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

Percentage Accurate: 99.5% → 99.8%

Time: 16.7s

Alternatives: 9

Speedup: 1.0×

• 44.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. exp-sqrt 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

   4. expm1-log1p-u 53.6%

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

   5. expm1-udef 37.9%

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

6. Applied egg-rr 37.9%

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

   1. expm1-def 53.6%

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

   2. expm1-log1p 99.8%

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

   3. fma-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+154} \lor \neg \left(t \leq 9.5 \cdot 10^{+290}\right):\\ \;\;\;\;\left(t \cdot \left(t_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot \left(2 \cdot {t}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= t 1.0)
     (* t_1 (sqrt (* z 2.0)))
     (if (or (<= t 9.5e+154) (not (<= t 9.5e+290)))
       (* (* t (* t_1 (sqrt 2.0))) (sqrt z))
       (* 0.5 (* x (sqrt (* z (* 2.0 (pow t 2.0))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else if ((t <= 9.5e+154) || !(t <= 9.5e+290)) {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	} else {
		tmp = 0.5 * (x * sqrt((z * (2.0 * pow(t, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.5d0 * x) - y
    if (t <= 1.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else if ((t <= 9.5d+154) .or. (.not. (t <= 9.5d+290))) then
        tmp = (t * (t_1 * sqrt(2.0d0))) * sqrt(z)
    else
        tmp = 0.5d0 * (x * sqrt((z * (2.0d0 * (t ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else if ((t <= 9.5e+154) || !(t <= 9.5e+290)) {
		tmp = (t * (t_1 * Math.sqrt(2.0))) * Math.sqrt(z);
	} else {
		tmp = 0.5 * (x * Math.sqrt((z * (2.0 * Math.pow(t, 2.0)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	tmp = 0
	if t <= 1.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	elif (t <= 9.5e+154) or not (t <= 9.5e+290):
		tmp = (t * (t_1 * math.sqrt(2.0))) * math.sqrt(z)
	else:
		tmp = 0.5 * (x * math.sqrt((z * (2.0 * math.pow(t, 2.0)))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (t <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	elseif ((t <= 9.5e+154) || !(t <= 9.5e+290))
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	else
		tmp = Float64(0.5 * Float64(x * sqrt(Float64(z * Float64(2.0 * (t ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	tmp = 0.0;
	if (t <= 1.0)
		tmp = t_1 * sqrt((z * 2.0));
	elseif ((t <= 9.5e+154) || ~((t <= 9.5e+290)))
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	else
		tmp = 0.5 * (x * sqrt((z * (2.0 * (t ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 9.5e+154], N[Not[LessEqual[t, 9.5e+290]], $MachinePrecision]], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[Sqrt[N[(z * N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 73.5%

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

   if 1 < t < 9.5000000000000001e154 or 9.4999999999999995e290 < t

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 50.0%

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

      5. expm1-udef 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. expm1-def 50.0%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 66.9%

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

       1. associate-*r* 66.9%

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

       2. distribute-rgt-out 66.9%

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

   11. Simplified 66.9%

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

   12. Taylor expanded in t around inf 55.0%

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

   if 9.5000000000000001e154 < t < 9.4999999999999995e290

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 44.4%

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

      5. expm1-udef 44.4%

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

   6. Applied egg-rr 44.4%

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

      1. expm1-def 44.4%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. distribute-rgt-out 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 81.5%

