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

Percentage Accurate: 99.4% → 99.8%

Time: 14.9s

Alternatives: 13

Speedup: 1.0×

• 43.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[N[Exp[N[(2.0 * t), $MachinePrecision]], $MachinePrecision], N[(0.5 * t), $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. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

8. Simplified 99.8%

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

   1. pow2 99.8%

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

   2. pow-exp 99.8%

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

   3. add-sqr-sqrt 99.8%

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

   4. unpow-prod-down 99.8%

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

   5. pow1/2 99.8%

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

   6. pow-unpow 99.8%

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

   7. *-commutative 99.8%

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

   8. pow1/2 99.8%

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

   9. pow-unpow 99.8%

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

   10. *-commutative 99.8%

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

   11. pow-prod-down 99.8%

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

   12. pow2 99.8%

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

   13. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. *-commutative 99.8%

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

12. Simplified 99.8%

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

    1. add-exp-log 99.8%

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

    2. log-pow 99.8%

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

    3. add-log-exp 99.8%

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

14. Applied egg-rr 99.8%

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

15. Final simplification 99.8%

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

Alternative 2: 59.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+136}:\\ \;\;\;\;t\_1 \cdot \left(y \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\right)\\ \mathbf{elif}\;t \leq 1.76 \cdot 10^{+174}:\\ \;\;\;\;y \cdot \sqrt{\left(z \cdot 2\right) \cdot {\left(t + 1\right)}^{t}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 1.3e+15)
     (* (- (* x 0.5) y) t_1)
     (if (<= t 1.02e+65)
       (sqrt (* (* z 2.0) (* y (- y x))))
       (if (<= t 1.05e+136)
         (* t_1 (* y (+ (* 0.5 (/ x y)) -1.0)))
         (if (<= t 1.76e+174)
           (* y (sqrt (* (* z 2.0) (pow (+ t 1.0) t))))
           (* t_1 (* x (- 0.5 (/ y x))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 1.3e+15) {
		tmp = ((x * 0.5) - y) * t_1;
	} else if (t <= 1.02e+65) {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	} else if (t <= 1.05e+136) {
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	} else if (t <= 1.76e+174) {
		tmp = y * sqrt(((z * 2.0) * pow((t + 1.0), t)));
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t <= 1.3d+15) then
        tmp = ((x * 0.5d0) - y) * t_1
    else if (t <= 1.02d+65) then
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    else if (t <= 1.05d+136) then
        tmp = t_1 * (y * ((0.5d0 * (x / y)) + (-1.0d0)))
    else if (t <= 1.76d+174) then
        tmp = y * sqrt(((z * 2.0d0) * ((t + 1.0d0) ** t)))
    else
        tmp = t_1 * (x * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 1.3e+15) {
		tmp = ((x * 0.5) - y) * t_1;
	} else if (t <= 1.02e+65) {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	} else if (t <= 1.05e+136) {
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	} else if (t <= 1.76e+174) {
		tmp = y * Math.sqrt(((z * 2.0) * Math.pow((t + 1.0), t)));
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 1.3e+15:
		tmp = ((x * 0.5) - y) * t_1
	elif t <= 1.02e+65:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	elif t <= 1.05e+136:
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0))
	elif t <= 1.76e+174:
		tmp = y * math.sqrt(((z * 2.0) * math.pow((t + 1.0), t)))
	else:
		tmp = t_1 * (x * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 1.3e+15)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	elseif (t <= 1.02e+65)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	elseif (t <= 1.05e+136)
		tmp = Float64(t_1 * Float64(y * Float64(Float64(0.5 * Float64(x / y)) + -1.0)));
	elseif (t <= 1.76e+174)
		tmp = Float64(y * sqrt(Float64(Float64(z * 2.0) * (Float64(t + 1.0) ^ t))));
	else
		tmp = Float64(t_1 * Float64(x * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 1.3e+15)
		tmp = ((x * 0.5) - y) * t_1;
	elseif (t <= 1.02e+65)
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	elseif (t <= 1.05e+136)
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	elseif (t <= 1.76e+174)
		tmp = y * sqrt(((z * 2.0) * ((t + 1.0) ^ t)));
	else
		tmp = t_1 * (x * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.3e+15], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 1.02e+65], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.05e+136], N[(t$95$1 * N[(y * N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.76e+174], N[(y * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[N[(t + 1.0), $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < 1.3e15

