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

Percentage Accurate: 99.4% → 99.8%

Time: 12.2s

Alternatives: 6

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (* (sqrt (* z 2.0)) (sqrt (pow (exp t_m) t_m))) (- (* x 0.5) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Power[N[Exp[t$95$m], $MachinePrecision], t$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.7%

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

   2. exp-sqrt 99.7%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (* z 2.0) (exp (pow t_m 2.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Exp[N[Power[t$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. associate-*r* 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. distribute-lft-in 75.5%

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

   5. exp-sqrt 75.5%

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

   6. pow-exp 75.5%

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

   7. sqrt-unprod 75.5%

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

   8. associate-*l* 75.5%

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

   9. pow-exp 75.5%

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

   10. pow2 75.5%

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

4. Applied egg-rr 75.5%

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

   1. distribute-lft-out 99.7%

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

   2. *-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (* (sqrt (* z 2.0)) (- (* x 0.5) y)) (exp (/ (* t_m t_m) 2.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t$95$m * t$95$m), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 4: 56.5% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (sqrt z) (* (- (* x 0.5) y) (sqrt 2.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0 58.2%

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

4. Final simplification 58.2%

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

Alternative 5: 56.5% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (sqrt 2.0) (* (- (* x 0.5) y) (sqrt z))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0 58.2%

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

   1. associate-*r* 58.1%

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

   2. *-commutative 58.1%

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

   3. associate-*l* 58.1%

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

5. Simplified 58.1%

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

6. Final simplification 58.1%

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

Alternative 6: 29.2% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m) :precision binary64 (* (sqrt z) (* 0.5 (* x (sqrt 2.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 47.8%

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

   2. pow2 47.8%

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

4. Applied egg-rr 47.8%

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

5. Taylor expanded in x around inf 61.0%

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

   1. *-commutative 61.0%

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

   2. associate-*l* 61.0%

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

   3. *-commutative 61.0%

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

   4. associate-*r* 61.0%

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

   5. *-commutative 61.0%

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

   6. *-commutative 61.0%

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

   7. associate-*r* 61.0%

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

7. Simplified 61.0%

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

8. Taylor expanded in t around 0 30.9%

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

   1. associate-*r* 30.9%

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

   2. *-commutative 30.9%

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

10. Simplified 30.9%

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

11. Final simplification 30.9%

    \[\leadsto \sqrt{z} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{2}\right)\right) \]
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 2024111 
(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))))