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

Percentage Accurate: 99.4% → 99.8%

Time: 10.4s

Alternatives: 8

Speedup: 1.0×

• 44.1% 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, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * (sqrt((z * 2.0)) * exp((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 = ((x * 0.5d0) - y) * (sqrt((z * 2.0d0)) * exp((0.5d0 * (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.exp((0.5 * (t * t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. add-exp-log 99.8%

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

   2. pow1/2 99.8%

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

   3. log-pow 99.8%

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

   4. add-log-exp 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 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 99.4%

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

2. Final simplification 99.4%

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

Alternative 3: 85.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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))) * ((0.5d0 * (t * t)) + 1.0d0)
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))) * ((0.5 * (t * t)) + 1.0);
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * ((0.5 * (t * t)) + 1.0);
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[(N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision] + 1.0), $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. Taylor expanded in t around 0 88.6%

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

   1. unpow2 88.6%

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

4. Simplified 88.6%

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

5. Final simplification 88.6%

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

Alternative 4: 86.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * (sqrt((z * 2.0)) * (1.0 + (t * (0.5 * 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)) * (1.0d0 + (t * (0.5d0 * 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)) * (1.0 + (t * (0.5 * t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. add-exp-log 99.8%

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

   2. pow1/2 99.8%

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

   3. log-pow 99.8%

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

   4. add-log-exp 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in t around 0 89.4%

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

   1. unpow2 89.4%

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

   2. associate-*r* 89.4%

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

8. Simplified 89.4%

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

9. Final simplification 89.4%

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

Alternative 5: 32.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0038:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -0.0038) (sqrt (* z (* 0.5 (* x x)))) (* x (sqrt (* 0.5 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.0038) {
		tmp = sqrt((z * (0.5 * (x * x))));
	} else {
		tmp = x * sqrt((0.5 * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.0038d0)) then
        tmp = sqrt((z * (0.5d0 * (x * x))))
    else
        tmp = x * sqrt((0.5d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.0038) {
		tmp = Math.sqrt((z * (0.5 * (x * x))));
	} else {
		tmp = x * Math.sqrt((0.5 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -0.0038:
		tmp = math.sqrt((z * (0.5 * (x * x))))
	else:
		tmp = x * math.sqrt((0.5 * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -0.0038)
		tmp = sqrt(Float64(z * Float64(0.5 * Float64(x * x))));
	else
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -0.0038)
		tmp = sqrt((z * (0.5 * (x * x))));
	else
		tmp = x * sqrt((0.5 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -0.0038], N[Sqrt[N[(z * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00379999999999999999

   1. Initial program 98.6%

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

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 20.0%

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

   5. Taylor expanded in x around inf 9.4%

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

      1. *-commutative 9.4%

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

      2. associate-*l* 9.4%

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

   7. Simplified 9.4%

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

      1. add-sqr-sqrt 6.8%

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

      2. sqrt-unprod 20.4%

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

      3. *-commutative 20.4%

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

      4. *-commutative 20.4%

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

      5. swap-sqr 23.1%

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

      6. swap-sqr 23.1%

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

      7. add-sqr-sqrt 23.1%

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

      8. metadata-eval 23.1%

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

      9. swap-sqr 23.1%

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

      10. add-sqr-sqrt 23.1%

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

   9. Applied egg-rr 23.1%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 0.25\right) \cdot \left(2 \cdot \left(x \cdot x\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*l* 23.1%

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

       2. unpow2 23.1%

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

       3. associate-*r* 23.1%

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

       4. metadata-eval 23.1%

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

       5. unpow2 23.1%

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

   11. Simplified 23.1%

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

   if -0.00379999999999999999 < t

   1. Initial program 99.7%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. 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. Simplified 99.7%

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

   4. Taylor expanded in t around 0 75.7%

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

   5. Taylor expanded in x around inf 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-*l* 44.2%

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

   7. Simplified 44.2%

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

      1. add-sqr-sqrt 20.0%

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

      2. sqrt-unprod 15.2%

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

      3. *-commutative 15.2%

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

      4. *-commutative 15.2%

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

      5. swap-sqr 15.6%

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

      6. swap-sqr 15.6%

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

      7. add-sqr-sqrt 15.6%

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

      8. metadata-eval 15.6%

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

      9. swap-sqr 15.6%

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

      10. add-sqr-sqrt 15.7%

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

   9. Applied egg-rr 15.7%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 0.25\right) \cdot \left(2 \cdot \left(x \cdot x\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*l* 15.7%

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

       2. unpow2 15.7%

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

       3. associate-*r* 15.7%

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

       4. metadata-eval 15.7%

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

       5. unpow2 15.7%

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

   11. Simplified 15.7%

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

       1. associate-*r* 15.7%

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

       2. sqrt-prod 15.7%

          \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{x \cdot x}} \]

       3. sqrt-prod 18.9%

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

       4. add-sqr-sqrt 44.3%

          \[\leadsto \sqrt{z \cdot 0.5} \cdot \color{blue}{x} \]

