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

Percentage Accurate: 99.6% → 99.8%

Time: 11.9s

Alternatives: 10

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: 99.6% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5 + (-y)), $MachinePrecision] * N[Power[N[Sqrt[N[Exp[t$95$m], $MachinePrecision]], $MachinePrecision], t$95$m], $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. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. fma-neg 99.8%

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

   4. associate-*l/ 99.8%

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

   5. exp-prod 99.8%

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

   6. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t\_m \leq 11.6:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\left(-y\right) \cdot {\left(1 + 0.5 \cdot t\_m\right)}^{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t_m 11.6)
     (* t_1 (- (* x 0.5) y))
     (* t_1 (* (- y) (pow (+ 1.0 (* 0.5 t_m)) t_m))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t_m <= 11.6) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * (-y * pow((1.0 + (0.5 * t_m)), t_m));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t_m <= 11.6d0) then
        tmp = t_1 * ((x * 0.5d0) - y)
    else
        tmp = t_1 * (-y * ((1.0d0 + (0.5d0 * t_m)) ** t_m))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t_m <= 11.6) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * (-y * Math.pow((1.0 + (0.5 * t_m)), t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t_m <= 11.6:
		tmp = t_1 * ((x * 0.5) - y)
	else:
		tmp = t_1 * (-y * math.pow((1.0 + (0.5 * t_m)), t_m))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t_m <= 11.6)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(t_1 * Float64(Float64(-y) * (Float64(1.0 + Float64(0.5 * t_m)) ^ t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t_m <= 11.6)
		tmp = t_1 * ((x * 0.5) - y);
	else
		tmp = t_1 * (-y * ((1.0 + (0.5 * t_m)) ^ t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$m, 11.6], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[((-y) * N[Power[N[(1.0 + N[(0.5 * t$95$m), $MachinePrecision]), $MachinePrecision], t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 11.5999999999999996

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.7%

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

      3. fma-neg 99.7%

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

      4. associate-*l/ 99.7%

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

      5. exp-prod 99.8%

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

      6. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 71.7%

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

   if 11.5999999999999996 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-neg 100.0%

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

      4. associate-*l/ 100.0%

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

      5. exp-prod 100.0%

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

      6. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 76.7%

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

   6. Taylor expanded in t around 0 76.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 11.6:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(\left(-y\right) \cdot {\left(1 + 0.5 \cdot t\right)}^{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t\_m \leq 11.6:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot y\right) \cdot \left(-{\left(1 + 0.5 \cdot t\_m\right)}^{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t_m 11.6)
     (* t_1 (- (* x 0.5) y))
     (* (* t_1 y) (- (pow (+ 1.0 (* 0.5 t_m)) t_m))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t_m <= 11.6) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = (t_1 * y) * -pow((1.0 + (0.5 * t_m)), t_m);
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t_m <= 11.6d0) then
        tmp = t_1 * ((x * 0.5d0) - y)
    else
        tmp = (t_1 * y) * -((1.0d0 + (0.5d0 * t_m)) ** t_m)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t_m <= 11.6) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = (t_1 * y) * -Math.pow((1.0 + (0.5 * t_m)), t_m);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t_m <= 11.6:
		tmp = t_1 * ((x * 0.5) - y)
	else:
		tmp = (t_1 * y) * -math.pow((1.0 + (0.5 * t_m)), t_m)
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t_m <= 11.6)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(Float64(t_1 * y) * Float64(-(Float64(1.0 + Float64(0.5 * t_m)) ^ t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t_m <= 11.6)
		tmp = t_1 * ((x * 0.5) - y);
	else
		tmp = (t_1 * y) * -((1.0 + (0.5 * t_m)) ^ t_m);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$m, 11.6], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * y), $MachinePrecision] * (-N[Power[N[(1.0 + N[(0.5 * t$95$m), $MachinePrecision]), $MachinePrecision], t$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 11.5999999999999996

