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

Percentage Accurate: 99.5% → 98.9%

Time: 9.9s

Alternatives: 7

Speedup: 1.0×

• 44.3% 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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \left(\left(x \cdot 0.5\right) \cdot t_1 - t_1 \cdot y\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (* (- (* (* x 0.5) t_1) (* t_1 y)) (exp (/ (* t t) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	return (((x * 0.5) * t_1) - (t_1 * y)) * 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
    real(8) :: t_1
    t_1 = sqrt((z * 2.0d0))
    code = (((x * 0.5d0) * t_1) - (t_1 * y)) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	return (((x * 0.5) * t_1) - (t_1 * y)) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	return (((x * 0.5) * t_1) - (t_1 * y)) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	return Float64(Float64(Float64(Float64(x * 0.5) * t_1) - Float64(t_1 * y)) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = (((x * 0.5) * t_1) - (t_1 * y)) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 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. sub-neg 99.8%

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

   3. distribute-rgt-in 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 37.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;t_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(t_1 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 2.1e+19)
     (* t_1 (- y))
     (* (exp (/ (* t t) 2.0)) (* t_1 y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 2.1e+19) {
		tmp = t_1 * -y;
	} else {
		tmp = exp(((t * t) / 2.0)) * (t_1 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((t * t) <= 2.1d+19) then
        tmp = t_1 * -y
    else
        tmp = exp(((t * t) / 2.0d0)) * (t_1 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 2.1e+19) {
		tmp = t_1 * -y;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (t_1 * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 2.1e+19:
		tmp = t_1 * -y
	else:
		tmp = math.exp(((t * t) / 2.0)) * (t_1 * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 2.1e+19)
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(t_1 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 2.1e+19)
		tmp = t_1 * -y;
	else
		tmp = exp(((t * t) / 2.0)) * (t_1 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2.1e+19], N[(t$95$1 * (-y)), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.1e19

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 58.5%

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

      1. mul-1-neg 58.5%

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

      2. distribute-rgt-neg-in 58.5%

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

      3. associate-*l* 58.6%

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

   4. Simplified 58.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 2.9%

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

      3. sqr-neg 2.9%

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

      4. add-sqr-sqrt 2.9%

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

      5. sqrt-prod 2.9%

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

      6. *-commutative 2.9%

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

   6. Applied egg-rr 2.9%

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

   7. Taylor expanded in t around 0 2.9%

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

      1. associate-*l* 2.9%

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

   9. Simplified 2.9%

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

       1. add-sqr-sqrt 1.0%

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

       2. sqrt-unprod 19.6%

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

       3. sqr-neg 19.6%

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

       4. sqrt-unprod 25.6%

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

       5. add-sqr-sqrt 55.2%

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

       6. sqrt-prod 55.3%

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

       7. distribute-lft-neg-out 55.3%

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

       8. sqrt-prod 55.2%

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

       9. neg-sub0 55.2%

          \[\leadsto \color{blue}{0 - y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

       10. sqrt-prod 55.3%

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

   11. Applied egg-rr 55.3%

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

       1. neg-sub0 55.3%

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

       2. distribute-rgt-neg-in 55.3%

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

   13. Simplified 55.3%

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

   if 2.1e19 < (*.f64 t 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. Taylor expanded in x around 0 71.5%

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

      1. mul-1-neg 71.5%

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

      2. distribute-rgt-neg-in 71.5%

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

      3. associate-*l* 71.5%

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

   4. Simplified 71.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 25.2%

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

      3. sqr-neg 25.2%

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

      4. add-sqr-sqrt 25.2%

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

      5. sqrt-prod 25.2%

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

      6. *-commutative 25.2%

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

   6. Applied egg-rr 25.2%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot y\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 4: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $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. Taylor expanded in x around 0 64.8%

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

   1. mul-1-neg 64.8%

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

   2. distribute-rgt-neg-in 64.8%

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

   3. associate-*l* 64.8%

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

4. Simplified 64.8%

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

   1. *-commutative 64.8%

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

   2. distribute-lft-neg-out 64.8%

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

   3. sqrt-prod 64.9%

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

6. Applied egg-rr 64.9%

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

7. Final simplification 64.9%

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

Alternative 5: 32.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 250000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 250000.0)
   (* (sqrt (* z 2.0)) (- y))
   (sqrt (* (* z 2.0) (pow y 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 250000.0) {
		tmp = sqrt((z * 2.0)) * -y;
	} else {
		tmp = sqrt(((z * 2.0) * pow(y, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 250000.0d0) then
        tmp = sqrt((z * 2.0d0)) * -y
    else
        tmp = sqrt(((z * 2.0d0) * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 250000.0) {
		tmp = Math.sqrt((z * 2.0)) * -y;
	} else {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 250000.0:
		tmp = math.sqrt((z * 2.0)) * -y
	else:
		tmp = math.sqrt(((z * 2.0) * math.pow(y, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 250000.0)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(-y));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 250000.0)
		tmp = sqrt((z * 2.0)) * -y;
	else
		tmp = sqrt(((z * 2.0) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 250000.0], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e5