   1. Initial program 98.9%

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

      1. associate-*l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.4%

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

   if 1.3e15 < t < 1.02000000000000005e65

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 3.8%

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

      1. add-sqr-sqrt 2.6%

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

      2. sqrt-unprod 21.4%

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

      3. *-commutative 21.4%

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

      4. *-commutative 21.4%

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

      5. swap-sqr 40.6%

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

      6. add-sqr-sqrt 40.6%

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

      7. pow2 40.6%

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

      8. fmm-def 40.6%

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

   5. Applied egg-rr 40.6%

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

      1. fmm-undef 40.6%

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

      2. *-commutative 40.6%

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

   7. Simplified 40.6%

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

   8. Taylor expanded in x around 0 30.6%

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

      1. +-commutative 30.6%

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

      2. unpow2 30.6%

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

      3. associate-*r* 30.6%

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

      4. distribute-rgt-in 40.6%

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

      5. mul-1-neg 40.6%

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

      6. unsub-neg 40.6%

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

   10. Simplified 40.6%

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

   if 1.02000000000000005e65 < t < 1.05e136

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 12.7%

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

   4. Taylor expanded in y around inf 36.1%

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

   if 1.05e136 < t < 1.76e174

   1. Initial program 100.0%

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

      1. exp-sqrt 100.0%

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

      2. pow-exp 100.0%

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

      3. pow1/2 100.0%

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

      4. pow-pow 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 50.0%

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

      1. mul-1-neg 50.0%

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

   7. Simplified 50.0%

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

   8. Taylor expanded in t around 0 50.0%

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

      1. +-commutative 50.0%

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

   10. Simplified 50.0%

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

       1. pow1 50.0%

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

       2. associate-*l* 50.0%

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

       3. add-sqr-sqrt 30.0%

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

       4. sqrt-unprod 50.0%

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

       5. sqr-neg 50.0%

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

       6. sqrt-unprod 40.0%

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

       7. add-sqr-sqrt 50.0%

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

       8. pow1/2 50.0%

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

       9. pow-unpow 50.0%

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

       10. pow-prod-down 50.0%

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

   12. Applied egg-rr 50.0%

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

       1. unpow1 50.0%

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

       2. unpow1/2 50.0%

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

   14. Simplified 50.0%

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

   if 1.76e174 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 12.0%

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

   4. Taylor expanded in x around inf 23.7%

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

      1. mul-1-neg 23.7%

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

      2. unsub-neg 23.7%

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

   6. Simplified 23.7%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(y \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\right)\\ \mathbf{elif}\;t \leq 1.76 \cdot 10^{+174}:\\ \;\;\;\;y \cdot \sqrt{\left(z \cdot 2\right) \cdot {\left(t + 1\right)}^{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \lor \neg \left(t \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{\left(z \cdot 2\right) \cdot {\left(t + 1\right)}^{t}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t 9.5) (not (<= t 1.35e+154)))
   (* (- (* x 0.5) y) (sqrt (* (* z 2.0) (fma t t 1.0))))
   (* y (- (sqrt (* (* z 2.0) (pow (+ t 1.0) t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= 9.5) || !(t <= 1.35e+154)) {
		tmp = ((x * 0.5) - y) * sqrt(((z * 2.0) * fma(t, t, 1.0)));
	} else {
		tmp = y * -sqrt(((z * 2.0) * pow((t + 1.0), t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= 9.5) || !(t <= 1.35e+154))
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(Float64(z * 2.0) * fma(t, t, 1.0))));
	else
		tmp = Float64(y * Float64(-sqrt(Float64(Float64(z * 2.0) * (Float64(t + 1.0) ^ t)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, 9.5], N[Not[LessEqual[t, 1.35e+154]], $MachinePrecision]], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * (-N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[N[(t + 1.0), $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5 or 1.35000000000000003e154 < t

