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

Percentage Accurate: 99.5% → 99.8%

Time: 14.1s

Alternatives: 11

Speedup: 1.0×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (pow (exp t) t) (* 2.0 z)))))

C

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

Fortran

NOTE: t should be positive before calling this function
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(((exp(t) ** t) * (2.0d0 * z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[Power[N[Exp[t], $MachinePrecision], t], $MachinePrecision] * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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. expm1-log1p-u 98.3%

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

   2. expm1-udef 78.3%

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

   3. sqrt-unprod 78.3%

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

   4. associate-*l* 78.3%

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

   5. exp-prod 78.3%

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

5. Applied egg-rr 78.3%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*r* 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (* 2.0 z) (exp (* t t))))))

C

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

Fortran

NOTE: t should be positive before calling this function
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

t = Math.abs(t);
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

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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. expm1-log1p-u 98.3%

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

   2. expm1-udef 78.3%

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

   3. sqrt-unprod 78.3%

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

   4. associate-*l* 78.3%

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

   5. exp-prod 78.3%

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

5. Applied egg-rr 78.3%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*r* 99.8%

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

7. Simplified 99.8%

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

   1. pow-exp 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;2 \cdot z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\left(t_1 \cdot \sqrt{2 \cdot z}\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* 2.0 z) 5e+80)
     (* (* t_1 (sqrt (* 2.0 z))) (+ 1.0 (* 0.5 (* t t))))
     (* t_1 (sqrt (* (* 2.0 z) (+ (* t t) 1.0)))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((2.0 * z) <= 5e+80) {
		tmp = (t_1 * sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	} else {
		tmp = t_1 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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 = (x * 0.5d0) - y
    if ((2.0d0 * z) <= 5d+80) then
        tmp = (t_1 * sqrt((2.0d0 * z))) * (1.0d0 + (0.5d0 * (t * t)))
    else
        tmp = t_1 * sqrt(((2.0d0 * z) * ((t * t) + 1.0d0)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((2.0 * z) <= 5e+80) {
		tmp = (t_1 * Math.sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	} else {
		tmp = t_1 * Math.sqrt(((2.0 * z) * ((t * t) + 1.0)));
	}
	return tmp;
}

Python

t = abs(t)
def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (2.0 * z) <= 5e+80:
		tmp = (t_1 * math.sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)))
	else:
		tmp = t_1 * math.sqrt(((2.0 * z) * ((t * t) + 1.0)))
	return tmp

Julia

t = abs(t)
function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(2.0 * z) <= 5e+80)
		tmp = Float64(Float64(t_1 * sqrt(Float64(2.0 * z))) * Float64(1.0 + Float64(0.5 * Float64(t * t))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * Float64(Float64(t * t) + 1.0))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((2.0 * z) <= 5e+80)
		tmp = (t_1 * sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	else
		tmp = t_1 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(2.0 * z), $MachinePrecision], 5e+80], N[(N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 2) < 4.99999999999999961e80

   1. Initial program 99.1%

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

   2. Taylor expanded in t around 0 81.7%

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

      1. unpow2 81.7%

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

   4. Simplified 81.7%

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

   if 4.99999999999999961e80 < (*.f64 z 2)

   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. Step-by-step derivation

      1. associate-*l* 99.9%

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

      2. exp-sqrt 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 96.8%

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

      2. expm1-udef 96.8%

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

      3. sqrt-unprod 96.8%

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

      4. associate-*l* 96.8%

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

      5. exp-prod 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. expm1-def 96.8%

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

      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. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. unpow2 98.1%

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

   10. Simplified 98.1%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(t \cdot t + 1\right)} \cdot \left(2 \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \end{array} \]

