt((0.5 * z));
	} else if (x <= 1e-62) {
		tmp = y * -sqrt((z * 2.0));
	} else {
		tmp = sqrt((2.0 * (0.25 * (z * pow(x, 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 <= (-1.6d+14)) then
        tmp = x * sqrt((0.5d0 * z))
    else if (x <= 1d-62) then
        tmp = y * -sqrt((z * 2.0d0))
    else
        tmp = sqrt((2.0d0 * (0.25d0 * (z * (x ** 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 <= -1.6e+14) {
		tmp = x * Math.sqrt((0.5 * z));
	} else if (x <= 1e-62) {
		tmp = y * -Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt((2.0 * (0.25 * (z * Math.pow(x, 2.0)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.6e+14:
		tmp = x * math.sqrt((0.5 * z))
	elif x <= 1e-62:
		tmp = y * -math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt((2.0 * (0.25 * (z * math.pow(x, 2.0)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.6e+14)
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	elseif (x <= 1e-62)
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.25 * Float64(z * (x ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.6e+14)
		tmp = x * sqrt((0.5 * z));
	elseif (x <= 1e-62)
		tmp = y * -sqrt((z * 2.0));
	else
		tmp = sqrt((2.0 * (0.25 * (z * (x ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.6e+14], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-62], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.25 * N[(z * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e14

   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. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. *-commutative 56.7%

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

      3. fma-neg 56.7%

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

      4. associate-*l* 56.6%

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

      5. fma-neg 56.6%

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

      6. *-commutative 56.6%

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

   7. Simplified 56.6%

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

      1. associate-*r* 56.7%

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

      2. *-commutative 56.7%

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

      3. add-sqr-sqrt 0.3%

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

      4. sqrt-unprod 3.4%

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

      5. *-commutative 3.4%

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

      6. associate-*r* 3.4%

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

      7. *-commutative 3.4%

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

      8. associate-*r* 3.4%

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

      9. swap-sqr 3.4%

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

      10. rem-square-sqrt 3.4%

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

      11. *-commutative 3.4%

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

      12. *-commutative 3.4%

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

      13. swap-sqr 3.4%

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

      14. add-sqr-sqrt 3.4%

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

      15. unpow2 3.4%

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

   9. Applied egg-rr 3.4%

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

   10. Taylor expanded in x around inf 2.0%

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

       1. associate-*r* 2.0%

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

       2. unpow2 2.0%

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

       3. associate-*l* 2.0%

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

       4. *-commutative 2.0%

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

       5. associate-*l* 2.0%

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

       6. unpow2 2.0%

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

   12. Simplified 2.0%

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

       1. associate-*r* 2.0%

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

       2. sqrt-prod 2.0%

          \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{{x}^{2}}} \]

       3. *-commutative 2.0%

          \[\leadsto \sqrt{\color{blue}{0.5 \cdot z}} \cdot \sqrt{{x}^{2}} \]

       4. unpow2 2.0%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \sqrt{\color{blue}{x \cdot x}} \]

       5. sqrt-prod 0.0%

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

       6. add-sqr-sqrt 47.4%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \color{blue}{x} \]

   14. Applied egg-rr 47.4%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot x} \]

   if -1.6e14 < x < 1e-62

   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. *-commutative 98.1%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 47.6%

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

      1. *-commutative 47.6%

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

      2. *-commutative 47.6%

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

      3. fma-neg 47.6%

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

      4. associate-*l* 47.6%

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

      5. fma-neg 47.6%

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

      6. *-commutative 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in x around 0 40.7%