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 90.7%

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

       1. +-commutative 90.7%

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

       2. unpow2 90.7%

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

       3. fma-define 90.7%

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

   11. Simplified 90.7%

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

   if 9.5 < t < 1.35000000000000003e154

   1. Initial program 94.2%

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

      1. exp-sqrt 94.2%

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

      2. pow-exp 94.3%

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

      3. pow1/2 94.3%

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

      4. pow-pow 94.3%

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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in x around 0 78.7%

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

      1. mul-1-neg 78.7%

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

   7. Simplified 78.7%

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

   8. Taylor expanded in t around 0 73.0%

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

      1. +-commutative 73.0%

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

   10. Simplified 73.0%

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

       1. associate-*l* 79.1%

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

       2. distribute-lft-neg-out 79.1%

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

       3. pow1/2 79.1%

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

       4. pow-unpow 79.1%

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

       5. pow-prod-down 79.1%

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

   12. Applied egg-rr 79.1%

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

       1. distribute-lft-neg-in 79.1%

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

       2. unpow1/2 79.1%

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

   14. Simplified 79.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \lor \neg \left(t \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{\left(z \cdot 2\right) \cdot {\left(t + 1\right)}^{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Exp[N[(2.0 * N[(t * N[(0.5 * t), $MachinePrecision]), $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. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

8. Simplified 99.8%

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

   1. pow2 99.8%

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

   2. pow-exp 99.8%

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

   3. add-sqr-sqrt 99.8%

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

   4. unpow-prod-down 99.8%

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

   5. pow1/2 99.8%

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

   6. pow-unpow 99.8%

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

   7. *-commutative 99.8%

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

   8. pow1/2 99.8%

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

   9. pow-unpow 99.8%

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

   10. *-commutative 99.8%

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

   11. pow-prod-down 99.8%

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

   12. pow2 99.8%

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

   13. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. *-commutative 99.8%

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

12. Simplified 99.8%

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

    1. add-exp-log 99.8%

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

    2. log-pow 99.8%

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

    3. add-log-exp 99.8%

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

14. Applied egg-rr 99.8%

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

    1. pow-exp 99.8%

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

    2. associate-*l* 99.8%

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

16. Applied egg-rr 99.8%

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

17. Final simplification 99.8%

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

Alternative 5: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{\left(z \cdot 2\right) \cdot {\left(t + 1\right)}^{t}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 9.5)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (* y (- (sqrt (* (* z 2.0) (pow (+ t 1.0) t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.5) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = y * -sqrt(((z * 2.0) * pow((t + 1.0), t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 9.5d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = y * -sqrt(((z * 2.0d0) * ((t + 1.0d0) ** t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.5) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = y * -Math.sqrt(((z * 2.0) * Math.pow((t + 1.0), t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 9.5:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = y * -math.sqrt(((z * 2.0) * math.pow((t + 1.0), t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 9.5)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(y * Float64(-sqrt(Float64(Float64(z * 2.0) * (Float64(t + 1.0) ^ t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 9.5)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = y * -sqrt(((z * 2.0) * ((t + 1.0) ^ t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 9.5], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * (-N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[N[(t + 1.0), $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 70.8%

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

   if 9.5 < t

   1. Initial program 97.0%

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

      1. exp-sqrt 97.0%

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

      2. pow-exp 97.1%

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

      3. pow1/2 97.1%

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

      4. pow-pow 97.1%

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in x around 0 74.5%

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

      1. mul-1-neg 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in t around 0 71.5%