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.7%

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

      3. fma-neg 99.7%

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

      4. associate-*l/ 99.7%

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

      5. exp-prod 99.8%

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

      6. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 71.7%

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

   if 11.5999999999999996 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-neg 100.0%

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

      4. associate-*l/ 100.0%

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

      5. exp-prod 100.0%

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

      6. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 76.7%

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

   6. Taylor expanded in t around 0 76.7%

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

   7. Taylor expanded in t around -inf 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-*r* 76.7%

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

      3. associate-*r* 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*l* 76.7%

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

      6. sqrt-prod 76.7%

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

      7. *-commutative 76.7%

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

      8. *-commutative 76.7%

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

      9. sub-neg 76.7%

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

      10. distribute-lft-neg-in 76.7%

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

      11. metadata-eval 76.7%

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

      12. *-commutative 76.7%

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

   9. Applied egg-rr 76.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 11.6:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{z \cdot 2} \cdot y\right) \cdot \left(-{\left(1 + 0.5 \cdot t\right)}^{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 15.5:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= t_m 15.5)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (* x (* (* (sqrt 2.0) (sqrt z)) (- 0.5 (/ y x))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 15.5) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = x * ((sqrt(2.0) * sqrt(z)) * (0.5 - (y / x)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 15.5d0) then
        tmp = sqrt((z * 2.0d0)) * ((x * 0.5d0) - y)
    else
        tmp = x * ((sqrt(2.0d0) * sqrt(z)) * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 15.5) {
		tmp = Math.sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = x * ((Math.sqrt(2.0) * Math.sqrt(z)) * (0.5 - (y / x)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	tmp = 0
	if t_m <= 15.5:
		tmp = math.sqrt((z * 2.0)) * ((x * 0.5) - y)
	else:
		tmp = x * ((math.sqrt(2.0) * math.sqrt(z)) * (0.5 - (y / x)))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 15.5)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(x * Float64(Float64(sqrt(2.0) * sqrt(z)) * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 15.5)
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	else
		tmp = x * ((sqrt(2.0) * sqrt(z)) * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[t$95$m, 15.5], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 15.5

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.7%

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

      3. fma-neg 99.7%

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

      4. associate-*l/ 99.7%

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

      5. exp-prod 99.8%

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

      6. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 71.7%

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

   if 15.5 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-neg 100.0%

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

      4. associate-*l/ 100.0%

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

      5. exp-prod 100.0%

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

      6. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 10.9%

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

      1. *-commutative 10.9%

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

      2. associate-*l* 10.9%

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

   7. Simplified 10.9%

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

   8. Taylor expanded in x around inf 26.4%

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

      1. +-commutative 26.4%

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

      2. mul-1-neg 26.4%

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

      3. unsub-neg 26.4%

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

      4. *-commutative 26.4%

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

      5. associate-*l/ 22.4%

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

      6. *-commutative 22.4%

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

      7. *-commutative 22.4%

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

      8. associate-*r* 22.4%

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

      9. associate-/l* 26.4%

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

      10. distribute-lft-out-- 26.4%

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

      11. *-commutative 26.4%

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

   10. Simplified 26.4%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 15.5:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2800000000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(x \cdot \left(\sqrt{2} \cdot \frac{y}{-x}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= t_m 2800000000.0)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (* (sqrt z) (* x (* (sqrt 2.0) (/ y (- x)))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 2800000000.0) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = sqrt(z) * (x * (sqrt(2.0) * (y / -x)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 2800000000.0d0) then
        tmp = sqrt((z * 2.0d0)) * ((x * 0.5d0) - y)
    else
        tmp = sqrt(z) * (x * (sqrt(2.0d0) * (y / -x)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 2800000000.0) {
		tmp = Math.sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = Math.sqrt(z) * (x * (Math.sqrt(2.0) * (y / -x)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	tmp = 0
	if t_m <= 2800000000.0:
		tmp = math.sqrt((z * 2.0)) * ((x * 0.5) - y)
	else:
		tmp = math.sqrt(z) * (x * (math.sqrt(2.0) * (y / -x)))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 2800000000.0)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(sqrt(z) * Float64(x * Float64(sqrt(2.0) * Float64(y / Float64(-x)))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 2800000000.0)
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	else
		tmp = sqrt(z) * (x * (sqrt(2.0) * (y / -x)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[t$95$m, 2800000000.0], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * N[(y / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8e9

