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

Percentage Accurate: 99.4% → 99.8%

Time: 13.5s

Alternatives: 8

Speedup: 1.0×

• 44.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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. expm1-log1p-u 98.5%

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

   2. expm1-udef 73.6%

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

   3. sqrt-unprod 73.6%

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

   4. associate-*l* 73.6%

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

   5. exp-prod 73.6%

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

5. Applied egg-rr 73.6%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-*l* 99.9%

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

   5. *-commutative 99.9%

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

   6. exp-prod 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= (* t t) 4.4e-5)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* y (- t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 4.4e-5) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * (y * -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 = sqrt((2.0d0 * z))
    if ((t * t) <= 4.4d-5) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * (y * -t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 4.4e-5) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (y * -t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (t * t) <= 4.4e-5:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * (y * -t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 4.4e-5)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * Float64(-t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((t * t) <= 4.4e-5)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * (y * -t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 4.4e-5], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * (-t$95$1)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 4.3999999999999999e-5

   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 99.1%

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

      1. *-commutative 99.1%

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

      2. sqrt-prod 99.3%

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

   6. Applied egg-rr 99.3%

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

   if 4.3999999999999999e-5 < (*.f64 t t)

   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. add-cube-cbrt 99.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot \sqrt[3]{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right) \cdot \sqrt[3]{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. pow3 99.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)}^{3}} \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied egg-rr 99.2%

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

      1. unpow3 99.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}} \cdot \sqrt[3]{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right) \cdot \sqrt[3]{\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. add-cube-cbrt 99.2%

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

      3. sqrt-prod 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. add-cbrt-cube 82.9%

         \[\leadsto \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. add-sqr-sqrt 82.9%

         \[\leadsto \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \sqrt[3]{\color{blue}{z} \cdot \sqrt{z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. cbrt-prod 99.2%

         \[\leadsto \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{\sqrt{z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sqrt{z}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right) \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sqrt{z}}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   6. Taylor expanded in x around 0 75.6%

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

   8. Simplified 75.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{2 \cdot z}\right)\right)\\ \end{array} \]

Alternative 3: 43.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \left(0.5 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= y -8.6e-33)
     (sqrt (* 2.0 (* z (* y y))))
     (if (<= y 1.85e+42) (* x (* 0.5 t_1)) (* y (- t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if (y <= -8.6e-33) {
		tmp = sqrt((2.0 * (z * (y * y))));
	} else if (y <= 1.85e+42) {
		tmp = x * (0.5 * t_1);
	} else {
		tmp = y * -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 = sqrt((2.0d0 * z))
    if (y <= (-8.6d-33)) then
        tmp = sqrt((2.0d0 * (z * (y * y))))
    else if (y <= 1.85d+42) then
        tmp = x * (0.5d0 * t_1)
    else
        tmp = y * -t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if (y <= -8.6e-33) {
		tmp = Math.sqrt((2.0 * (z * (y * y))));
	} else if (y <= 1.85e+42) {
		tmp = x * (0.5 * t_1);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if y <= -8.6e-33:
		tmp = math.sqrt((2.0 * (z * (y * y))))
	elif y <= 1.85e+42:
		tmp = x * (0.5 * t_1)
	else:
		tmp = y * -t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (y <= -8.6e-33)
		tmp = sqrt(Float64(2.0 * Float64(z * Float64(y * y))));
	elseif (y <= 1.85e+42)
		tmp = Float64(x * Float64(0.5 * t_1));
	else
		tmp = Float64(y * Float64(-t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if (y <= -8.6e-33)
		tmp = sqrt((2.0 * (z * (y * y))));
	elseif (y <= 1.85e+42)
		tmp = x * (0.5 * t_1);
	else
		tmp = y * -t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -8.6e-33], N[Sqrt[N[(2.0 * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.85e+42], N[(x * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.60000000000000062e-33

   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 55.9%

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

      1. associate-*l* 56.0%

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

      2. *-commutative 56.0%

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

      3. fma-neg 56.0%

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

   4. Simplified 56.0%

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

   5. Taylor expanded in x around 0 43.4%

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

      1. mul-1-neg 43.4%

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

      2. distribute-rgt-neg-in 43.4%

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

   7. Simplified 43.4%

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

      1. add-sqr-sqrt 43.2%

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

      2. sqrt-unprod 58.4%

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

      3. swap-sqr 58.3%

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

      4. add-sqr-sqrt 58.4%

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

      5. *-commutative 58.4%

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

      6. *-commutative 58.4%

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

      7. swap-sqr 56.9%

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

      8. sqr-neg 56.9%

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

      9. add-sqr-sqrt 56.9%

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

   9. Applied egg-rr 56.9%

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

   if -8.60000000000000062e-33 < y < 1.84999999999999998e42

   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. Taylor expanded in t around 0 55.3%