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

      1. +-commutative 71.5%

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

   10. Simplified 71.5%

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

       1. associate-*l* 76.3%

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

       2. distribute-lft-neg-out 76.3%

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

       3. pow1/2 76.3%

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

       4. pow-unpow 76.3%

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

       5. pow-prod-down 76.3%

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

   12. Applied egg-rr 76.3%

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

       1. distribute-lft-neg-in 76.3%

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

       2. unpow1/2 76.3%

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

   14. Simplified 76.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{\left(z \cdot 2\right) \cdot {\left(t + 1\right)}^{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 7: 59.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+174}:\\ \;\;\;\;t\_1 \cdot \left(y \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 1.6e+14)
     (* (- (* x 0.5) y) t_1)
     (if (<= t 4.4e+64)
       (sqrt (* (* z 2.0) (* y (- y x))))
       (if (<= t 1.85e+174)
         (* t_1 (* y (+ (* 0.5 (/ x y)) -1.0)))
         (* t_1 (* x (- 0.5 (/ y x)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 1.6e+14) {
		tmp = ((x * 0.5) - y) * t_1;
	} else if (t <= 4.4e+64) {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	} else if (t <= 1.85e+174) {
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t <= 1.6d+14) then
        tmp = ((x * 0.5d0) - y) * t_1
    else if (t <= 4.4d+64) then
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    else if (t <= 1.85d+174) then
        tmp = t_1 * (y * ((0.5d0 * (x / y)) + (-1.0d0)))
    else
        tmp = t_1 * (x * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 1.6e+14) {
		tmp = ((x * 0.5) - y) * t_1;
	} else if (t <= 4.4e+64) {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	} else if (t <= 1.85e+174) {
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 1.6e+14:
		tmp = ((x * 0.5) - y) * t_1
	elif t <= 4.4e+64:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	elif t <= 1.85e+174:
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0))
	else:
		tmp = t_1 * (x * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 1.6e+14)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	elseif (t <= 4.4e+64)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	elseif (t <= 1.85e+174)
		tmp = Float64(t_1 * Float64(y * Float64(Float64(0.5 * Float64(x / y)) + -1.0)));
	else
		tmp = Float64(t_1 * Float64(x * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 1.6e+14)
		tmp = ((x * 0.5) - y) * t_1;
	elseif (t <= 4.4e+64)
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	elseif (t <= 1.85e+174)
		tmp = t_1 * (y * ((0.5 * (x / y)) + -1.0));
	else
		tmp = t_1 * (x * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.6e+14], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 4.4e+64], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.85e+174], N[(t$95$1 * N[(y * N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.6e14

   1. Initial program 98.9%

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

      1. associate-*l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.4%

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

   if 1.6e14 < t < 4.40000000000000004e64

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 3.8%

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

      1. add-sqr-sqrt 2.6%

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

      2. sqrt-unprod 21.4%

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

      3. *-commutative 21.4%

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

      4. *-commutative 21.4%

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

      5. swap-sqr 40.6%

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

      6. add-sqr-sqrt 40.6%

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

      7. pow2 40.6%

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

      8. fmm-def 40.6%

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

   5. Applied egg-rr 40.6%

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

      1. fmm-undef 40.6%

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

      2. *-commutative 40.6%

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

   7. Simplified 40.6%

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

   8. Taylor expanded in x around 0 30.6%

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

      1. +-commutative 30.6%

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

      2. unpow2 30.6%

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

      3. associate-*r* 30.6%

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

      4. distribute-rgt-in 40.6%

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

      5. mul-1-neg 40.6%

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

      6. unsub-neg 40.6%

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

   10. Simplified 40.6%

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

   if 4.40000000000000004e64 < t < 1.8500000000000001e174

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 8.8%

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

   4. Taylor expanded in y around inf 29.1%

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

   if 1.8500000000000001e174 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 12.3%