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

      1. mul-1-neg 40.7%

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

      2. distribute-lft-neg-out 40.7%

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

      3. *-commutative 40.7%

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

   10. Simplified 40.7%

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

       1. associate-*r* 40.8%

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

       2. distribute-rgt-neg-out 40.8%

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

       3. add-sqr-sqrt 24.7%

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

       4. sqrt-unprod 27.8%

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

       5. sqr-neg 27.8%

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

       6. sqrt-unprod 0.7%

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

       7. add-sqr-sqrt 1.4%

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

       8. *-commutative 1.4%

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

       9. add-sqr-sqrt 0.7%

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

       10. sqrt-unprod 27.8%

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

       11. sqr-neg 27.8%

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

       12. sqrt-unprod 24.7%

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

       13. add-sqr-sqrt 40.8%

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

       14. sqrt-unprod 40.9%

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

       15. *-commutative 40.9%

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

   12. Applied egg-rr 40.9%

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

       1. distribute-rgt-neg-in 40.9%

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

       2. *-commutative 40.9%

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

   14. Simplified 40.9%

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

   if 1e-62 < x

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 55.8%

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

      1. *-commutative 55.8%

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

      2. *-commutative 55.8%

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

      3. fma-neg 55.8%

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

      4. associate-*l* 55.7%

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

      5. fma-neg 55.7%

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

      6. *-commutative 55.7%

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

   7. Simplified 55.7%

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

      1. associate-*r* 55.8%

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

      2. *-commutative 55.8%

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

      3. add-sqr-sqrt 46.2%

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

      4. sqrt-unprod 58.7%

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

      5. *-commutative 58.7%

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

      6. associate-*r* 58.7%

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

      7. *-commutative 58.7%

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

      8. associate-*r* 58.7%

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

      9. swap-sqr 58.6%

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

      10. rem-square-sqrt 58.8%

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

      11. *-commutative 58.8%

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

      12. *-commutative 58.8%

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

      13. swap-sqr 61.3%

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

      14. add-sqr-sqrt 61.4%

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

      15. unpow2 61.4%

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

   9. Applied egg-rr 61.4%

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

   10. Taylor expanded in x around inf 52.9%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(0.25 \cdot \left({x}^{2} \cdot z\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{elif}\;x \leq 10^{-62}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.25 \cdot \left(z \cdot {x}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 520000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, -y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.25 \cdot \left(z \cdot {x}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot \left(y - x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 520000000.0)
   (* (fma 0.5 x (- y)) (sqrt (* z 2.0)))
   (if (<= t 1.4e+239)
     (sqrt (* 2.0 (* 0.25 (* z (pow x 2.0)))))
     (sqrt (* 2.0 (* z (* y (- y x))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 520000000.0) {
		tmp = fma(0.5, x, -y) * sqrt((z * 2.0));
	} else if (t <= 1.4e+239) {
		tmp = sqrt((2.0 * (0.25 * (z * pow(x, 2.0)))));
	} else {
		tmp = sqrt((2.0 * (z * (y * (y - x)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 520000000.0)
		tmp = Float64(fma(0.5, x, Float64(-y)) * sqrt(Float64(z * 2.0)));
	elseif (t <= 1.4e+239)
		tmp = sqrt(Float64(2.0 * Float64(0.25 * Float64(z * (x ^ 2.0)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(z * Float64(y * Float64(y - x)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 520000000.0], N[(N[(0.5 * x + (-y)), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e+239], N[Sqrt[N[(2.0 * N[(0.25 * N[(z * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(z * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.2e8

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. *-commutative 64.1%

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

      3. fma-neg 64.1%

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

      4. associate-*l* 64.0%

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

      5. fma-neg 64.0%

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

      6. *-commutative 64.0%

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

   7. Simplified 64.0%

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

      1. *-commutative 64.0%

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

      2. associate-*r* 64.1%

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

      3. sub-neg 64.1%

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

      4. distribute-lft-in 64.1%

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

      5. *-commutative 64.1%

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

      6. associate-*l* 64.1%

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

      7. *-commutative 64.1%

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

      8. associate-*l* 64.1%

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

      9. *-commutative 64.1%

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

      10. sqrt-unprod 64.2%

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

      11. *-commutative 64.2%

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

      12. sqrt-unprod 64.3%

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

   9. Applied egg-rr 64.3%

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

       1. *-commutative 64.3%

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

       2. *-commutative 64.3%

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

       3. associate-*r* 64.3%

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

       4. *-commutative 64.3%

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

       5. distribute-rgt-in 64.3%

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

       6. fma-udef 64.3%

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

       7. *-commutative 64.3%

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

       8. *-commutative 64.3%

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

   11. Simplified 64.3%

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

   if 5.2e8 < t < 1.40000000000000001e239

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 20.2%

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

      1. *-commutative 20.2%

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

      2. *-commutative 20.2%

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

      3. fma-neg 20.2%

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

      4. associate-*l* 20.2%

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

      5. fma-neg 20.2%

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

      6. *-commutative 20.2%

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

   7. Simplified 20.2%

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

      1. associate-*r* 20.2%

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

      2. *-commutative 20.2%

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

      3. add-sqr-sqrt 8.3%

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

      4. sqrt-unprod 24.9%

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

      5. *-commutative 24.9%

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

      6. associate-*r* 24.9%

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

      7. *-commutative 24.9%

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

      8. associate-*r* 24.9%

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

      9. swap-sqr 24.9%

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

      10. rem-square-sqrt 24.9%

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

      11. *-commutative 24.9%

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

      12. *-commutative 24.9%

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

      13. swap-sqr 26.8%

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

      14. add-sqr-sqrt 26.8%

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

      15. unpow2 26.8%

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

   9. Applied egg-rr 26.8%

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

   10. Taylor expanded in x around inf 23.0%

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

   if 1.40000000000000001e239 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 11.0%