Alternative 4: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 0.0095:\\ \;\;\;\;t_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 0.0095)
     (* t_1 (sqrt (* (* 2.0 z) (+ (* t t) 1.0))))
     (* t_1 (* (sqrt (* 2.0 z)) (* 0.5 (* t t)))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 0.0095) {
		tmp = t_1 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	} else {
		tmp = t_1 * (sqrt((2.0 * z)) * (0.5 * (t * t)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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 = (x * 0.5d0) - y
    if ((t * t) <= 0.0095d0) then
        tmp = t_1 * sqrt(((2.0d0 * z) * ((t * t) + 1.0d0)))
    else
        tmp = t_1 * (sqrt((2.0d0 * z)) * (0.5d0 * (t * t)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 0.0095) {
		tmp = t_1 * Math.sqrt(((2.0 * z) * ((t * t) + 1.0)));
	} else {
		tmp = t_1 * (Math.sqrt((2.0 * z)) * (0.5 * (t * t)));
	}
	return tmp;
}

Python

t = abs(t)
def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (t * t) <= 0.0095:
		tmp = t_1 * math.sqrt(((2.0 * z) * ((t * t) + 1.0)))
	else:
		tmp = t_1 * (math.sqrt((2.0 * z)) * (0.5 * (t * t)))
	return tmp

Julia

t = abs(t)
function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 0.0095)
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * Float64(Float64(t * t) + 1.0))));
	else
		tmp = Float64(t_1 * Float64(sqrt(Float64(2.0 * z)) * Float64(0.5 * Float64(t * t))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((t * t) <= 0.0095)
		tmp = t_1 * sqrt(((2.0 * z) * ((t * t) + 1.0)));
	else
		tmp = t_1 * (sqrt((2.0 * z)) * (0.5 * (t * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 0.0095], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 0.00949999999999999976

   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. Step-by-step derivation

      1. expm1-log1p-u 96.7%

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

      2. expm1-udef 56.6%

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

      3. sqrt-unprod 56.6%

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

      4. associate-*l* 56.6%

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

      5. exp-prod 56.6%

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

   5. Applied egg-rr 56.6%

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

      1. expm1-def 96.7%

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

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow2 99.4%

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

   10. Simplified 99.4%

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

   if 0.00949999999999999976 < (*.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. associate-*l* 99.9%

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

      2. associate-*l/ 99.9%

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

      3. exp-prod 100.0%

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

      4. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1/2 100.0%

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

      3. log-pow 100.0%

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

      4. add-log-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in t around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. associate-*l* 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in t around inf 77.1%

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

       1. unpow2 77.1%

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

   11. Simplified 77.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.0095:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\\ \end{array} \]

Alternative 5: 86.8% accurate, 1.8× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (- (* x 0.5) y) (* (sqrt (* 2.0 z)) (+ 1.0 (* t (* 0.5 t))))))

C

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

Fortran

NOTE: t should be positive before calling this function
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)) * (1.0d0 + (t * (0.5d0 * t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * z), $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. associate-*l/ 99.8%

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

   3. exp-prod 99.8%

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

   4. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

   1. add-exp-log 99.8%

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

   2. pow1/2 99.8%

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

   3. log-pow 99.8%

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

   4. add-log-exp 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in t around 0 88.2%

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

   1. *-commutative 88.2%

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

   2. unpow2 88.2%

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

   3. associate-*l* 88.2%

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

8. Simplified 88.2%

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

9. Final simplification 88.2%

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

Alternative 6: 84.8% accurate, 1.9× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (* 2.0 z) (+ (* t t) 1.0)))))

C

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

Fortran

NOTE: t should be positive before calling this function
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) * ((t * t) + 1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. 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 98.3%

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

   2. expm1-udef 78.3%

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

   3. sqrt-unprod 78.3%

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

   4. associate-*l* 78.3%

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

   5. exp-prod 78.3%

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

5. Applied egg-rr 78.3%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*r* 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 85.9%

