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

Percentage Accurate: 99.4% → 99.4%

Time: 14.0s

Alternatives: 6

Speedup: 1.0×

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * sqrt(exp((t * t)));
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[Sqrt[N[Exp[N[(t * t), $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. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 59.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1.7 \cdot 10^{+44}:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {t\_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.7e+44)
     (* t_1 (sqrt (* z 2.0)))
     (sqrt (* (* z 2.0) (pow t_1 2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.7e+44) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = sqrt(((z * 2.0) * pow(t_1, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 1.7d+44) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else
        tmp = sqrt(((z * 2.0d0) * (t_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.7e+44) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(t_1, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 1.7e+44:
		tmp = t_1 * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(((z * 2.0) * math.pow(t_1, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.7e+44)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * (t_1 ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 1.7e+44)
		tmp = t_1 * sqrt((z * 2.0));
	else
		tmp = sqrt(((z * 2.0) * (t_1 ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1.7e+44], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.7e44

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 70.9%

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

   if 1.7e44 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 28.7%

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

      1. add-sqr-sqrt 14.3%

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

      2. sqrt-unprod 26.6%

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

      3. *-commutative 26.6%

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

      4. *-commutative 26.6%

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

      5. swap-sqr 37.0%

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

      6. add-sqr-sqrt 37.0%

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

      7. pow2 37.0%

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

   5. Applied egg-rr 37.0%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+44}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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%

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

Alternative 4: 43.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+95} \lor \neg \left(x \leq 1.6 \cdot 10^{-35}\right):\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \sqrt{z \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.06e+95) (not (<= x 1.6e-35)))
   (* x (sqrt (* 0.5 z)))
   (- (* y (sqrt (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.06e+95) || !(x <= 1.6e-35)) {
		tmp = x * sqrt((0.5 * z));
	} else {
		tmp = -(y * sqrt((z * 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 ((x <= (-1.06d+95)) .or. (.not. (x <= 1.6d-35))) then
        tmp = x * sqrt((0.5d0 * z))
    else
        tmp = -(y * sqrt((z * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.06e+95) || !(x <= 1.6e-35)) {
		tmp = x * Math.sqrt((0.5 * z));
	} else {
		tmp = -(y * Math.sqrt((z * 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.06e+95) or not (x <= 1.6e-35):
		tmp = x * math.sqrt((0.5 * z))
	else:
		tmp = -(y * math.sqrt((z * 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.06e+95) || !(x <= 1.6e-35))
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	else
		tmp = Float64(-Float64(y * sqrt(Float64(z * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.06e+95) || ~((x <= 1.6e-35)))
		tmp = x * sqrt((0.5 * z));
	else
		tmp = -(y * sqrt((z * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.06e+95], N[Not[LessEqual[x, 1.6e-35]], $MachinePrecision]], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(y * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -1.06000000000000001e95 or 1.5999999999999999e-35 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 68.1%

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

   4. Taylor expanded in x around inf 51.7%

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

      1. *-commutative 51.7%

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

      2. associate-*l* 51.8%

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

      3. *-commutative 51.8%

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

      4. associate-*r* 51.8%

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

      5. associate-*l* 51.8%

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

   6. Simplified 51.8%

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

      1. pow1 51.8%

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

      2. add-sqr-sqrt 51.7%

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

      3. sqrt-unprod 51.8%

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

      4. swap-sqr 51.6%

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

      5. add-sqr-sqrt 51.6%

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

      6. swap-sqr 51.6%

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

      7. rem-square-sqrt 51.8%

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

      8. metadata-eval 51.8%

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

      9. metadata-eval 51.8%

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

   8. Applied egg-rr 51.8%

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

      1. unpow1 51.8%

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

   10. Simplified 51.8%

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

   if -1.06000000000000001e95 < x < 1.5999999999999999e-35

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 57.0%

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

   4. Taylor expanded in x around 0 45.2%

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

      1. mul-1-neg 45.2%

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

   6. Simplified 45.2%

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

      1. pow1 45.2%

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

      2. associate-*l* 45.2%

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

      3. sqrt-prod 45.3%

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

      4. rem-exp-log 43.1%

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

      5. *-commutative 43.1%

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

      6. rem-exp-log 45.3%

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

   8. Applied egg-rr 45.3%

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

      1. unpow1 45.3%

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

      2. *-commutative 45.3%

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

      3. *-commutative 45.3%

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

   10. Simplified 45.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+95} \lor \neg \left(x \leq 1.6 \cdot 10^{-35}\right):\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-y \cdot \sqrt{z \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 61.8%

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

4. Final simplification 61.8%

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

Alternative 6: 29.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \sqrt{0.5 \cdot z} \end{array} \]

FPCore

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

C

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

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 * sqrt((0.5d0 * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 61.8%

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

4. Taylor expanded in x around inf 31.4%

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

   1. *-commutative 31.4%

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

   2. associate-*l* 31.4%

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

   3. *-commutative 31.4%

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

   4. associate-*r* 31.4%

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

   5. associate-*l* 31.4%

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

6. Simplified 31.4%

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

   1. pow1 31.4%

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

   2. add-sqr-sqrt 31.3%

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

   3. sqrt-unprod 31.4%

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

   4. swap-sqr 31.3%

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

   5. add-sqr-sqrt 31.3%

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

   6. swap-sqr 31.3%

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

   7. rem-square-sqrt 31.4%

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

   8. metadata-eval 31.4%

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

   9. metadata-eval 31.4%

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

8. Applied egg-rr 31.4%

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

   1. unpow1 31.4%

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

10. Simplified 31.4%

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

11. Final simplification 31.4%

    \[\leadsto x \cdot \sqrt{0.5 \cdot z} \]
12. Add Preprocessing

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024059 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

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