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

      1. *-commutative 11.0%

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

      2. *-commutative 11.0%

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

      3. fma-neg 11.0%

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

      4. associate-*l* 11.0%

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

      5. fma-neg 11.0%

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

      6. *-commutative 11.0%

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

   7. Simplified 11.0%

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

      1. associate-*r* 11.0%

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

      2. *-commutative 11.0%

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

      3. add-sqr-sqrt 1.6%

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

      4. sqrt-unprod 8.1%

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

      5. *-commutative 8.1%

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

      6. associate-*r* 8.1%

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

      7. *-commutative 8.1%

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

      8. associate-*r* 8.1%

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

      9. swap-sqr 8.1%

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

      10. rem-square-sqrt 8.1%

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

      11. *-commutative 8.1%

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

      12. *-commutative 8.1%

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

      13. swap-sqr 8.0%

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

      14. add-sqr-sqrt 8.0%

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

      15. unpow2 8.0%

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

   9. Applied egg-rr 8.0%

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

   10. Taylor expanded in x around 0 7.7%

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

       1. +-commutative 7.7%

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

       2. unpow2 7.7%

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

       3. associate-*r* 7.7%

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

       4. distribute-rgt-out 7.7%

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

       5. mul-1-neg 7.7%

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

   12. Simplified 7.7%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 520000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, -y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.25 \cdot \left(z \cdot {x}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot \left(y - x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -45000000000000:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-62}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -45000000000000.0)
   (* x (sqrt (* 0.5 z)))
   (if (<= x 1.25e-62)
     (* y (- (sqrt (* z 2.0))))
     (sqrt (* z (* 0.5 (pow x 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -45000000000000.0) {
		tmp = x * sqrt((0.5 * z));
	} else if (x <= 1.25e-62) {
		tmp = y * -sqrt((z * 2.0));
	} else {
		tmp = sqrt((z * (0.5 * pow(x, 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 <= (-45000000000000.0d0)) then
        tmp = x * sqrt((0.5d0 * z))
    else if (x <= 1.25d-62) then
        tmp = y * -sqrt((z * 2.0d0))
    else
        tmp = sqrt((z * (0.5d0 * (x ** 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 <= -45000000000000.0) {
		tmp = x * Math.sqrt((0.5 * z));
	} else if (x <= 1.25e-62) {
		tmp = y * -Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt((z * (0.5 * Math.pow(x, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -45000000000000.0:
		tmp = x * math.sqrt((0.5 * z))
	elif x <= 1.25e-62:
		tmp = y * -math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt((z * (0.5 * math.pow(x, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -45000000000000.0)
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	elseif (x <= 1.25e-62)
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	else
		tmp = sqrt(Float64(z * Float64(0.5 * (x ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -45000000000000.0)
		tmp = x * sqrt((0.5 * z));
	elseif (x <= 1.25e-62)
		tmp = y * -sqrt((z * 2.0));
	else
		tmp = sqrt((z * (0.5 * (x ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -45000000000000.0], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e-62], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(z * N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5e13

   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. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. *-commutative 56.7%

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

      3. fma-neg 56.7%

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

      4. associate-*l* 56.6%

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

      5. fma-neg 56.6%

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

      6. *-commutative 56.6%

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

   7. Simplified 56.6%

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

      1. associate-*r* 56.7%

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

      2. *-commutative 56.7%

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

      3. add-sqr-sqrt 0.3%

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

      4. sqrt-unprod 3.4%

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

      5. *-commutative 3.4%

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

      6. associate-*r* 3.4%

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

      7. *-commutative 3.4%

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

      8. associate-*r* 3.4%

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

      9. swap-sqr 3.4%

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

      10. rem-square-sqrt 3.4%

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

      11. *-commutative 3.4%

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

      12. *-commutative 3.4%

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

      13. swap-sqr 3.4%

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

      14. add-sqr-sqrt 3.4%

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

      15. unpow2 3.4%

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

   9. Applied egg-rr 3.4%

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

   10. Taylor expanded in x around inf 2.0%

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

       1. associate-*r* 2.0%

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

       2. unpow2 2.0%

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

       3. associate-*l* 2.0%

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

       4. *-commutative 2.0%

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

       5. associate-*l* 2.0%

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

       6. unpow2 2.0%

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

   12. Simplified 2.0%

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

       1. associate-*r* 2.0%

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

       2. sqrt-prod 2.0%

          \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{{x}^{2}}} \]

       3. *-commutative 2.0%

          \[\leadsto \sqrt{\color{blue}{0.5 \cdot z}} \cdot \sqrt{{x}^{2}} \]

       4. unpow2 2.0%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \sqrt{\color{blue}{x \cdot x}} \]

       5. sqrt-prod 0.0%

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

       6. add-sqr-sqrt 47.4%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \color{blue}{x} \]

   14. Applied egg-rr 47.4%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot x} \]

   if -4.5e13 < x < 1.25e-62

   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. *-commutative 98.1%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 47.6%

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

      1. *-commutative 47.6%

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

      2. *-commutative 47.6%

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

      3. fma-neg 47.6%

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

      4. associate-*l* 47.6%

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

      5. fma-neg 47.6%

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

      6. *-commutative 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in x around 0 40.7%