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

   1. +-commutative 85.9%

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

   2. unpow2 85.9%

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

10. Simplified 85.9%

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

11. Final simplification 85.9%

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

Alternative 7: 43.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;y \leq -6200000000:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= y -6200000000.0)
     (sqrt (* z (* 2.0 (* y y))))
     (if (<= y 1.1e+14) (* (* x 0.5) t_1) (* t_1 (- y))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if (y <= -6200000000.0) {
		tmp = sqrt((z * (2.0 * (y * y))));
	} else if (y <= 1.1e+14) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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 <= (-6200000000.0d0)) then
        tmp = sqrt((z * (2.0d0 * (y * y))))
    else if (y <= 1.1d+14) then
        tmp = (x * 0.5d0) * t_1
    else
        tmp = t_1 * -y
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if (y <= -6200000000.0) {
		tmp = Math.sqrt((z * (2.0 * (y * y))));
	} else if (y <= 1.1e+14) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Python

t = abs(t)
def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if y <= -6200000000.0:
		tmp = math.sqrt((z * (2.0 * (y * y))))
	elif y <= 1.1e+14:
		tmp = (x * 0.5) * t_1
	else:
		tmp = t_1 * -y
	return tmp

Julia

t = abs(t)
function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (y <= -6200000000.0)
		tmp = sqrt(Float64(z * Float64(2.0 * Float64(y * y))));
	elseif (y <= 1.1e+14)
		tmp = Float64(Float64(x * 0.5) * t_1);
	else
		tmp = Float64(t_1 * Float64(-y));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if (y <= -6200000000.0)
		tmp = sqrt((z * (2.0 * (y * y))));
	elseif (y <= 1.1e+14)
		tmp = (x * 0.5) * t_1;
	else
		tmp = t_1 * -y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -6200000000.0], N[Sqrt[N[(z * N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.1e+14], N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * (-y)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2e9

   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 68.8%

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

   3. Taylor expanded in x around 0 54.7%

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

      1. mul-1-neg 54.7%

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

      2. distribute-rgt-neg-in 54.7%

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

   5. Simplified 54.7%

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

      1. add-sqr-sqrt 54.4%

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

      2. sqrt-unprod 55.4%

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

      3. *-commutative 55.4%

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

      4. *-commutative 55.4%

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

      5. swap-sqr 55.4%

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

      6. add-sqr-sqrt 55.4%

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

      7. swap-sqr 55.3%

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

      8. add-sqr-sqrt 55.5%

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

      9. sqr-neg 55.5%

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

   7. Applied egg-rr 55.5%

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

   if -6.2e9 < y < 1.1e14

   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 53.1%

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

      1. add-sqr-sqrt 27.9%

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

      2. sqrt-unprod 24.2%

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

      3. *-commutative 24.2%

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

      4. *-commutative 24.2%

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

      5. swap-sqr 23.4%

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

      6. add-sqr-sqrt 23.4%

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

      7. pow2 23.4%

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

   4. Applied egg-rr 23.4%

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

   5. Taylor expanded in x around inf 20.7%

      \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(0.25 \cdot {x}^{2}\right)}} \cdot 1 \]
   6. Step-by-step derivation

      1. unpow2 20.7%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \left(0.25 \cdot \color{blue}{\left(x \cdot x\right)}\right)} \cdot 1 \]

      2. *-commutative 20.7%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.25\right)}} \cdot 1 \]

      3. associate-*l* 20.7%

         \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.25\right)\right)}} \cdot 1 \]

   7. Simplified 20.7%

      \[\leadsto \sqrt{\left(z \cdot 2\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.25\right)\right)}} \cdot 1 \]
   8. Step-by-step derivation

      1. *-commutative 20.7%

         \[\leadsto \sqrt{\color{blue}{\left(x \cdot \left(x \cdot 0.25\right)\right) \cdot \left(z \cdot 2\right)}} \cdot 1 \]

      2. sqrt-prod 21.5%

         \[\leadsto \color{blue}{\left(\sqrt{x \cdot \left(x \cdot 0.25\right)} \cdot \sqrt{z \cdot 2}\right)} \cdot 1 \]

      3. sqrt-prod 22.1%

         \[\leadsto \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x \cdot 0.25}\right)} \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