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

   3. Taylor expanded in x around inf 46.0%

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

      1. *-commutative 46.0%

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

      2. associate-*r* 46.0%

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

      3. *-commutative 46.0%

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

   5. Simplified 46.0%

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

      1. expm1-log1p-u 29.7%

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

      2. expm1-udef 15.1%

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

      3. *-commutative 15.1%

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

      4. *-commutative 15.1%

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

      5. associate-*l* 15.1%

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

      6. *-commutative 15.1%

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

      7. sqrt-prod 15.1%

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

   7. Applied egg-rr 15.1%

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

      1. expm1-def 29.7%

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

      2. expm1-log1p 46.1%

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

      3. associate-*l* 46.1%

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

      4. *-commutative 46.1%

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

   9. Simplified 46.1%

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

   if 1.84999999999999998e42 < y

   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 57.2%

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

      1. associate-*l* 57.1%

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

      2. *-commutative 57.1%

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

      3. fma-neg 57.1%

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

   4. Simplified 57.1%

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

   5. Taylor expanded in x around 0 46.3%

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

      1. mul-1-neg 46.3%

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

      2. distribute-rgt-neg-in 46.3%

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

   7. Simplified 46.3%

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

      1. associate-*r* 46.3%

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

      2. distribute-rgt-neg-out 46.3%

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

      3. add-sqr-sqrt 46.3%

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

      4. sqr-neg 46.3%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.1%

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

      7. associate-*r* 1.1%

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

      8. *-commutative 1.1%

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

      9. associate-*l* 1.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 46.4%

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

      12. sqr-neg 46.4%

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

      13. add-sqr-sqrt 46.4%

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

      14. sqrt-prod 46.4%

          \[\leadsto -y \cdot \color{blue}{\sqrt{z \cdot 2}} \]

   9. Applied egg-rr 46.4%

      \[\leadsto \color{blue}{-y \cdot \sqrt{z \cdot 2}} \]
   10. Step-by-step derivation

       1. distribute-lft-neg-in 46.4%

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

       2. *-commutative 46.4%

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

       3. *-commutative 46.4%

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

   11. Simplified 46.4%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \end{array} \]

Alternative 4: 56.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.05e+30)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (sqrt (* 2.0 (* z (* y y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.05e+30) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt((2.0 * (z * (y * y))));
	}
	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 <= 2.05d+30) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt((2.0d0 * (z * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.05e+30) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt((2.0 * (z * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 2.05e+30:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt((2.0 * (z * (y * y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.05e+30)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(2.0 * Float64(z * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.05e+30)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt((2.0 * (z * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.05e+30], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.05000000000000003e30

   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 68.6%

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

      1. *-commutative 68.6%

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

      2. sqrt-prod 68.7%

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

   6. Applied egg-rr 68.7%

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

   if 2.05000000000000003e30 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 8.3%

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

      1. associate-*l* 8.3%

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

      2. *-commutative 8.3%

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

      3. fma-neg 8.3%

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

   4. Simplified 8.3%

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

   5. Taylor expanded in x around 0 6.8%

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

      1. mul-1-neg 6.8%

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

      2. distribute-rgt-neg-in 6.8%

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

   7. Simplified 6.8%

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

      1. add-sqr-sqrt 5.6%

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

      2. sqrt-unprod 21.8%

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

      3. swap-sqr 21.8%

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

      4. add-sqr-sqrt 21.8%

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

      5. *-commutative 21.8%

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

      6. *-commutative 21.8%

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

      7. swap-sqr 27.2%

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

      8. sqr-neg 27.2%

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

      9. add-sqr-sqrt 27.2%

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

   9. Applied egg-rr 27.2%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \end{array} \]

Alternative 5: 30.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.2e+30) (* y (- (sqrt (* 2.0 z)))) (sqrt (* 2.0 (* z (* y y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.2e+30) {
		tmp = y * -sqrt((2.0 * z));
	} else {
		tmp = sqrt((2.0 * (z * (y * y))));
	}
	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 <= 2.2d+30) then
        tmp = y * -sqrt((2.0d0 * z))
    else
        tmp = sqrt((2.0d0 * (z * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.2e+30) {
		tmp = y * -Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt((2.0 * (z * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 2.2e+30:
		tmp = y * -math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt((2.0 * (z * (y * y))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.2e+30)
		tmp = Float64(y * Float64(-sqrt(Float64(2.0 * z))));
	else
		tmp = sqrt(Float64(2.0 * Float64(z * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.2e+30)
		tmp = y * -sqrt((2.0 * z));
	else
		tmp = sqrt((2.0 * (z * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.2e+30], N[(y * (-N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(2.0 * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.2e30

