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

Percentage Accurate: 99.5% → 99.8%

Time: 13.7s

Alternatives: 8

Speedup: 1.0×

• 45.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 70.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\frac{t \cdot t}{2}}\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 6800000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_2\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+43} \lor \neg \left(t \leq 9 \cdot 10^{+186}\right):\\ \;\;\;\;t\_1 \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(y \cdot \left(-t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (exp (/ (* t t) 2.0))) (t_2 (sqrt (* z 2.0))))
   (if (<= t 6800000.0)
     (* (- (* x 0.5) y) t_2)
     (if (or (<= t 4e+43) (not (<= t 9e+186)))
       (* t_1 (* x (sqrt (* 0.5 z))))
       (* t_1 (* y (- t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = exp(((t * t) / 2.0));
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 6800000.0) {
		tmp = ((x * 0.5) - y) * t_2;
	} else if ((t <= 4e+43) || !(t <= 9e+186)) {
		tmp = t_1 * (x * sqrt((0.5 * z)));
	} else {
		tmp = t_1 * (y * -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 = exp(((t * t) / 2.0d0))
    t_2 = sqrt((z * 2.0d0))
    if (t <= 6800000.0d0) then
        tmp = ((x * 0.5d0) - y) * t_2
    else if ((t <= 4d+43) .or. (.not. (t <= 9d+186))) then
        tmp = t_1 * (x * sqrt((0.5d0 * z)))
    else
        tmp = t_1 * (y * -t_2)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.exp(((t * t) / 2.0));
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 6800000.0) {
		tmp = ((x * 0.5) - y) * t_2;
	} else if ((t <= 4e+43) || !(t <= 9e+186)) {
		tmp = t_1 * (x * Math.sqrt((0.5 * z)));
	} else {
		tmp = t_1 * (y * -t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.exp(((t * t) / 2.0))
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 6800000.0:
		tmp = ((x * 0.5) - y) * t_2
	elif (t <= 4e+43) or not (t <= 9e+186):
		tmp = t_1 * (x * math.sqrt((0.5 * z)))
	else:
		tmp = t_1 * (y * -t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = exp(Float64(Float64(t * t) / 2.0))
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 6800000.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_2);
	elseif ((t <= 4e+43) || !(t <= 9e+186))
		tmp = Float64(t_1 * Float64(x * sqrt(Float64(0.5 * z))));
	else
		tmp = Float64(t_1 * Float64(y * Float64(-t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = exp(((t * t) / 2.0));
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 6800000.0)
		tmp = ((x * 0.5) - y) * t_2;
	elseif ((t <= 4e+43) || ~((t <= 9e+186)))
		tmp = t_1 * (x * sqrt((0.5 * z)));
	else
		tmp = t_1 * (y * -t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 6800000.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$2), $MachinePrecision], If[Or[LessEqual[t, 4e+43], N[Not[LessEqual[t, 9e+186]], $MachinePrecision]], N[(t$95$1 * N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(y * (-t$95$2)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.8e6

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. exp-sqrt 99.7%

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

      3. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. pow-exp 99.7%

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

      6. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 69.5%

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

   if 6.8e6 < t < 4.00000000000000006e43 or 9.0000000000000009e186 < 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 x around inf 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*r* 74.1%

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

      5. associate-*l* 74.1%

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

   5. Simplified 74.1%

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

      1. add-sqr-sqrt 74.1%

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

      2. pow1 74.1%

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

      3. sqrt-unprod 74.1%

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

      4. *-commutative 74.1%

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

      5. *-commutative 74.1%

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

      6. swap-sqr 74.1%

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

      7. swap-sqr 74.1%

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

      8. rem-square-sqrt 74.1%

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

      9. metadata-eval 74.1%

         \[\leadsto \left(x \cdot {\left(\sqrt{\left(2 \cdot \color{blue}{0.25}\right) \cdot \left(\sqrt{z} \cdot \sqrt{z}\right)}\right)}^{1}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 74.1%

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

      11. add-sqr-sqrt 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. unpow1 74.1%

