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

Percentage Accurate: 99.5% → 99.8%

Time: 12.3s

Alternatives: 8

Speedup: 1.0×

• 45.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{\left(z \cdot 2\right) \cdot {t\_1}^{2}}\\ \mathbf{if}\;t \leq 19:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+133}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot \left(x \cdot {\left({\left(z \cdot 2\right)}^{1.5}\right)}^{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* (* z 2.0) (pow t_1 2.0)))))
   (if (<= t 19.0)
     (* t_1 (sqrt (* z 2.0)))
     (if (<= t 4.1e+133)
       t_2
       (if (<= t 3.1e+190)
         (* 0.5 (* x (pow (pow (* z 2.0) 1.5) 0.3333333333333333)))
         t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt(((z * 2.0) * pow(t_1, 2.0)));
	double tmp;
	if (t <= 19.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else if (t <= 4.1e+133) {
		tmp = t_2;
	} else if (t <= 3.1e+190) {
		tmp = 0.5 * (x * pow(pow((z * 2.0), 1.5), 0.3333333333333333));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt(((z * 2.0d0) * (t_1 ** 2.0d0)))
    if (t <= 19.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else if (t <= 4.1d+133) then
        tmp = t_2
    else if (t <= 3.1d+190) then
        tmp = 0.5d0 * (x * (((z * 2.0d0) ** 1.5d0) ** 0.3333333333333333d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt(((z * 2.0) * Math.pow(t_1, 2.0)));
	double tmp;
	if (t <= 19.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else if (t <= 4.1e+133) {
		tmp = t_2;
	} else if (t <= 3.1e+190) {
		tmp = 0.5 * (x * Math.pow(Math.pow((z * 2.0), 1.5), 0.3333333333333333));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt(((z * 2.0) * math.pow(t_1, 2.0)))
	tmp = 0
	if t <= 19.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	elif t <= 4.1e+133:
		tmp = t_2
	elif t <= 3.1e+190:
		tmp = 0.5 * (x * math.pow(math.pow((z * 2.0), 1.5), 0.3333333333333333))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(Float64(z * 2.0) * (t_1 ^ 2.0)))
	tmp = 0.0
	if (t <= 19.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	elseif (t <= 4.1e+133)
		tmp = t_2;
	elseif (t <= 3.1e+190)
		tmp = Float64(0.5 * Float64(x * ((Float64(z * 2.0) ^ 1.5) ^ 0.3333333333333333)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt(((z * 2.0) * (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t <= 19.0)
		tmp = t_1 * sqrt((z * 2.0));
	elseif (t <= 4.1e+133)
		tmp = t_2;
	elseif (t <= 3.1e+190)
		tmp = 0.5 * (x * (((z * 2.0) ^ 1.5) ^ 0.3333333333333333));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 19.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.1e+133], t$95$2, If[LessEqual[t, 3.1e+190], N[(0.5 * N[(x * N[Power[N[Power[N[(z * 2.0), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 19

   1. Initial program 99.2%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 86.8%

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

      5. exp-sqrt 86.8%

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

      6. pow-exp 86.8%

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

      7. sqrt-unprod 86.8%

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

      8. associate-*l* 86.8%

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

      9. pow-exp 86.8%

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

      10. pow2 86.8%

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

   4. Applied egg-rr 86.8%

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

      1. distribute-lft-out 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in t around 0 75.8%

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

   if 19 < t < 4.10000000000000004e133 or 3.1000000000000001e190 < t

   1. Initial program 98.0%

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

   3. Taylor expanded in t around 0 16.1%

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

      1. associate-*r* 16.1%

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

      2. sqrt-prod 16.1%

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

      3. *-commutative 16.1%

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

      4. *-commutative 16.1%

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

      5. pow1 16.1%

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

      6. metadata-eval 16.1%

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

      7. sqrt-pow1 19.8%

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

      8. *-commutative 19.8%

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

      9. sqrt-prod 25.5%

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

      10. pow1/2 25.5%

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

      11. pow1 25.5%

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

      12. pow1 25.5%

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

      13. associate-*l* 25.5%

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

      14. *-commutative 25.5%

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

   5. Applied egg-rr 25.5%

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

      1. unpow1/2 25.5%

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

      2. associate-*r* 25.5%

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

      3. *-commutative 25.5%

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

   7. Simplified 25.5%

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

   if 4.10000000000000004e133 < t < 3.1000000000000001e190

   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 20.0%

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

   4. Taylor expanded in x around inf 10.8%

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

      1. associate-*l* 10.8%

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

   6. Simplified 10.8%

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

      1. sqrt-prod 20.0%

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

      2. add-cbrt-cube 35.9%

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

      3. pow1/3 35.9%

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

      4. add-sqr-sqrt 35.9%

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

      5. pow1 35.9%

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

      6. pow1/2 35.9%

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

      7. pow-prod-up 35.9%

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

      8. metadata-eval 35.9%

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

   8. Applied egg-rr 18.8%

      \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{{\left({\left(2 \cdot z\right)}^{1.5}\right)}^{0.3333333333333333}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 19:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+190}:\\ \;\;\;\;0.5 \cdot \left(x \cdot {\left({\left(z \cdot 2\right)}^{1.5}\right)}^{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.2%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 86.8%

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

      5. exp-sqrt 86.8%

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

      6. pow-exp 86.8%

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

      7. sqrt-unprod 86.8%

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

      8. associate-*l* 86.8%

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

      9. pow-exp 86.8%

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

      10. pow2 86.8%

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

   4. Applied egg-rr 86.8%

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

      1. distribute-lft-out 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in t around 0 75.8%

