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

Percentage Accurate: 99.5% → 99.8%

Time: 15.5s

Alternatives: 6

Speedup: 1.0×

• 43.2% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. exp-sqrt 99.8%

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

   2. associate-*r* 99.0%

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

   3. *-commutative 99.0%

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

   4. expm1-log1p-u 54.1%

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

   5. expm1-udef 42.0%

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

6. Applied egg-rr 42.4%

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

   1. expm1-def 54.5%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. pow2 99.8%

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

   2. exp-prod 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. exp-sqrt 99.8%

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

   2. associate-*r* 99.0%

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

   3. *-commutative 99.0%

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

   4. expm1-log1p-u 54.1%

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

   5. expm1-udef 42.0%

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

6. Applied egg-rr 42.4%

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

   1. expm1-def 54.5%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. pow2 87.0%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 60.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathbf{if}\;t \leq 155000:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(t_1 \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (if (<= t 155000.0)
     (* t_1 (sqrt (* 2.0 z)))
     (sqrt (* (* 2.0 z) (* t_1 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 155000.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = sqrt(((2.0 * z) * (t_1 * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.5d0 * x) - y
    if (t <= 155000.0d0) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = sqrt(((2.0d0 * z) * (t_1 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	double tmp;
	if (t <= 155000.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(((2.0 * z) * (t_1 * t_1)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (0.5 * x) - y
	tmp = 0
	if t <= 155000.0:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(((2.0 * z) * (t_1 * t_1)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	tmp = 0.0
	if (t <= 155000.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(Float64(2.0 * z) * Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (0.5 * x) - y;
	tmp = 0.0;
	if (t <= 155000.0)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = sqrt(((2.0 * z) * (t_1 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 155000.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 155000

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.7%

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

      3. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around 0 74.8%

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

   if 155000 < t

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 17.0%

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

      1. sub-neg 17.0%

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

      2. *-commutative 17.0%

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

      3. sub-neg 17.0%

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

      4. add-sqr-sqrt 6.5%

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

      5. sqrt-unprod 19.3%

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

      6. swap-sqr 24.1%

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

      7. add-sqr-sqrt 24.1%

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

      8. *-commutative 24.1%

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

      9. pow2 24.1%

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

      10. *-commutative 24.1%

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

   7. Applied egg-rr 24.1%

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

      1. unpow2 24.1%

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

   9. Applied egg-rr 24.1%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 155000:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(\left(0.5 \cdot x - y\right) \cdot \left(0.5 \cdot x - y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt((2.0 * z)) * (((0.5 * x) - y) * (1.0 + (0.5 * (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 = sqrt((2.0d0 * z)) * (((0.5d0 * x) - y) * (1.0d0 + (0.5d0 * (t * t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 87.0%

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

   1. *-commutative 87.0%

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

7. Simplified 87.0%

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

   1. pow2 87.0%

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

9. Applied egg-rr 87.0%

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

10. Final simplification 87.0%

    \[\leadsto \sqrt{2 \cdot z} \cdot \left(\left(0.5 \cdot x - y\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right)\right) \]
11. Add Preprocessing

Alternative 5: 58.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 61.3%

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

6. Final simplification 61.3%

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

Alternative 6: 31.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. *-commutative 99.0%

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

   2. associate-*l* 99.8%

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

   3. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 61.3%

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

6. Taylor expanded in x around inf 33.4%

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

   1. associate-*l* 33.4%

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

8. Simplified 33.4%

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

   1. expm1-log1p-u 22.0%

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

   2. expm1-udef 15.9%

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

   3. sqrt-prod 15.9%

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

10. Applied egg-rr 15.9%

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

    1. expm1-def 22.0%

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

    2. expm1-log1p 33.5%

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

12. Simplified 33.5%

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

13. Final simplification 33.5%

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

Developer target: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024021 
(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))))