      4. sqrt-prod 22.1%

         \[\leadsto \left(\left(\sqrt{x} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{0.25}\right)}\right) \cdot \sqrt{z \cdot 2}\right) \cdot 1 \]

      5. metadata-eval 22.1%

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

      6. associate-*l* 22.1%

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

      7. add-sqr-sqrt 43.7%

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

   9. Applied egg-rr 43.7%

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

   if 1.1e14 < y

   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 64.1%

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

   3. Taylor expanded in x around 0 52.8%

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

      1. mul-1-neg 52.8%

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

      2. distribute-rgt-neg-in 52.8%

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

   5. Simplified 52.8%

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

      1. distribute-rgt-neg-out 52.8%

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

      2. distribute-lft-neg-out 52.8%

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

      3. add-sqr-sqrt 52.7%

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

      4. sqrt-unprod 56.9%

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

      5. sqr-neg 56.9%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.8%

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

      8. *-commutative 0.8%

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

      9. associate-*r* 0.8%

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

      10. sqrt-prod 0.8%

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

      11. *-commutative 0.8%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 57.0%

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

      14. sqr-neg 57.0%

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

      15. sqrt-unprod 52.6%

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

      16. add-sqr-sqrt 52.8%

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

   7. Applied egg-rr 52.8%

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

      1. distribute-rgt-neg-in 52.8%

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

   9. Simplified 52.8%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6200000000:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot y\right)\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(-y\right)\\ \end{array} \]

Alternative 8: 37.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.05 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t)
 :precision binary64
 (if (<= x 3.05e+72) (* (sqrt (* 2.0 z)) (- y)) (sqrt (* 0.5 (* z (* x x))))))

C

t = abs(t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.05e+72) {
		tmp = sqrt((2.0 * z)) * -y;
	} else {
		tmp = sqrt((0.5 * (z * (x * x))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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 <= 3.05d+72) then
        tmp = sqrt((2.0d0 * z)) * -y
    else
        tmp = sqrt((0.5d0 * (z * (x * x))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 3.05e+72) {
		tmp = Math.sqrt((2.0 * z)) * -y;
	} else {
		tmp = Math.sqrt((0.5 * (z * (x * x))));
	}
	return tmp;
}

Python

t = abs(t)
def code(x, y, z, t):
	tmp = 0
	if x <= 3.05e+72:
		tmp = math.sqrt((2.0 * z)) * -y
	else:
		tmp = math.sqrt((0.5 * (z * (x * x))))
	return tmp

Julia

t = abs(t)
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 3.05e+72)
		tmp = Float64(sqrt(Float64(2.0 * z)) * Float64(-y));
	else
		tmp = sqrt(Float64(0.5 * Float64(z * Float64(x * x))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 3.05e+72)
		tmp = sqrt((2.0 * z)) * -y;
	else
		tmp = sqrt((0.5 * (z * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := If[LessEqual[x, 3.05e+72], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(z * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.04999999999999996e72

   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 60.7%

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

   3. Taylor expanded in x around 0 38.4%

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

      1. mul-1-neg 38.4%

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

      2. distribute-rgt-neg-in 38.4%

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

   5. Simplified 38.4%

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

      1. distribute-rgt-neg-out 38.4%

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

      2. distribute-lft-neg-out 38.4%

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

      3. add-sqr-sqrt 18.0%

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

      4. sqrt-unprod 19.9%

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

      5. sqr-neg 19.9%

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

      6. sqrt-unprod 1.6%

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

      7. add-sqr-sqrt 2.6%

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

      8. *-commutative 2.6%

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

      9. associate-*r* 2.6%

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

      10. sqrt-prod 2.6%

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

      11. *-commutative 2.6%

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

      12. add-sqr-sqrt 1.6%

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

      13. sqrt-unprod 19.9%

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

      14. sqr-neg 19.9%

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

      15. sqrt-unprod 17.9%

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

      16. add-sqr-sqrt 38.5%

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

   7. Applied egg-rr 38.5%

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

      1. distribute-rgt-neg-in 38.5%

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

   9. Simplified 38.5%

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

   if 3.04999999999999996e72 < 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 56.3%