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

   if 1 < t

   1. Initial program 98.4%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-lft-in 47.5%

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

      5. exp-sqrt 47.5%

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

      6. pow-exp 47.5%

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

      7. sqrt-unprod 47.5%

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

      8. associate-*l* 47.5%

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

      9. pow-exp 47.5%

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

      10. pow2 47.5%

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

   4. Applied egg-rr 47.5%

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

      1. distribute-lft-out 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 68.8%

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

      1. distribute-lft-out 68.8%

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

      2. *-commutative 68.8%

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

   9. Simplified 68.8%

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

   10. Taylor expanded in t around inf 53.0%

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

       1. *-commutative 53.0%

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

       2. associate-*r* 53.0%

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

   12. Simplified 53.0%

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

4. Final simplification 70.4%

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

Alternative 4: 64.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.2%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 86.8%

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

      5. exp-sqrt 86.8%

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

      6. pow-exp 86.8%

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

      7. sqrt-unprod 86.8%

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

      8. associate-*l* 86.8%

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

      9. pow-exp 86.8%

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

      10. pow2 86.8%

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

   4. Applied egg-rr 86.8%

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

      1. distribute-lft-out 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in t around 0 75.8%

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

   if 1 < t

   1. Initial program 98.4%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-lft-in 47.5%

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

      5. exp-sqrt 47.5%

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

      6. pow-exp 47.5%

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

      7. sqrt-unprod 47.5%

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

      8. associate-*l* 47.5%

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

      9. pow-exp 47.5%

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

      10. pow2 47.5%

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

   4. Applied egg-rr 47.5%

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

      1. distribute-lft-out 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 68.8%

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

      1. distribute-lft-out 68.8%

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

      2. *-commutative 68.8%

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

   9. Simplified 68.8%

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

   10. Taylor expanded in t around inf 53.0%

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

       1. associate-*l* 49.9%

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

       2. *-commutative 49.9%

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

   12. Simplified 49.9%

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

4. Final simplification 69.6%

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

Alternative 5: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 2.2 \cdot 10^{+42}:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 2.2e+42)
     (* t_1 (sqrt (* z 2.0)))
     (* t_1 (cbrt (pow (* z 2.0) 1.5))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 2.2e+42) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = t_1 * cbrt(pow((z * 2.0), 1.5));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 2.2e+42) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = t_1 * Math.cbrt(Math.pow((z * 2.0), 1.5));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 2.2e+42)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(t_1 * cbrt((Float64(z * 2.0) ^ 1.5)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 2.2e+42], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[Power[N[(z * 2.0), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.2000000000000001e42

   1. Initial program 99.2%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 84.5%

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

      5. exp-sqrt 84.5%

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

      6. pow-exp 84.5%

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

      7. sqrt-unprod 84.5%

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

      8. associate-*l* 84.5%

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

      9. pow-exp 84.5%

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

      10. pow2 84.5%

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

   4. Applied egg-rr 84.5%

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

      1. distribute-lft-out 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in t around 0 73.1%

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

   if 2.2000000000000001e42 < t

   1. Initial program 98.1%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-lft-in 50.0%

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

      5. exp-sqrt 50.0%

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

      6. pow-exp 50.0%

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

      7. sqrt-unprod 50.0%

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

      8. associate-*l* 50.0%

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

      9. pow-exp 50.0%

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

      10. pow2 50.0%

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

   4. Applied egg-rr 50.0%

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

      1. distribute-lft-out 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t around 0 17.1%

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

      1. *-commutative 17.1%

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

      2. *-commutative 17.1%

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

      3. associate-*l* 17.1%

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

   9. Simplified 17.1%

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

       1. sqrt-prod 17.1%

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

       2. add-cbrt-cube 29.7%

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

       3. pow1/3 29.7%

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

       4. add-sqr-sqrt 29.7%

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

       5. pow1 29.7%

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

       6. pow1/2 29.7%

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

       7. pow-prod-up 29.7%

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

       8. metadata-eval 29.7%

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

   11. Applied egg-rr 29.7%

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

       1. unpow1/3 29.7%

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

   13. Simplified 29.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{+42}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-*r* 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. *-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 77.5%

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

   5. exp-sqrt 77.5%

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

   6. pow-exp 77.5%

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

   7. sqrt-unprod 77.5%

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

   8. associate-*l* 77.5%

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

   9. pow-exp 77.5%

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

   10. pow2 77.5%

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

4. Applied egg-rr 77.5%

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

   1. distribute-lft-out 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

   1. pow2 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 7: 57.2% 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 99.0%

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

   1. associate-*r* 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. *-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 77.5%

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

   5. exp-sqrt 77.5%

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

   6. pow-exp 77.5%

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

   7. sqrt-unprod 77.5%

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

   8. associate-*l* 77.5%

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

   9. pow-exp 77.5%

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

   10. pow2 77.5%

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

4. Applied egg-rr 77.5%

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

   1. distribute-lft-out 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in t around 0 61.7%

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

8. Final simplification 61.7%

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

Alternative 8: 30.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0 61.5%

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

4. Taylor expanded in x around inf 31.6%

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

   1. associate-*l* 31.6%

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

6. Simplified 31.6%

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

   1. sqrt-prod 31.7%

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

   2. pow1/2 31.7%

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

8. Applied egg-rr 31.7%

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

   1. unpow1/2 31.7%

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

10. Simplified 31.7%

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

11. Final simplification 31.7%

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

Developer target: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024101 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

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