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

   4. Taylor expanded in x around inf 24.6%

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

      1. mul-1-neg 24.6%

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

      2. unsub-neg 24.6%

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

   6. Simplified 24.6%

      \[\leadsto \left(\color{blue}{\left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{+14}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(y \cdot \left(0.5 \cdot \frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{+68} \lor \neg \left(x \cdot 0.5 \leq 5 \cdot 10^{+14}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= (* x 0.5) -4e+68) (not (<= (* x 0.5) 5e+14)))
     (* (* x 0.5) t_1)
     (* y (- t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (((x * 0.5) <= -4e+68) || !((x * 0.5) <= 5e+14)) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (((x * 0.5d0) <= (-4d+68)) .or. (.not. ((x * 0.5d0) <= 5d+14))) then
        tmp = (x * 0.5d0) * t_1
    else
        tmp = y * -t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (((x * 0.5) <= -4e+68) || !((x * 0.5) <= 5e+14)) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if ((x * 0.5) <= -4e+68) or not ((x * 0.5) <= 5e+14):
		tmp = (x * 0.5) * t_1
	else:
		tmp = y * -t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((Float64(x * 0.5) <= -4e+68) || !(Float64(x * 0.5) <= 5e+14))
		tmp = Float64(Float64(x * 0.5) * t_1);
	else
		tmp = Float64(y * Float64(-t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (((x * 0.5) <= -4e+68) || ~(((x * 0.5) <= 5e+14)))
		tmp = (x * 0.5) * t_1;
	else
		tmp = y * -t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -4e+68], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 5e+14]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -3.99999999999999981e68 or 5e14 < (*.f64 x #s(literal 1/2 binary64))

   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. Add Preprocessing

   3. Taylor expanded in t around 0 61.4%

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

   4. Taylor expanded in x around inf 56.6%

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

   if -3.99999999999999981e68 < (*.f64 x #s(literal 1/2 binary64)) < 5e14

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0 52.5%

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

   4. Taylor expanded in x around 0 43.8%

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

      1. mul-1-neg 82.2%

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

   6. Simplified 43.8%

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

      1. *-rgt-identity 43.8%

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

      2. distribute-lft-neg-out 43.8%

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

      3. neg-sub0 43.8%

         \[\leadsto \color{blue}{0 - y \cdot \sqrt{z \cdot 2}} \]

   8. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{0 - y \cdot \sqrt{z \cdot 2}} \]
   9. Step-by-step derivation

      1. neg-sub0 43.8%

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

      2. *-commutative 43.8%

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

      3. distribute-rgt-neg-in 43.8%

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

      4. *-commutative 43.8%

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

   10. Simplified 43.8%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{+68} \lor \neg \left(x \cdot 0.5 \leq 5 \cdot 10^{+14}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 1.35e+15)
     (* (- (* x 0.5) y) t_1)
     (if (<= t 2.9e+41)
       (sqrt (* (* z 2.0) (* y (- y x))))
       (* t_1 (* x (- 0.5 (/ y x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 1.35e+15) {
		tmp = ((x * 0.5) - y) * t_1;
	} else if (t <= 2.9e+41) {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t <= 1.35d+15) then
        tmp = ((x * 0.5d0) - y) * t_1
    else if (t <= 2.9d+41) then
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    else
        tmp = t_1 * (x * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 1.35e+15) {
		tmp = ((x * 0.5) - y) * t_1;
	} else if (t <= 2.9e+41) {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 1.35e+15:
		tmp = ((x * 0.5) - y) * t_1
	elif t <= 2.9e+41:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	else:
		tmp = t_1 * (x * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 1.35e+15)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	elseif (t <= 2.9e+41)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	else
		tmp = Float64(t_1 * Float64(x * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 1.35e+15)
		tmp = ((x * 0.5) - y) * t_1;
	elseif (t <= 2.9e+41)
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	else
		tmp = t_1 * (x * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.35e+15], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 2.9e+41], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.35e15

