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

Percentage Accurate: 99.5% → 99.5%

Time: 12.6s

Alternatives: 5

Speedup: 1.0×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% 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]

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 2: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\frac{t \cdot t}{2}}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-21} \lor \neg \left(x \leq 1.08 \cdot 10^{-13}\right):\\ \;\;\;\;t_1 \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-y \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (exp (/ (* t t) 2.0))))
   (if (or (<= x -4.2e-21) (not (<= x 1.08e-13)))
     (* t_1 (* x (sqrt (* 0.5 z))))
     (* t_1 (- (* y (sqrt (* z 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = exp(((t * t) / 2.0));
	double tmp;
	if ((x <= -4.2e-21) || !(x <= 1.08e-13)) {
		tmp = t_1 * (x * sqrt((0.5 * z)));
	} else {
		tmp = t_1 * -(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) :: t_1
    real(8) :: tmp
    t_1 = exp(((t * t) / 2.0d0))
    if ((x <= (-4.2d-21)) .or. (.not. (x <= 1.08d-13))) then
        tmp = t_1 * (x * sqrt((0.5d0 * z)))
    else
        tmp = t_1 * -(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 t_1 = Math.exp(((t * t) / 2.0));
	double tmp;
	if ((x <= -4.2e-21) || !(x <= 1.08e-13)) {
		tmp = t_1 * (x * Math.sqrt((0.5 * z)));
	} else {
		tmp = t_1 * -(y * Math.sqrt((z * 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.exp(((t * t) / 2.0))
	tmp = 0
	if (x <= -4.2e-21) or not (x <= 1.08e-13):
		tmp = t_1 * (x * math.sqrt((0.5 * z)))
	else:
		tmp = t_1 * -(y * math.sqrt((z * 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = exp(Float64(Float64(t * t) / 2.0))
	tmp = 0.0
	if ((x <= -4.2e-21) || !(x <= 1.08e-13))
		tmp = Float64(t_1 * Float64(x * sqrt(Float64(0.5 * z))));
	else
		tmp = Float64(t_1 * Float64(-Float64(y * sqrt(Float64(z * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = exp(((t * t) / 2.0));
	tmp = 0.0;
	if ((x <= -4.2e-21) || ~((x <= 1.08e-13)))
		tmp = t_1 * (x * sqrt((0.5 * z)));
	else
		tmp = t_1 * -(y * sqrt((z * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -4.2e-21], N[Not[LessEqual[x, 1.08e-13]], $MachinePrecision]], N[(t$95$1 * N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * (-N[(y * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000025e-21 or 1.0799999999999999e-13 < 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. Step-by-step derivation

      1. add-sqr-sqrt 52.8%

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

      2. sqrt-unprod 41.1%

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

      3. *-commutative 41.1%

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

      4. *-commutative 41.1%

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

      5. swap-sqr 40.3%

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

      6. add-sqr-sqrt 40.3%

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

      7. pow2 40.3%

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

      8. fma-neg 40.3%

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

   4. Applied egg-rr 40.3%

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

      1. associate-*l* 40.3%

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

      2. fma-neg 40.3%

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

      3. *-commutative 40.3%

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

   6. Simplified 40.3%

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

   7. Taylor expanded in x around inf 39.6%

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

      1. associate-*r* 39.6%

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

   9. Simplified 39.6%

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

   10. Taylor expanded in x around 0 84.0%

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

       1. expm1-log1p-u 50.7%

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

       2. expm1-udef 40.5%

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

       3. associate-*l* 40.5%

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

       4. sqrt-unprod 40.5%

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

   12. Applied egg-rr 40.5%

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

       1. expm1-def 50.7%

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

       2. expm1-log1p 84.1%

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

   14. Simplified 84.1%

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

   if -4.20000000000000025e-21 < x < 1.0799999999999999e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 86.0%

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

      1. mul-1-neg 86.0%

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

      2. associate-*l* 85.9%

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

   5. Simplified 85.9%

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

      1. *-commutative 85.9%

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

      2. sqrt-prod 86.2%

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

   7. Applied egg-rr 86.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-21} \lor \neg \left(x \leq 1.08 \cdot 10^{-13}\right):\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(-y \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. 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. Step-by-step derivation

   1. add-sqr-sqrt 49.8%

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

   2. sqrt-unprod 37.3%

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

   3. *-commutative 37.3%

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

   4. *-commutative 37.3%

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

   5. swap-sqr 35.2%

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

   6. add-sqr-sqrt 35.2%

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

   7. pow2 35.2%

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

   8. fma-neg 35.2%

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

4. Applied egg-rr 35.2%

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

   1. associate-*l* 35.2%

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

   2. fma-neg 35.2%

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

   3. *-commutative 35.2%

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

6. Simplified 35.2%

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

7. Taylor expanded in x around inf 25.1%

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

   1. associate-*r* 25.1%

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

9. Simplified 25.1%

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

10. Taylor expanded in x around 0 55.8%

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

    1. expm1-log1p-u 39.4%

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

    2. expm1-udef 21.1%

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

    3. associate-*l* 21.1%

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

    4. sqrt-unprod 21.1%

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

12. Applied egg-rr 21.1%

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

    1. expm1-def 39.4%

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

    2. expm1-log1p 55.9%

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

14. Simplified 55.9%

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

15. Final simplification 55.9%

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

Alternative 4: 30.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[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. Step-by-step derivation

   1. add-sqr-sqrt 49.8%

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

   2. sqrt-unprod 37.3%

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

   3. *-commutative 37.3%

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

   4. *-commutative 37.3%

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

   5. swap-sqr 35.2%

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

   6. add-sqr-sqrt 35.2%

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

   7. pow2 35.2%

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

   8. fma-neg 35.2%

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

4. Applied egg-rr 35.2%

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

   1. associate-*l* 35.2%

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

   2. fma-neg 35.2%

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

   3. *-commutative 35.2%

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

6. Simplified 35.2%

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

7. Taylor expanded in x around inf 25.1%

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

   1. associate-*r* 25.1%

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

9. Simplified 25.1%

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

10. Taylor expanded in x around 0 55.8%

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

11. Taylor expanded in t around 0 26.8%

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

    1. associate-*l* 26.7%

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

    2. *-commutative 26.7%

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

13. Simplified 26.7%

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

14. Final simplification 26.7%

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

Alternative 5: 30.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $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. Step-by-step derivation

   1. add-sqr-sqrt 49.8%

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

   2. sqrt-unprod 37.3%

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

   3. *-commutative 37.3%

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

   4. *-commutative 37.3%

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

   5. swap-sqr 35.2%

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

   6. add-sqr-sqrt 35.2%

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

   7. pow2 35.2%

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

   8. fma-neg 35.2%

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

4. Applied egg-rr 35.2%

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

   1. associate-*l* 35.2%

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

   2. fma-neg 35.2%

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

   3. *-commutative 35.2%

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

6. Simplified 35.2%

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

7. Taylor expanded in x around inf 25.1%

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

   1. associate-*r* 25.1%

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

9. Simplified 25.1%

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

10. Taylor expanded in x around 0 55.8%

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

11. Taylor expanded in t around 0 26.8%

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

12. Final simplification 26.8%

    \[\leadsto \left(x \cdot \sqrt{0.5}\right) \cdot \sqrt{z} \]
13. 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 2024019 
(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))))