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

      1. mul-1-neg 40.7%

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

      2. distribute-lft-neg-out 40.7%

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

      3. *-commutative 40.7%

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

   10. Simplified 40.7%

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

       1. associate-*r* 40.8%

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

       2. distribute-rgt-neg-out 40.8%

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

       3. add-sqr-sqrt 24.7%

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

       4. sqrt-unprod 27.8%

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

       5. sqr-neg 27.8%

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

       6. sqrt-unprod 0.7%

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

       7. add-sqr-sqrt 1.4%

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

       8. *-commutative 1.4%

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

       9. add-sqr-sqrt 0.7%

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

       10. sqrt-unprod 27.8%

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

       11. sqr-neg 27.8%

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

       12. sqrt-unprod 24.7%

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

       13. add-sqr-sqrt 40.8%

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

       14. sqrt-unprod 40.9%

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

       15. *-commutative 40.9%

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

   12. Applied egg-rr 40.9%

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

       1. distribute-rgt-neg-in 40.9%

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

       2. *-commutative 40.9%

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

   14. Simplified 40.9%

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

   if 1.25e-62 < x

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 55.8%

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

      1. *-commutative 55.8%

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

      2. *-commutative 55.8%

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

      3. fma-neg 55.8%

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

      4. associate-*l* 55.7%

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

      5. fma-neg 55.7%

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

      6. *-commutative 55.7%

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

   7. Simplified 55.7%

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

      1. associate-*r* 55.8%

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

      2. *-commutative 55.8%

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

      3. add-sqr-sqrt 46.2%

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

      4. sqrt-unprod 58.7%

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

      5. *-commutative 58.7%

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

      6. associate-*r* 58.7%

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

      7. *-commutative 58.7%

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

      8. associate-*r* 58.7%

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

      9. swap-sqr 58.6%

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

      10. rem-square-sqrt 58.8%

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

      11. *-commutative 58.8%

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

      12. *-commutative 58.8%

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

      13. swap-sqr 61.3%

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

      14. add-sqr-sqrt 61.4%

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

      15. unpow2 61.4%

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

   9. Applied egg-rr 61.4%

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

   10. Taylor expanded in x around inf 52.9%

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

       1. associate-*r* 52.9%

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

       2. unpow2 52.9%

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

       3. associate-*l* 52.9%

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

       4. *-commutative 52.9%

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

       5. associate-*l* 52.9%

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

       6. unpow2 52.9%

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

   12. Simplified 52.9%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -45000000000000:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-62}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 150000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, -y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(0.5 \cdot x - y\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 150000.0)
   (* (fma 0.5 x (- y)) (sqrt (* z 2.0)))
   (sqrt (* (* z 2.0) (pow (- (* 0.5 x) y) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 150000.0) {
		tmp = fma(0.5, x, -y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt(((z * 2.0) * pow(((0.5 * x) - y), 2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 150000.0)
		tmp = Float64(fma(0.5, x, Float64(-y)) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * (Float64(Float64(0.5 * x) - y) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 150000.0], N[(N[(0.5 * x + (-y)), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e5

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. *-commutative 64.4%

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

      3. fma-neg 64.4%

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

      4. associate-*l* 64.4%

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

      5. fma-neg 64.4%

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

      6. *-commutative 64.4%

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

   7. Simplified 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-*r* 64.4%

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

      3. sub-neg 64.4%

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

      4. distribute-lft-in 64.4%

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

      5. *-commutative 64.4%

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

      6. associate-*l* 64.4%

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

      7. *-commutative 64.4%

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

      8. associate-*l* 64.4%

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

      9. *-commutative 64.4%

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

      10. sqrt-unprod 64.5%

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

      11. *-commutative 64.5%

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

      12. sqrt-unprod 64.6%

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

   9. Applied egg-rr 64.6%

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

       1. *-commutative 64.6%

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

       2. *-commutative 64.6%

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

       3. associate-*r* 64.6%

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

       4. *-commutative 64.6%

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

       5. distribute-rgt-in 64.6%

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

       6. fma-udef 64.6%

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

       7. *-commutative 64.6%

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

       8. *-commutative 64.6%

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

   11. Simplified 64.6%

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

   if 1.5e5 < t

   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. *-commutative 98.5%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 17.9%