   1. Initial program 98.9%

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

      1. associate-*l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.4%

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

   if 1.35e15 < t < 2.89999999999999988e41

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 3.8%

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

      1. add-sqr-sqrt 3.2%

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

      2. sqrt-unprod 26.6%

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

      3. *-commutative 26.6%

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

      4. *-commutative 26.6%

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

      5. swap-sqr 50.6%

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

      6. add-sqr-sqrt 50.6%

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

      7. pow2 50.6%

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

      8. fmm-def 50.6%

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

   5. Applied egg-rr 50.6%

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

      1. fmm-undef 50.6%

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

      2. *-commutative 50.6%

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

   7. Simplified 50.6%

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

   8. Taylor expanded in x around 0 38.1%

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

      1. +-commutative 38.1%

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

      2. unpow2 38.1%

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

      3. associate-*r* 38.1%

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

      4. distribute-rgt-in 50.6%

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

      5. mul-1-neg 50.6%

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

      6. unsub-neg 50.6%

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

   10. Simplified 50.6%

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

   if 2.89999999999999988e41 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 10.3%

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

   4. Taylor expanded in x around inf 20.1%

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

      1. mul-1-neg 20.1%

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

      2. unsub-neg 20.1%

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

   6. Simplified 20.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.26e+15)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (sqrt (* (* z 2.0) (* y (- y x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.26e+15) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.26d+15) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.26e+15) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.26e+15:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.26e+15)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.26e+15)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.26e+15], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.26e15

   1. Initial program 98.9%

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

      1. associate-*l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 69.4%

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

   if 1.26e15 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 9.4%

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

      1. add-sqr-sqrt 5.8%

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

      2. sqrt-unprod 22.7%

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

      3. *-commutative 22.7%

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

      4. *-commutative 22.7%

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

      5. swap-sqr 31.3%

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

      6. add-sqr-sqrt 31.3%

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

      7. pow2 31.3%

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

      8. fmm-def 31.3%

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

   5. Applied egg-rr 31.3%

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

      1. fmm-undef 31.3%

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

      2. *-commutative 31.3%

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

   7. Simplified 31.3%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. +-commutative 18.8%

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

      2. unpow2 18.8%

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

      3. associate-*r* 18.8%

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

      4. distribute-rgt-in 22.4%

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

      5. mul-1-neg 22.4%

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

      6. unsub-neg 22.4%

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

   10. Simplified 22.4%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{+15}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * 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. associate-*l* 99.8%

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

   2. remove-double-neg 99.8%

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in t around 0 56.2%

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

10. Final simplification 56.2%

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

Alternative 12: 29.8% accurate, 2.0× speedup

Math

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

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 * Float64(-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. Add Preprocessing

3. Taylor expanded in t around 0 56.2%

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

4. Taylor expanded in x around 0 28.4%

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

   1. mul-1-neg 62.9%

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

6. Simplified 28.4%

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

   1. *-rgt-identity 28.4%

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

   2. distribute-lft-neg-out 28.4%

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

   3. neg-sub0 28.4%

      \[\leadsto \color{blue}{0 - y \cdot \sqrt{z \cdot 2}} \]

8. Applied egg-rr 28.4%

   \[\leadsto \color{blue}{0 - y \cdot \sqrt{z \cdot 2}} \]
9. Step-by-step derivation

   1. neg-sub0 28.4%

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

   2. *-commutative 28.4%

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

   3. distribute-rgt-neg-in 28.4%

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

   4. *-commutative 28.4%

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

10. Simplified 28.4%

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

11. Final simplification 28.4%

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

Alternative 13: 2.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in t around 0 56.2%

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

4. Taylor expanded in x around 0 28.4%

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

   1. mul-1-neg 28.4%

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

6. Simplified 28.4%

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

   1. add-sqr-sqrt 13.4%

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

   2. sqrt-unprod 15.3%

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

   3. sqr-neg 15.3%

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

   4. sqrt-unprod 1.7%

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

   5. add-sqr-sqrt 2.8%

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

   6. *-commutative 2.8%

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

   7. *-commutative 2.8%

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

   8. associate-*r* 2.8%

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

   9. sqrt-prod 2.8%

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

8. Applied egg-rr 2.8%

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

9. Final simplification 2.8%

   \[\leadsto y \cdot \sqrt{z \cdot 2} \]
10. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024186 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))