   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. Taylor expanded in x around 0 62.6%

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

      1. mul-1-neg 62.6%

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

      2. distribute-rgt-neg-in 62.6%

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

      3. associate-*l* 62.7%

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

   4. Simplified 62.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 10.3%

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

      3. sqr-neg 10.3%

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

      4. add-sqr-sqrt 10.3%

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

      5. sqrt-prod 10.3%

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

      6. *-commutative 10.3%

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

   6. Applied egg-rr 10.3%

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

   7. Taylor expanded in t around 0 2.5%

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

      1. associate-*l* 2.5%

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

   9. Simplified 2.5%

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

       1. add-sqr-sqrt 0.8%

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

       2. sqrt-unprod 18.6%

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

       3. sqr-neg 18.6%

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

       4. sqrt-unprod 19.7%

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

       5. add-sqr-sqrt 40.4%

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

       6. sqrt-prod 40.4%

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

       7. distribute-lft-neg-out 40.4%

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

       8. sqrt-prod 40.4%

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

       9. neg-sub0 40.4%

          \[\leadsto \color{blue}{0 - y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

       10. sqrt-prod 40.4%

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

   11. Applied egg-rr 40.4%

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

       1. neg-sub0 40.4%

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

       2. distribute-rgt-neg-in 40.4%

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

   13. Simplified 40.4%

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

   if 2.5e5 < 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. Taylor expanded in x around 0 71.4%

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

      1. mul-1-neg 71.4%

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

      2. distribute-rgt-neg-in 71.4%

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

      3. associate-*l* 71.4%

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

   4. Simplified 71.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 23.8%

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

      3. sqr-neg 23.8%

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

      4. add-sqr-sqrt 23.8%

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

      5. sqrt-prod 23.8%

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

      6. *-commutative 23.8%

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

   6. Applied egg-rr 23.8%

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

   7. Taylor expanded in t around 0 1.1%

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

      1. associate-*l* 1.1%

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

   9. Simplified 1.1%

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

       1. rem-square-sqrt 0.5%

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

       2. sqrt-unprod 13.8%

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

       3. sqrt-prod 13.8%

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

       4. *-commutative 13.8%

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

       5. sqrt-prod 13.8%

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

       6. *-commutative 13.8%

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

       7. swap-sqr 16.9%

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

       8. add-sqr-sqrt 16.9%

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

       9. pow2 16.9%

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

   11. Applied egg-rr 16.9%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 250000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \]

Alternative 6: 30.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $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. Taylor expanded in x around 0 64.8%

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

   1. mul-1-neg 64.8%

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

   2. distribute-rgt-neg-in 64.8%

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

   3. associate-*l* 64.8%

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

4. Simplified 64.8%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 13.6%

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

   3. sqr-neg 13.6%

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

   4. add-sqr-sqrt 13.6%

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

   5. sqrt-prod 13.6%

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

   6. *-commutative 13.6%

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

6. Applied egg-rr 13.6%

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

7. Taylor expanded in t around 0 2.1%

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

   1. associate-*l* 2.1%

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

9. Simplified 2.1%

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

    1. add-sqr-sqrt 0.8%

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

    2. sqrt-unprod 16.3%

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

    3. sqr-neg 16.3%

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

    4. sqrt-unprod 15.5%

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

    5. add-sqr-sqrt 32.7%

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

    6. sqrt-prod 32.8%

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

    7. distribute-lft-neg-out 32.8%

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

    8. sqrt-prod 32.7%

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

    9. neg-sub0 32.7%

       \[\leadsto \color{blue}{0 - y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

    10. sqrt-prod 32.8%

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

11. Applied egg-rr 32.8%

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

    1. neg-sub0 32.8%

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

    2. distribute-rgt-neg-in 32.8%

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

13. Simplified 32.8%

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

14. Final simplification 32.8%

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

Alternative 7: 2.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * y), $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. Taylor expanded in x around 0 64.8%

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

   1. mul-1-neg 64.8%

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

   2. distribute-rgt-neg-in 64.8%

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

   3. associate-*l* 64.8%

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

4. Simplified 64.8%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 13.6%

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

   3. sqr-neg 13.6%

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

   4. add-sqr-sqrt 13.6%

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

   5. sqrt-prod 13.6%

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

   6. *-commutative 13.6%

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

6. Applied egg-rr 13.6%

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

7. Taylor expanded in t around 0 2.1%

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

   1. associate-*l* 2.1%

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

9. Simplified 2.1%

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

    1. sqrt-prod 2.1%

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

    2. expm1-log1p-u 2.1%

       \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot z}\right)\right)} \]

    3. expm1-udef 2.3%

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

11. Applied egg-rr 2.3%

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

    1. expm1-def 2.1%

       \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot z}\right)\right)} \]

    2. expm1-log1p 2.1%

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

13. Simplified 2.1%

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

14. Final simplification 2.1%

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

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 2023310 
(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))))