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

   13. Taylor expanded in t around inf 81.5%

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

       1. associate-*r* 81.5%

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

       2. *-commutative 81.5%

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

   15. Simplified 81.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+154} \lor \neg \left(t \leq 9.5 \cdot 10^{+290}\right):\\ \;\;\;\;\left(t \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot \left(2 \cdot {t}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+160}:\\ \;\;\;\;\left(t \cdot \left(t_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot \left(2 + 2 \cdot {t}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= t 1.0)
     (* t_1 (sqrt (* z 2.0)))
     (if (<= t 1.85e+160)
       (* (* t (* t_1 (sqrt 2.0))) (sqrt z))
       (* y (- (sqrt (* z (+ 2.0 (* 2.0 (pow t 2.0)))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else if (t <= 1.85e+160) {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	} else {
		tmp = y * -sqrt((z * (2.0 + (2.0 * pow(t, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.5d0 * x) - y
    if (t <= 1.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else if (t <= 1.85d+160) then
        tmp = (t * (t_1 * sqrt(2.0d0))) * sqrt(z)
    else
        tmp = y * -sqrt((z * (2.0d0 + (2.0d0 * (t ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else if (t <= 1.85e+160) {
		tmp = (t * (t_1 * Math.sqrt(2.0))) * Math.sqrt(z);
	} else {
		tmp = y * -Math.sqrt((z * (2.0 + (2.0 * Math.pow(t, 2.0)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	tmp = 0
	if t <= 1.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	elif t <= 1.85e+160:
		tmp = (t * (t_1 * math.sqrt(2.0))) * math.sqrt(z)
	else:
		tmp = y * -math.sqrt((z * (2.0 + (2.0 * math.pow(t, 2.0)))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (t <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	elseif (t <= 1.85e+160)
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	else
		tmp = Float64(y * Float64(-sqrt(Float64(z * Float64(2.0 + Float64(2.0 * (t ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	tmp = 0.0;
	if (t <= 1.0)
		tmp = t_1 * sqrt((z * 2.0));
	elseif (t <= 1.85e+160)
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	else
		tmp = y * -sqrt((z * (2.0 + (2.0 * (t ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.85e+160], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(y * (-N[Sqrt[N[(z * N[(2.0 + N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 73.5%

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

   if 1 < t < 1.85000000000000008e160

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 47.4%

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

      5. expm1-udef 47.4%

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

   6. Applied egg-rr 47.4%

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

      1. expm1-def 47.4%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 65.1%

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

       1. associate-*r* 65.1%

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

       2. distribute-rgt-out 65.1%

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

   11. Simplified 65.1%

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

   12. Taylor expanded in t around inf 50.0%

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

   if 1.85000000000000008e160 < t

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 48.3%

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

      5. expm1-udef 48.3%

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

   6. Applied egg-rr 48.3%

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

      1. expm1-def 48.3%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. distribute-rgt-out 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 93.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+160}:\\ \;\;\;\;\left(t \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot \left(2 + 2 \cdot {t}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{+58}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+292}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot \left(2 \cdot {t}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(y \cdot t\right)\right) \cdot \left(-\sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 9.2e+58)
   (* (- (* 0.5 x) y) (sqrt (* z 2.0)))
   (if (<= t 2e+292)
     (* 0.5 (* x (sqrt (* z (* 2.0 (pow t 2.0))))))
     (* (* (sqrt 2.0) (* y t)) (- (sqrt z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.2e+58) {
		tmp = ((0.5 * x) - y) * sqrt((z * 2.0));
	} else if (t <= 2e+292) {
		tmp = 0.5 * (x * sqrt((z * (2.0 * pow(t, 2.0)))));
	} else {
		tmp = (sqrt(2.0) * (y * t)) * -sqrt(z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 9.2d+58) then
        tmp = ((0.5d0 * x) - y) * sqrt((z * 2.0d0))
    else if (t <= 2d+292) then
        tmp = 0.5d0 * (x * sqrt((z * (2.0d0 * (t ** 2.0d0)))))
    else
        tmp = (sqrt(2.0d0) * (y * t)) * -sqrt(z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 9.2e+58) {
		tmp = ((0.5 * x) - y) * Math.sqrt((z * 2.0));
	} else if (t <= 2e+292) {
		tmp = 0.5 * (x * Math.sqrt((z * (2.0 * Math.pow(t, 2.0)))));
	} else {
		tmp = (Math.sqrt(2.0) * (y * t)) * -Math.sqrt(z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 9.2e+58:
		tmp = ((0.5 * x) - y) * math.sqrt((z * 2.0))
	elif t <= 2e+292:
		tmp = 0.5 * (x * math.sqrt((z * (2.0 * math.pow(t, 2.0)))))
	else:
		tmp = (math.sqrt(2.0) * (y * t)) * -math.sqrt(z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 9.2e+58)
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(z * 2.0)));
	elseif (t <= 2e+292)
		tmp = Float64(0.5 * Float64(x * sqrt(Float64(z * Float64(2.0 * (t ^ 2.0))))));
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(y * t)) * Float64(-sqrt(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 9.2e+58)
		tmp = ((0.5 * x) - y) * sqrt((z * 2.0));
	elseif (t <= 2e+292)
		tmp = 0.5 * (x * sqrt((z * (2.0 * (t ^ 2.0)))));
	else
		tmp = (sqrt(2.0) * (y * t)) * -sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 9.2e+58], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+292], N[(0.5 * N[(x * N[Sqrt[N[(z * N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(y * t), $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[z], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.2000000000000001e58