   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. Taylor expanded in t around 0 68.6%

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

      1. associate-*l* 68.6%

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

      2. *-commutative 68.6%

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

      3. fma-neg 68.6%

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

   4. Simplified 68.6%

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

   5. Taylor expanded in x around 0 34.1%

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

      1. mul-1-neg 34.1%

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

      2. distribute-rgt-neg-in 34.1%

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

   7. Simplified 34.1%

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

      1. associate-*r* 34.0%

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

      2. distribute-rgt-neg-out 34.0%

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

      3. add-sqr-sqrt 34.0%

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

      4. sqr-neg 34.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 2.4%

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

      7. associate-*r* 2.4%

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

      8. *-commutative 2.4%

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

      9. associate-*l* 2.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 34.1%

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

      12. sqr-neg 34.1%

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

      13. add-sqr-sqrt 34.1%

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

      14. sqrt-prod 34.2%

          \[\leadsto -y \cdot \color{blue}{\sqrt{z \cdot 2}} \]

   9. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{-y \cdot \sqrt{z \cdot 2}} \]
   10. Step-by-step derivation

       1. distribute-lft-neg-in 34.2%

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

       2. *-commutative 34.2%

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

       3. *-commutative 34.2%

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

   11. Simplified 34.2%

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

   if 2.2e30 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 8.3%

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

      1. associate-*l* 8.3%

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

      2. *-commutative 8.3%

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

      3. fma-neg 8.3%

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

   4. Simplified 8.3%

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

   5. Taylor expanded in x around 0 6.8%

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

      1. mul-1-neg 6.8%

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

      2. distribute-rgt-neg-in 6.8%

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

   7. Simplified 6.8%

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

      1. add-sqr-sqrt 5.6%

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

      2. sqrt-unprod 21.8%

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

      3. swap-sqr 21.8%

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

      4. add-sqr-sqrt 21.8%

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

      5. *-commutative 21.8%

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

      6. *-commutative 21.8%

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

      7. swap-sqr 27.2%

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

      8. sqr-neg 27.2%

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

      9. add-sqr-sqrt 27.2%

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

   9. Applied egg-rr 27.2%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z \cdot \left(y \cdot y\right)\right)}\\ \end{array} \]

Alternative 6: 36.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot z\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 6.5e-100) (* y (- (sqrt (* 2.0 z)))) (sqrt (* (* 0.5 z) (* x x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 6.5e-100) {
		tmp = y * -sqrt((2.0 * z));
	} else {
		tmp = sqrt(((0.5 * z) * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 6.5d-100) then
        tmp = y * -sqrt((2.0d0 * z))
    else
        tmp = sqrt(((0.5d0 * z) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 6.5e-100) {
		tmp = y * -Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(((0.5 * z) * (x * x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 6.5e-100:
		tmp = y * -math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(((0.5 * z) * (x * x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 6.5e-100)
		tmp = Float64(y * Float64(-sqrt(Float64(2.0 * z))));
	else
		tmp = sqrt(Float64(Float64(0.5 * z) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 6.5e-100)
		tmp = y * -sqrt((2.0 * z));
	else
		tmp = sqrt(((0.5 * z) * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 6.5e-100], N[(y * (-N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(N[(0.5 * z), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.50000000000000013e-100

   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. Taylor expanded in t around 0 55.7%

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

      1. associate-*l* 55.8%

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

      2. *-commutative 55.8%

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

      3. fma-neg 55.8%

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

   4. Simplified 55.8%

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

   5. Taylor expanded in x around 0 33.7%

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

      1. mul-1-neg 33.7%

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

      2. distribute-rgt-neg-in 33.7%

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

   7. Simplified 33.7%

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

      1. associate-*r* 33.7%

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

      2. distribute-rgt-neg-out 33.7%

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

      3. add-sqr-sqrt 33.7%

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

      4. sqr-neg 33.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 2.2%

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

      7. associate-*r* 2.2%

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

      8. *-commutative 2.2%

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

      9. associate-*l* 2.2%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 33.7%

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

      12. sqr-neg 33.7%

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

      13. add-sqr-sqrt 33.7%

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

      14. sqrt-prod 33.8%

          \[\leadsto -y \cdot \color{blue}{\sqrt{z \cdot 2}} \]