   13. Applied egg-rr 44.3%

       \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0038:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \end{array} \]

Alternative 6: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.55e+94) (sqrt (* (* z 2.0) (* y y))) (* x (sqrt (* 0.5 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.55e+94) {
		tmp = sqrt(((z * 2.0) * (y * y)));
	} else {
		tmp = x * sqrt((0.5 * z));
	}
	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 (y <= (-1.55d+94)) then
        tmp = sqrt(((z * 2.0d0) * (y * y)))
    else
        tmp = x * sqrt((0.5d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.55e+94) {
		tmp = Math.sqrt(((z * 2.0) * (y * y)));
	} else {
		tmp = x * Math.sqrt((0.5 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.55e+94:
		tmp = math.sqrt(((z * 2.0) * (y * y)))
	else:
		tmp = x * math.sqrt((0.5 * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.55e+94)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * y)));
	else
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.55e+94)
		tmp = sqrt(((z * 2.0) * (y * y)));
	else
		tmp = x * sqrt((0.5 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.55e+94], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.54999999999999996e94

   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. Taylor expanded in t around 0 70.1%

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

      1. add-sqr-sqrt 62.8%

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

      2. sqrt-unprod 69.8%

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

      3. *-commutative 69.8%

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

      4. *-commutative 69.8%

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

      5. swap-sqr 67.4%

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

      6. add-sqr-sqrt 67.4%

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

      7. pow2 67.4%

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

   4. Applied egg-rr 67.4%

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

   5. Taylor expanded in x around 0 67.3%

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

      1. unpow2 67.3%

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

   7. Simplified 67.3%

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

   if -1.54999999999999996e94 < y

   1. Initial program 99.3%

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

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 57.9%

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

   5. Taylor expanded in x around inf 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-*l* 36.3%

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

   7. Simplified 36.3%

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

      1. add-sqr-sqrt 16.3%

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

      2. sqrt-unprod 14.7%

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

      3. *-commutative 14.7%

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

      4. *-commutative 14.7%

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

      5. swap-sqr 16.0%

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

      6. swap-sqr 16.0%

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

      7. add-sqr-sqrt 16.0%

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

      8. metadata-eval 16.0%

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

      9. swap-sqr 16.0%

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

      10. add-sqr-sqrt 16.0%

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

   9. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{\sqrt{\left(z \cdot 0.25\right) \cdot \left(2 \cdot \left(x \cdot x\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*l* 16.0%

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

       2. unpow2 16.0%

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

       3. associate-*r* 16.0%

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

       4. metadata-eval 16.0%

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

       5. unpow2 16.0%

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

   11. Simplified 16.0%

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

       1. associate-*r* 16.0%

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

       2. sqrt-prod 16.5%

          \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{x \cdot x}} \]

       3. sqrt-prod 15.4%

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

       4. add-sqr-sqrt 36.4%

          \[\leadsto \sqrt{z \cdot 0.5} \cdot \color{blue}{x} \]

   13. Applied egg-rr 36.4%

       \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \end{array} \]

Alternative 7: 57.3% 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 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 59.8%

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

   1. *-commutative 59.8%

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

   2. sqrt-prod 60.0%

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

6. Applied egg-rr 60.0%

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

7. Final simplification 60.0%

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

Alternative 8: 30.2% 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 99.4%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 59.8%

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

5. Taylor expanded in x around inf 34.3%

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

   1. *-commutative 34.3%

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

   2. associate-*l* 34.3%

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

7. Simplified 34.3%

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

   1. add-sqr-sqrt 16.2%

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

   2. sqrt-unprod 16.7%

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

   3. *-commutative 16.7%

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

   4. *-commutative 16.7%

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

   5. swap-sqr 17.8%

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

   6. swap-sqr 17.8%

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

   7. add-sqr-sqrt 17.8%

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

   8. metadata-eval 17.8%

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

   9. swap-sqr 17.7%

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

   10. add-sqr-sqrt 17.8%

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

9. Applied egg-rr 17.8%

   \[\leadsto \color{blue}{\sqrt{\left(z \cdot 0.25\right) \cdot \left(2 \cdot \left(x \cdot x\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*l* 17.8%

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

    2. unpow2 17.8%

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

    3. associate-*r* 17.8%

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

    4. metadata-eval 17.8%

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

    5. unpow2 17.8%

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

11. Simplified 17.8%

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

    1. associate-*r* 17.8%

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

    2. sqrt-prod 16.0%

       \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot \sqrt{x \cdot x}} \]

    3. sqrt-prod 15.4%

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

    4. add-sqr-sqrt 34.3%

       \[\leadsto \sqrt{z \cdot 0.5} \cdot \color{blue}{x} \]

13. Applied egg-rr 34.3%

    \[\leadsto \color{blue}{\sqrt{z \cdot 0.5} \cdot x} \]

14. Final simplification 34.3%

    \[\leadsto x \cdot \sqrt{0.5 \cdot z} \]

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 2023192 
(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))))