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

      1. *-commutative 17.9%

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

      2. *-commutative 17.9%

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

      3. fma-neg 17.9%

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

      4. associate-*l* 17.9%

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

      5. fma-neg 17.9%

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

      6. *-commutative 17.9%

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

   7. Simplified 17.9%

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

      1. associate-*r* 17.9%

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

      2. *-commutative 17.9%

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

      3. add-sqr-sqrt 6.7%

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

      4. sqrt-unprod 20.8%

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

      5. *-commutative 20.8%

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

      6. associate-*r* 20.8%

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

      7. *-commutative 20.8%

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

      8. associate-*r* 20.8%

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

      9. swap-sqr 20.8%

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

      10. rem-square-sqrt 20.8%

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

      11. *-commutative 20.8%

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

      12. *-commutative 20.8%

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

      13. swap-sqr 23.6%

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

      14. add-sqr-sqrt 23.6%

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

      15. unpow2 23.6%

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

   9. Applied egg-rr 23.6%

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

   10. Taylor expanded in z around 0 23.6%

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

       1. associate-*r* 23.6%

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

       2. *-commutative 23.6%

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

       3. *-commutative 23.6%

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

       4. *-commutative 23.6%

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

   12. Simplified 23.6%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 150000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, -y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(0.5 \cdot x - y\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot \left(y - x\right)\right)\right)}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-96} \lor \neg \left(y \leq 5.3 \cdot 10^{-20}\right) \land y \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.8e+56)
   (sqrt (* 2.0 (* z (* y (- y x)))))
   (if (or (<= y 1.25e-96) (and (not (<= y 5.3e-20)) (<= y 1.56e+29)))
     (* x (sqrt (* 0.5 z)))
     (* y (- (sqrt (* z 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.8e+56) {
		tmp = sqrt((2.0 * (z * (y * (y - x)))));
	} else if ((y <= 1.25e-96) || (!(y <= 5.3e-20) && (y <= 1.56e+29))) {
		tmp = x * sqrt((0.5 * z));
	} else {
		tmp = y * -sqrt((z * 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 (y <= (-4.8d+56)) then
        tmp = sqrt((2.0d0 * (z * (y * (y - x)))))
    else if ((y <= 1.25d-96) .or. (.not. (y <= 5.3d-20)) .and. (y <= 1.56d+29)) then
        tmp = x * sqrt((0.5d0 * z))
    else
        tmp = y * -sqrt((z * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.8e+56) {
		tmp = Math.sqrt((2.0 * (z * (y * (y - x)))));
	} else if ((y <= 1.25e-96) || (!(y <= 5.3e-20) && (y <= 1.56e+29))) {
		tmp = x * Math.sqrt((0.5 * z));
	} else {
		tmp = y * -Math.sqrt((z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -4.8e+56:
		tmp = math.sqrt((2.0 * (z * (y * (y - x)))))
	elif (y <= 1.25e-96) or (not (y <= 5.3e-20) and (y <= 1.56e+29)):
		tmp = x * math.sqrt((0.5 * z))
	else:
		tmp = y * -math.sqrt((z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.8e+56)
		tmp = sqrt(Float64(2.0 * Float64(z * Float64(y * Float64(y - x)))));
	elseif ((y <= 1.25e-96) || (!(y <= 5.3e-20) && (y <= 1.56e+29)))
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	else
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -4.8e+56)
		tmp = sqrt((2.0 * (z * (y * (y - x)))));
	elseif ((y <= 1.25e-96) || (~((y <= 5.3e-20)) && (y <= 1.56e+29)))
		tmp = x * sqrt((0.5 * z));
	else
		tmp = y * -sqrt((z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.8e+56], N[Sqrt[N[(2.0 * N[(z * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[y, 1.25e-96], And[N[Not[LessEqual[y, 5.3e-20]], $MachinePrecision], LessEqual[y, 1.56e+29]]], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.80000000000000027e56

   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. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 51.0%

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

      1. *-commutative 51.0%

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

      2. *-commutative 51.0%

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

      3. fma-neg 51.0%

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

      4. associate-*l* 50.9%

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

      5. fma-neg 50.9%

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

      6. *-commutative 50.9%

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

   7. Simplified 50.9%

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

      1. associate-*r* 51.0%

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

      2. *-commutative 51.0%

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

      3. add-sqr-sqrt 47.0%

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

      4. sqrt-unprod 57.6%

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

      5. *-commutative 57.6%

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

      6. associate-*r* 57.6%

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

      7. *-commutative 57.6%

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

      8. associate-*r* 57.6%

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

      9. swap-sqr 57.6%

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

      10. rem-square-sqrt 57.7%

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

      11. *-commutative 57.7%

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

      12. *-commutative 57.7%

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

      13. swap-sqr 63.0%

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

      14. add-sqr-sqrt 63.0%

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

      15. unpow2 63.0%

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

   9. Applied egg-rr 63.0%

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

   10. Taylor expanded in x around 0 59.4%

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

       1. +-commutative 59.4%

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

       2. unpow2 59.4%

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

       3. associate-*r* 59.4%

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

       4. distribute-rgt-out 61.2%

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

       5. mul-1-neg 61.2%

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

   12. Simplified 61.2%

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

   if -4.80000000000000027e56 < y < 1.24999999999999999e-96 or 5.3000000000000002e-20 < y < 1.5599999999999999e29

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 48.8%

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

      1. *-commutative 48.8%

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

      2. *-commutative 48.8%

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

      3. fma-neg 48.8%

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

      4. associate-*l* 48.8%

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

      5. fma-neg 48.8%

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

      6. *-commutative 48.8%

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

   7. Simplified 48.8%

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

      1. associate-*r* 48.8%

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

      2. *-commutative 48.8%

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

      3. add-sqr-sqrt 23.1%

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

      4. sqrt-unprod 24.4%

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

      5. *-commutative 24.4%

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

      6. associate-*r* 24.4%

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

      7. *-commutative 24.4%

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

      8. associate-*r* 24.4%

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

      9. swap-sqr 24.3%

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

      10. rem-square-sqrt 24.5%

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

      11. *-commutative 24.5%

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

      12. *-commutative 24.5%

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

      13. swap-sqr 22.8%

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

      14. add-sqr-sqrt 22.9%

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

      15. unpow2 22.9%

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

   9. Applied egg-rr 22.9%

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

   10. Taylor expanded in x around inf 20.3%

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

       1. associate-*r* 20.3%

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

       2. unpow2 20.3%

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

       3. associate-*l* 20.3%

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

       4. *-commutative 20.3%

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

       5. associate-*l* 20.3%

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

       6. unpow2 20.3%

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

   12. Simplified 20.3%

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

       1. associate-*r* 20.3%

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

       2. sqrt-prod 18.7%

          \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{{x}^{2}}} \]