   9. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{-y \cdot \sqrt{z \cdot 2}} \]
   10. Step-by-step derivation

       1. distribute-lft-neg-in 33.8%

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

       2. *-commutative 33.8%

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

       3. *-commutative 33.8%

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

   11. Simplified 33.8%

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

   if 6.50000000000000013e-100 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0 56.1%

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

   3. Taylor expanded in x around inf 39.5%

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

      1. *-commutative 39.5%

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

      2. associate-*r* 39.5%

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

      3. *-commutative 39.5%

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

   5. Simplified 39.5%

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

      1. add-sqr-sqrt 39.4%

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

      2. sqrt-unprod 44.8%

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

      3. *-commutative 44.8%

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

      4. *-commutative 44.8%

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

      5. swap-sqr 45.9%

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

      6. add-sqr-sqrt 45.9%

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

      7. *-commutative 45.9%

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

      8. *-commutative 45.9%

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

      9. swap-sqr 45.9%

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

      10. add-sqr-sqrt 46.0%

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

      11. pow2 46.0%

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

   7. Applied egg-rr 46.0%

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

      1. associate-*r* 46.0%

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

      2. *-commutative 46.0%

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

      3. unpow2 46.0%

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

      4. swap-sqr 46.0%

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

      5. metadata-eval 46.0%

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

   9. Simplified 46.0%

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

   10. Taylor expanded in z around 0 46.0%

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

       1. associate-*r* 46.0%

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

       2. unpow2 46.0%

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

   12. Simplified 46.0%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-100}:\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot z\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 7: 29.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in t around 0 55.8%

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

   1. associate-*l* 55.9%

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

   2. *-commutative 55.9%

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

   3. fma-neg 55.9%

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

4. Simplified 55.9%

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

5. Taylor expanded in x around 0 28.3%

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

   1. mul-1-neg 28.3%

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

   2. distribute-rgt-neg-in 28.3%

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

7. Simplified 28.3%

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

   1. associate-*r* 28.3%

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

   2. distribute-rgt-neg-out 28.3%

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

   3. add-sqr-sqrt 28.3%

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

   4. sqr-neg 28.3%

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

   5. sqrt-unprod 0.0%

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

   6. add-sqr-sqrt 2.2%

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

   7. associate-*r* 2.2%

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

   8. *-commutative 2.2%

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

   9. associate-*l* 2.2%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-unprod 28.4%

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

   12. sqr-neg 28.4%

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

   13. add-sqr-sqrt 28.4%

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

   14. sqrt-prod 28.4%

       \[\leadsto -y \cdot \color{blue}{\sqrt{z \cdot 2}} \]

9. Applied egg-rr 28.4%

   \[\leadsto \color{blue}{-y \cdot \sqrt{z \cdot 2}} \]
10. Step-by-step derivation

    1. distribute-lft-neg-in 28.4%

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

    2. *-commutative 28.4%

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

    3. *-commutative 28.4%

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

11. Simplified 28.4%

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

12. Final simplification 28.4%

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

Alternative 8: 2.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \sqrt{2 \cdot z} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in t around 0 55.8%

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

   1. associate-*l* 55.9%

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

   2. *-commutative 55.9%

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

   3. fma-neg 55.9%

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

4. Simplified 55.9%

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

5. Taylor expanded in x around 0 28.3%

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

   1. mul-1-neg 28.3%

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

   2. distribute-rgt-neg-in 28.3%

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

7. Simplified 28.3%

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

   1. expm1-log1p-u 18.0%

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

   2. expm1-udef 12.3%

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

   3. *-commutative 12.3%

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

   4. associate-*l* 12.3%

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

   5. add-sqr-sqrt 0.0%

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

   6. sqrt-unprod 1.9%

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

   7. sqr-neg 1.9%

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

   8. add-sqr-sqrt 1.9%

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

   9. sqrt-prod 1.9%

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

9. Applied egg-rr 1.9%

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

    1. expm1-def 1.9%

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

    2. expm1-log1p 2.2%

       \[\leadsto \color{blue}{y \cdot \sqrt{z \cdot 2}} \]

    3. *-commutative 2.2%

       \[\leadsto y \cdot \sqrt{\color{blue}{2 \cdot z}} \]

11. Simplified 2.2%

    \[\leadsto \color{blue}{y \cdot \sqrt{2 \cdot z}} \]

12. Final simplification 2.2%

    \[\leadsto y \cdot \sqrt{2 \cdot z} \]

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 2023202 
(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))))