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

      1. add-sqr-sqrt 49.1%

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

      2. sqrt-unprod 41.3%

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

      3. *-commutative 41.3%

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

      4. *-commutative 41.3%

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

      5. swap-sqr 51.5%

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

      6. add-sqr-sqrt 51.5%

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

      7. pow2 51.5%

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

   4. Applied egg-rr 51.5%

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

   5. Taylor expanded in x around inf 51.5%

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

      1. unpow2 51.5%

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

   7. Simplified 51.5%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.05 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternative 9: 56.8% accurate, 2.0× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* (- (* x 0.5) y) (sqrt (* 2.0 z))))

C

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

Fortran

NOTE: t should be positive before calling this function
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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * 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. Step-by-step derivation

   1. expm1-log1p-u 98.3%

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

   2. expm1-udef 78.3%

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

   3. sqrt-unprod 78.3%

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

   4. associate-*l* 78.3%

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

   5. exp-prod 78.3%

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

5. Applied egg-rr 78.3%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*r* 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 60.0%

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

9. Final simplification 60.0%

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

Alternative 10: 30.2% accurate, 2.0× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* (sqrt (* 2.0 z)) (- y)))

C

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

Fortran

NOTE: t should be positive before calling this function
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((2.0d0 * z)) * -y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-y)), $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 59.9%

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

3. Taylor expanded in x around 0 33.4%

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

   1. mul-1-neg 33.4%

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

   2. distribute-rgt-neg-in 33.4%

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

5. Simplified 33.4%

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

   1. distribute-rgt-neg-out 33.4%

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

   2. distribute-lft-neg-out 33.4%

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

   3. add-sqr-sqrt 16.0%

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

   4. sqrt-unprod 18.1%

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

   5. sqr-neg 18.1%

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

   6. sqrt-unprod 1.4%

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

   7. add-sqr-sqrt 2.5%

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

   8. *-commutative 2.5%

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

   9. associate-*r* 2.5%

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

   10. sqrt-prod 2.5%

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

   11. *-commutative 2.5%

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

   12. add-sqr-sqrt 1.4%

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

   13. sqrt-unprod 18.1%

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

   14. sqr-neg 18.1%

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

   15. sqrt-unprod 16.0%

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

   16. add-sqr-sqrt 33.4%

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

7. Applied egg-rr 33.4%

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

   1. distribute-rgt-neg-in 33.4%

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

9. Simplified 33.4%

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

10. Final simplification 33.4%

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

Alternative 11: 2.4% accurate, 2.0× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (x y z t) :precision binary64 (* y (sqrt (* 2.0 z))))

C

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

Fortran

NOTE: t should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[x_, y_, z_, t_] := N[(y * N[Sqrt[N[(2.0 * 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. Taylor expanded in t around 0 59.9%

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

3. Taylor expanded in x around 0 33.4%

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

   1. mul-1-neg 33.4%

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

   2. distribute-rgt-neg-in 33.4%

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

5. Simplified 33.4%

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

   1. expm1-log1p-u 19.5%

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

   2. expm1-udef 14.9%

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

   3. *-commutative 14.9%

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

   4. associate-*r* 14.9%

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

   5. sqrt-prod 14.9%

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

   6. *-commutative 14.9%

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

   7. add-sqr-sqrt 14.0%

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

   8. sqrt-unprod 15.7%

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

   9. sqr-neg 15.7%

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

   10. sqrt-unprod 1.2%

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

   11. add-sqr-sqrt 1.9%

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

7. Applied egg-rr 1.9%

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

   1. expm1-def 1.9%

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

   2. expm1-log1p 2.5%

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

   3. *-commutative 2.5%

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

9. Simplified 2.5%

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

10. Final simplification 2.5%

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

Developer target: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023224 
(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))))