       3. *-commutative 18.7%

          \[\leadsto \sqrt{\color{blue}{0.5 \cdot z}} \cdot \sqrt{{x}^{2}} \]

       4. unpow2 18.7%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \sqrt{\color{blue}{x \cdot x}} \]

       5. sqrt-prod 19.7%

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

       6. add-sqr-sqrt 43.3%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \color{blue}{x} \]

   14. Applied egg-rr 43.3%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot x} \]

   if 1.24999999999999999e-96 < y < 5.3000000000000002e-20 or 1.5599999999999999e29 < y

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 58.7%

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

      1. *-commutative 58.7%

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

      2. *-commutative 58.7%

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

      3. fma-neg 58.7%

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

      4. associate-*l* 58.6%

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

      5. fma-neg 58.6%

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

      6. *-commutative 58.6%

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

   7. Simplified 58.6%

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

   8. Taylor expanded in x around 0 48.1%

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

      1. mul-1-neg 48.1%

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

      2. distribute-lft-neg-out 48.1%

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

      3. *-commutative 48.1%

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

   10. Simplified 48.1%

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

       1. associate-*r* 48.1%

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

       2. distribute-rgt-neg-out 48.1%

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

       3. add-sqr-sqrt 48.1%

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

       4. sqrt-unprod 54.6%

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

       5. sqr-neg 54.6%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 1.1%

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

       8. *-commutative 1.1%

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

       9. add-sqr-sqrt 0.0%

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

       10. sqrt-unprod 54.6%

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

       11. sqr-neg 54.6%

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

       12. sqrt-unprod 48.1%

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

       13. add-sqr-sqrt 48.1%

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

       14. sqrt-unprod 48.2%

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

       15. *-commutative 48.2%

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

   12. Applied egg-rr 48.2%

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

       1. distribute-rgt-neg-in 48.2%

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

       2. *-commutative 48.2%

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

   14. Simplified 48.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot \left(y - x\right)\right)\right)}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-96} \lor \neg \left(y \leq 5.3 \cdot 10^{-20}\right) \land y \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10000000000000 \lor \neg \left(x \leq 3.7 \cdot 10^{+146}\right):\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -10000000000000.0) (not (<= x 3.7e+146)))
   (* x (sqrt (* 0.5 z)))
   (* y (- (sqrt (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -10000000000000.0) || !(x <= 3.7e+146)) {
		tmp = x * sqrt((0.5 * z));
	} else {
		tmp = y * -sqrt((z * 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 <= (-10000000000000.0d0)) .or. (.not. (x <= 3.7d+146))) then
        tmp = x * sqrt((0.5d0 * z))
    else
        tmp = y * -sqrt((z * 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 <= -10000000000000.0) || !(x <= 3.7e+146)) {
		tmp = x * Math.sqrt((0.5 * z));
	} else {
		tmp = y * -Math.sqrt((z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -10000000000000.0) or not (x <= 3.7e+146):
		tmp = x * math.sqrt((0.5 * z))
	else:
		tmp = y * -math.sqrt((z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -10000000000000.0) || !(x <= 3.7e+146))
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	else
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -10000000000000.0) || ~((x <= 3.7e+146)))
		tmp = x * sqrt((0.5 * z));
	else
		tmp = y * -sqrt((z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -10000000000000.0], N[Not[LessEqual[x, 3.7e+146]], $MachinePrecision]], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e13 or 3.70000000000000004e146 < x

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 58.7%

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

      1. *-commutative 58.7%

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

      2. *-commutative 58.7%

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

      3. fma-neg 58.7%

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

      4. associate-*l* 58.6%

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

      5. fma-neg 58.6%

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

      6. *-commutative 58.6%

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

   7. Simplified 58.6%

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

      1. associate-*r* 58.7%

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

      2. *-commutative 58.7%

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

      3. add-sqr-sqrt 19.4%

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

      4. sqrt-unprod 24.2%

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

      5. *-commutative 24.2%

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

      6. associate-*r* 24.2%

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

      7. *-commutative 24.2%

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

      8. associate-*r* 24.2%

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

      9. swap-sqr 24.2%

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

      10. rem-square-sqrt 24.2%

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

      11. *-commutative 24.2%

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

      12. *-commutative 24.2%

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

      13. swap-sqr 25.1%

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

      14. add-sqr-sqrt 25.1%

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

      15. unpow2 25.1%

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

   9. Applied egg-rr 25.1%

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

   10. Taylor expanded in x around inf 24.2%

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

       1. associate-*r* 24.2%

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

       2. unpow2 24.2%

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

       3. associate-*l* 24.2%

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

       4. *-commutative 24.2%

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

       5. associate-*l* 24.2%

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

       6. unpow2 24.2%

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

   12. Simplified 24.2%

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

       1. associate-*r* 24.2%

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

       2. sqrt-prod 25.1%

          \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{{x}^{2}}} \]

       3. *-commutative 25.1%

          \[\leadsto \sqrt{\color{blue}{0.5 \cdot z}} \cdot \sqrt{{x}^{2}} \]

       4. unpow2 25.1%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \sqrt{\color{blue}{x \cdot x}} \]