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 71.0%

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

   if 9.2000000000000001e58 < t < 2e292

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 50.0%

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

      5. expm1-udef 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. expm1-def 50.0%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 88.1%

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

       1. associate-*r* 88.1%

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

       2. distribute-rgt-out 88.1%

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

   11. Simplified 88.1%

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

   12. Taylor expanded in x around inf 69.3%

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

   13. Taylor expanded in t around inf 69.3%

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

       1. associate-*r* 69.3%

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

       2. *-commutative 69.3%

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

   15. Simplified 69.3%

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

   if 2e292 < t

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 83.3%

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

      5. expm1-udef 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. expm1-def 83.3%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. distribute-rgt-out 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 68.7%

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

       1. associate-*l* 37.8%

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

       2. *-commutative 37.8%

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

       3. associate-*r* 37.8%

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

       4. *-commutative 37.8%

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

   14. Simplified 37.8%

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

   15. Taylor expanded in x around 0 68.5%

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

       1. mul-1-neg 68.5%

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

       2. *-commutative 68.5%

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

       3. associate-*r* 68.5%

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

   17. Simplified 68.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{+58}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+292}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot \left(2 \cdot {t}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(y \cdot t\right)\right) \cdot \left(-\sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y, z, t):
	return math.exp(((t * t) / 2.0)) * (((0.5 * x) - 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(0.5 * x) - y) * sqrt(Float64(z * 2.0))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = exp(((t * t) / 2.0)) * (((0.5 * x) - 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[(0.5 * x), $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%

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

Alternative 6: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. exp-sqrt 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

   4. expm1-log1p-u 53.6%

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

   5. expm1-udef 37.9%

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

6. Applied egg-rr 37.9%

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

   1. expm1-def 53.6%

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

   2. expm1-log1p 99.8%

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

   3. fma-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in t around 0 87.9%

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

    1. associate-*r* 87.9%

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

    2. distribute-rgt-out 87.9%

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

11. Simplified 87.9%

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

12. Final simplification 87.9%

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

Alternative 7: 64.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 73.5%

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

   if 1 < t

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. exp-sqrt 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 47.8%

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

      5. expm1-udef 47.8%

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

   6. Applied egg-rr 47.8%

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

      1. expm1-def 47.8%

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

      2. expm1-log1p 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 80.2%

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

       1. associate-*r* 80.2%

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

       2. distribute-rgt-out 80.2%

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

   11. Simplified 80.2%

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

   12. Taylor expanded in t around inf 61.8%

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

       1. associate-*l* 52.2%

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

       2. *-commutative 52.2%

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

       3. associate-*r* 52.2%

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

       4. *-commutative 52.2%

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

   14. Simplified 52.2%

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

       1. *-commutative 11.8%

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

       2. sqrt-prod 11.8%

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

   16. Applied egg-rr 52.2%

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

4. Final simplification 67.9%

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

Alternative 8: 57.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 60.6%

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

6. Final simplification 60.6%

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

Alternative 9: 29.8% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 60.6%

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

6. Taylor expanded in x around inf 32.7%

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

   1. associate-*l* 32.8%

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

8. Simplified 32.8%

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

   1. *-commutative 32.8%

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

   2. sqrt-prod 32.9%

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

10. Applied egg-rr 32.9%

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

11. Final simplification 32.9%

    \[\leadsto 0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right) \]
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 2024020 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

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