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

      2. *-commutative 74.1%

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

   9. Simplified 74.1%

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

   if 4.00000000000000006e43 < t < 9.0000000000000009e186

   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 x around 0 80.6%

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

      1. associate-*r* 80.6%

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

      2. *-commutative 80.6%

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

      3. associate-*r* 80.6%

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

      4. neg-mul-1 80.6%

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

   5. Simplified 80.6%

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

      1. *-commutative 80.6%

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

      2. distribute-lft-neg-out 80.6%

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

      3. distribute-lft-neg-out 80.6%

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

      4. add-sqr-sqrt 50.0%

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

      5. sqrt-unprod 50.0%

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

      6. sqr-neg 50.0%

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

      7. sqrt-unprod 8.3%

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

      8. add-sqr-sqrt 16.7%

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

      9. associate-*l* 16.7%

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

      10. add-sqr-sqrt 8.3%

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

      11. sqrt-unprod 50.0%

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

      12. sqr-neg 50.0%

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

      13. sqrt-unprod 50.0%

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

      14. add-sqr-sqrt 80.6%

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

      15. sqrt-prod 80.6%

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

      16. *-commutative 80.6%

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

   7. Applied egg-rr 80.6%

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

      1. distribute-rgt-neg-in 80.6%

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

      2. *-commutative 80.6%

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

   9. Simplified 80.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6800000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+43} \lor \neg \left(t \leq 9 \cdot 10^{+186}\right):\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6800000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 6800000.0)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (* (exp (/ (* t t) 2.0)) (* x (sqrt (* 0.5 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 6800000.0) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = exp(((t * t) / 2.0)) * (x * sqrt((0.5 * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 6800000.0d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = exp(((t * t) / 2.0d0)) * (x * sqrt((0.5d0 * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 6800000.0) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (x * Math.sqrt((0.5 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 6800000.0:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.exp(((t * t) / 2.0)) * (x * math.sqrt((0.5 * z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 6800000.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(x * sqrt(Float64(0.5 * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 6800000.0)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = exp(((t * t) / 2.0)) * (x * sqrt((0.5 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 6800000.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.8e6

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. exp-sqrt 99.7%

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

      3. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. pow-exp 99.7%

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

      6. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 69.5%

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

   if 6.8e6 < 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 x around inf 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*l* 68.3%

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

      3. *-commutative 68.3%

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

      4. associate-*r* 68.3%

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

      5. associate-*l* 68.3%

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

   5. Simplified 68.3%

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

      1. add-sqr-sqrt 68.3%

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

      2. pow1 68.3%

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

      3. sqrt-unprod 68.3%

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

      4. *-commutative 68.3%

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

      5. *-commutative 68.3%

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

      6. swap-sqr 68.3%

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

      7. swap-sqr 68.3%

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

      8. rem-square-sqrt 68.3%

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

      9. metadata-eval 68.3%

         \[\leadsto \left(x \cdot {\left(\sqrt{\left(2 \cdot \color{blue}{0.25}\right) \cdot \left(\sqrt{z} \cdot \sqrt{z}\right)}\right)}^{1}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. metadata-eval 68.3%

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

      11. add-sqr-sqrt 68.3%

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

   7. Applied egg-rr 68.3%

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

      1. unpow1 68.3%

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

      2. *-commutative 68.3%

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

   9. Simplified 68.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6800000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 28:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 28.0)
     (* 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 <= 28.0) {
		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 <= 28.0) {
		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 <= 28.0)
		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, 28.0], 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 < 28

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. exp-sqrt 99.7%

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

      3. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. pow-exp 99.7%

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

      6. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 69.8%

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

   if 28 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 16.5%

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

       1. add-cbrt-cube 23.6%

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

       2. pow1/3 23.6%

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

       3. add-sqr-sqrt 23.6%

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

       4. pow1 23.6%

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

       5. pow1/2 23.6%

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

       6. pow-prod-up 23.6%

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

       7. metadata-eval 23.6%

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

   11. Applied egg-rr 23.6%

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

       1. unpow1/3 23.6%

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

   13. Simplified 23.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 28:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

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 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 56.4% 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 99.8%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. pow-exp 99.8%

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

   6. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 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 56.4%

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

10. Final simplification 56.4%

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

Alternative 7: 30.0% 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 99.8%

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

3. Taylor expanded in x around 0 62.1%

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

   1. associate-*r* 62.1%

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

   2. *-commutative 62.1%

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

   3. associate-*r* 62.1%

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

   4. neg-mul-1 62.1%

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

5. Simplified 62.1%

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

6. Taylor expanded in t around 0 29.1%

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

   1. mul-1-neg 29.1%

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

   2. associate-*l* 29.0%

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

   3. *-commutative 29.0%

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

   4. distribute-rgt-neg-in 29.0%

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

   5. distribute-rgt-neg-in 29.0%

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

8. Simplified 29.0%

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

10. Applied egg-rr 29.1%

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

    1. distribute-lft-neg-in 29.1%

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

    2. *-commutative 29.1%

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

    3. *-commutative 29.1%

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

12. Simplified 29.1%

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

13. Final simplification 29.1%

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

Alternative 8: 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 99.8%

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

3. Taylor expanded in x around 0 62.1%

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

   1. associate-*r* 62.1%

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

   2. *-commutative 62.1%

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

   3. associate-*r* 62.1%

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

   4. neg-mul-1 62.1%

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

5. Simplified 62.1%

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

6. Taylor expanded in t around 0 29.1%

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

   1. mul-1-neg 29.1%

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

   2. associate-*l* 29.0%

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

   3. *-commutative 29.0%

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

   4. distribute-rgt-neg-in 29.0%

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

   5. distribute-rgt-neg-in 29.0%

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

8. Simplified 29.0%

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

10. Applied egg-rr 3.2%

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

    1. log1p-undefine 3.2%

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

    2. rem-exp-log 3.5%

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

    3. +-commutative 3.5%

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

    4. associate--l+ 3.5%

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

    5. metadata-eval 3.5%

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

    6. +-rgt-identity 3.5%

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

    7. *-commutative 3.5%

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

12. Simplified 3.5%

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

13. Final simplification 3.5%

    \[\leadsto y \cdot \sqrt{z \cdot 2} \]
14. 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 2024085 
(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))))