       5. sqrt-prod 19.2%

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

       6. add-sqr-sqrt 50.6%

          \[\leadsto \sqrt{0.5 \cdot z} \cdot \color{blue}{x} \]

   14. Applied egg-rr 50.6%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot x} \]

   if -1e13 < x < 3.70000000000000004e146

   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. *-commutative 98.5%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 48.2%

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

      1. *-commutative 48.2%

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

      2. *-commutative 48.2%

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

      3. fma-neg 48.2%

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

      4. associate-*l* 48.2%

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

      5. fma-neg 48.2%

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

      6. *-commutative 48.2%

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

   7. Simplified 48.2%

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

   8. Taylor expanded in x around 0 37.2%

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

      1. mul-1-neg 37.2%

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

      2. distribute-lft-neg-out 37.2%

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

      3. *-commutative 37.2%

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

   10. Simplified 37.2%

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

       1. associate-*r* 37.2%

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

       2. distribute-rgt-neg-out 37.2%

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

       3. add-sqr-sqrt 21.6%

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

       4. sqrt-unprod 24.1%

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

       5. sqr-neg 24.1%

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

       6. sqrt-unprod 0.6%

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

       7. add-sqr-sqrt 1.6%

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

       8. *-commutative 1.6%

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

       9. add-sqr-sqrt 0.6%

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

       10. sqrt-unprod 24.1%

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

       11. sqr-neg 24.1%

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

       12. sqrt-unprod 21.6%

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

       13. add-sqr-sqrt 37.2%

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

       14. sqrt-unprod 37.4%

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

       15. *-commutative 37.4%

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

   12. Applied egg-rr 37.4%

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

       1. distribute-rgt-neg-in 37.4%

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

       2. *-commutative 37.4%

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

   14. Simplified 37.4%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000000000 \lor \neg \left(x \leq 3.7 \cdot 10^{+146}\right):\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 2.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*l* 99.9%

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

   3. exp-sqrt 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in t around 0 52.4%

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

   1. *-commutative 52.4%

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

   2. *-commutative 52.4%

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

   3. fma-neg 52.4%

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

   4. associate-*l* 52.4%

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

   5. fma-neg 52.4%

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

   6. *-commutative 52.4%

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

7. Simplified 52.4%

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

8. Taylor expanded in x around 0 27.0%

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

   1. mul-1-neg 27.0%

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

   2. distribute-lft-neg-out 27.0%

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

   3. *-commutative 27.0%

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

10. Simplified 27.0%

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

    1. expm1-log1p-u 13.6%

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

    2. expm1-udef 9.4%

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

    3. associate-*r* 9.4%

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

    4. *-commutative 9.4%

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

    5. add-sqr-sqrt 8.8%

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

    6. sqrt-unprod 13.0%

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

    7. sqr-neg 13.0%

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

    8. sqrt-unprod 0.9%

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

    9. add-sqr-sqrt 1.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{y} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)} - 1 \]

    10. sqrt-unprod 1.4%

        \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\sqrt{2 \cdot z}}\right)} - 1 \]

    11. *-commutative 1.4%

        \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{z \cdot 2}}\right)} - 1 \]

12. Applied egg-rr 1.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \sqrt{z \cdot 2}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 1.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \sqrt{z \cdot 2}\right)\right)} \]

    2. expm1-log1p 1.7%

       \[\leadsto \color{blue}{y \cdot \sqrt{z \cdot 2}} \]

    3. *-commutative 1.7%

       \[\leadsto y \cdot \sqrt{\color{blue}{2 \cdot z}} \]

14. Simplified 1.7%

    \[\leadsto \color{blue}{y \cdot \sqrt{2 \cdot z}} \]

15. Final simplification 1.7%

    \[\leadsto y \cdot \sqrt{z \cdot 2} \]
16. Add Preprocessing

Alternative 10: 29.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \sqrt{0.5 \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (sqrt (* 0.5 z))))

C

double code(double x, double y, double z, double t) {
	return x * sqrt((0.5 * 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 * sqrt((0.5d0 * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * Math.sqrt((0.5 * z));
}

Python

def code(x, y, z, t):
	return x * math.sqrt((0.5 * z))

Julia

function code(x, y, z, t)
	return Float64(x * sqrt(Float64(0.5 * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * sqrt((0.5 * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Step-by-step derivation

   1. *-commutative 99.1%

      \[\leadsto \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   2. associate-*l* 99.9%

      \[\leadsto \color{blue}{\sqrt{z \cdot 2} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot e^{\frac{t \cdot t}{2}}\right)} \]

   3. exp-sqrt 99.9%

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{e^{t \cdot t}}}\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sqrt{z \cdot 2} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{e^{t \cdot t}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 52.4%

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 52.4%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \sqrt{z}} \]

   2. *-commutative 52.4%

      \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{x \cdot 0.5} - y\right)\right) \cdot \sqrt{z} \]

   3. fma-neg 52.4%

      \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)}\right) \cdot \sqrt{z} \]

   4. associate-*l* 52.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\mathsf{fma}\left(x, 0.5, -y\right) \cdot \sqrt{z}\right)} \]

   5. fma-neg 52.4%

      \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\left(x \cdot 0.5 - y\right)} \cdot \sqrt{z}\right) \]

   6. *-commutative 52.4%

      \[\leadsto \sqrt{2} \cdot \left(\left(\color{blue}{0.5 \cdot x} - y\right) \cdot \sqrt{z}\right) \]

7. Simplified 52.4%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right)} \]
8. Step-by-step derivation

   1. associate-*r* 52.4%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \sqrt{z}} \]

   2. *-commutative 52.4%

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)} \]

   3. add-sqr-sqrt 22.1%

      \[\leadsto \color{blue}{\sqrt{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)} \cdot \sqrt{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)}} \]

   4. sqrt-unprod 27.0%

      \[\leadsto \color{blue}{\sqrt{\left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)\right) \cdot \left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)\right)}} \]

   5. *-commutative 27.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \sqrt{z}\right)} \cdot \left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)\right)} \]

   6. associate-*r* 27.0%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right)\right)} \cdot \left(\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)\right)} \]

   7. *-commutative 27.0%

      \[\leadsto \sqrt{\left(\sqrt{2} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right)\right) \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \sqrt{z}\right)}} \]

   8. associate-*r* 27.0%

      \[\leadsto \sqrt{\left(\sqrt{2} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right)\right)}} \]

   9. swap-sqr 27.0%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(\left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right) \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right)\right)}} \]

   10. rem-square-sqrt 27.1%

       \[\leadsto \sqrt{\color{blue}{2} \cdot \left(\left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right) \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right)\right)} \]

   11. *-commutative 27.1%

       \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt{z} \cdot \left(0.5 \cdot x - y\right)\right)} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{z}\right)\right)} \]

   12. *-commutative 27.1%

       \[\leadsto \sqrt{2 \cdot \left(\left(\sqrt{z} \cdot \left(0.5 \cdot x - y\right)\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \left(0.5 \cdot x - y\right)\right)}\right)} \]

   13. swap-sqr 27.4%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)\right)}} \]

   14. add-sqr-sqrt 27.4%

       \[\leadsto \sqrt{2 \cdot \left(\color{blue}{z} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)\right)} \]

   15. unpow2 27.4%

       \[\leadsto \sqrt{2 \cdot \left(z \cdot \color{blue}{{\left(0.5 \cdot x - y\right)}^{2}}\right)} \]

9. Applied egg-rr 27.4%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(z \cdot {\left(\mathsf{fma}\left(0.5, x, -y\right)\right)}^{2}\right)}} \]

10. Taylor expanded in x around inf 17.9%

    \[\leadsto \sqrt{\color{blue}{0.5 \cdot \left({x}^{2} \cdot z\right)}} \]
11. Step-by-step derivation

    1. associate-*r* 17.9%

       \[\leadsto \sqrt{\color{blue}{\left(0.5 \cdot {x}^{2}\right) \cdot z}} \]

    2. unpow2 17.9%

       \[\leadsto \sqrt{\left(0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot z} \]

    3. associate-*l* 17.9%

       \[\leadsto \sqrt{\color{blue}{\left(\left(0.5 \cdot x\right) \cdot x\right)} \cdot z} \]

    4. *-commutative 17.9%

       \[\leadsto \sqrt{\color{blue}{z \cdot \left(\left(0.5 \cdot x\right) \cdot x\right)}} \]

    5. associate-*l* 17.9%

       \[\leadsto \sqrt{z \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot x\right)\right)}} \]

    6. unpow2 17.9%

       \[\leadsto \sqrt{z \cdot \left(0.5 \cdot \color{blue}{{x}^{2}}\right)} \]

12. Simplified 17.9%

    \[\leadsto \sqrt{\color{blue}{z \cdot \left(0.5 \cdot {x}^{2}\right)}} \]
13. Step-by-step derivation

    1. associate-*r* 17.9%

       \[\leadsto \sqrt{\color{blue}{\left(z \cdot 0.5\right) \cdot {x}^{2}}} \]

    2. sqrt-prod 15.6%

       \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{{x}^{2}}} \]

    3. *-commutative 15.6%

       \[\leadsto \sqrt{\color{blue}{0.5 \cdot z}} \cdot \sqrt{{x}^{2}} \]

    4. unpow2 15.6%

       \[\leadsto \sqrt{0.5 \cdot z} \cdot \sqrt{\color{blue}{x \cdot x}} \]

    5. sqrt-prod 13.1%

       \[\leadsto \sqrt{0.5 \cdot z} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

    6. add-sqr-sqrt 27.6%

       \[\leadsto \sqrt{0.5 \cdot z} \cdot \color{blue}{x} \]

14. Applied egg-rr 27.6%

    \[\leadsto \color{blue}{\sqrt{0.5 \cdot z} \cdot x} \]

15. Final simplification 27.6%

    \[\leadsto x \cdot \sqrt{0.5 \cdot z} \]
16. Add Preprocessing

Developer target: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(exp(1.0), ((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (exp(1.0d0) ** ((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.pow(Math.exp(1.0), ((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.exp(1.0), ((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * (exp(1.0) ^ Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(1.0) ^ ((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[1.0], $MachinePrecision], N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024026 
(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))))