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.7%

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

      3. fma-neg 99.7%

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

      4. associate-*l/ 99.7%

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

      5. exp-prod 99.8%

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

      6. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 70.6%

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

   if 2.8e9 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-neg 100.0%

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

      4. associate-*l/ 100.0%

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

      5. exp-prod 100.0%

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

      6. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 11.1%

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

   6. Taylor expanded in x around inf 20.7%

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

      1. +-commutative 20.7%

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

      2. mul-1-neg 20.7%

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

      3. unsub-neg 20.7%

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

      4. *-commutative 20.7%

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

      5. *-commutative 20.7%

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

      6. associate-/l* 20.7%

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

      7. distribute-lft-out-- 20.7%

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

   8. Simplified 20.7%

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

   9. Taylor expanded in y around inf 16.9%

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

       1. mul-1-neg 16.9%

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

       2. distribute-neg-frac2 16.9%

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

   11. Simplified 16.9%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2800000000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(x \cdot \left(\sqrt{2} \cdot \frac{y}{-x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 7: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 9500000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (if (<= t_m 9500000.0)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (sqrt (* (* z 2.0) (pow y 2.0)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 9500000.0) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = sqrt(((z * 2.0) * pow(y, 2.0)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 9500000.0d0) then
        tmp = sqrt((z * 2.0d0)) * ((x * 0.5d0) - y)
    else
        tmp = sqrt(((z * 2.0d0) * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 9500000.0) {
		tmp = Math.sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	tmp = 0
	if t_m <= 9500000.0:
		tmp = math.sqrt((z * 2.0)) * ((x * 0.5) - y)
	else:
		tmp = math.sqrt(((z * 2.0) * math.pow(y, 2.0)))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 9500000.0)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 9500000.0)
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	else
		tmp = sqrt(((z * 2.0) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := If[LessEqual[t$95$m, 9500000.0], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5e6

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.7%

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

      3. fma-neg 99.7%

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

      4. associate-*l/ 99.7%

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

      5. exp-prod 99.8%

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

      6. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 70.9%

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

   if 9.5e6 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-neg 100.0%

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

      4. associate-*l/ 100.0%

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

      5. exp-prod 100.0%

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

      6. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 77.5%

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

   6. Taylor expanded in t around 0 7.3%

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

      1. add-sqr-sqrt 4.2%

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

      2. sqrt-unprod 13.9%

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

      3. swap-sqr 19.3%

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

      4. add-sqr-sqrt 19.3%

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

      5. mul-1-neg 19.3%

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

      6. mul-1-neg 19.3%

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

      7. sqr-neg 19.3%

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

      8. pow2 19.3%

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

   8. Applied egg-rr 19.3%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9500000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. fma-neg 99.8%

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

   4. associate-*l/ 99.8%

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

   5. exp-prod 99.8%

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

   6. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around 0 54.3%

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

6. Final simplification 54.3%

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

Alternative 9: 30.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Sqrt[N[(z * 2.0), $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. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. fma-neg 99.8%

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

   4. associate-*l/ 99.8%

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

   5. exp-prod 99.8%

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

   6. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 66.8%

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

6. Taylor expanded in t around 0 32.4%

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

   1. *-commutative 32.4%

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

   2. associate-*l* 32.4%

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

   3. add-sqr-sqrt 16.5%

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

   4. sqrt-unprod 16.6%

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

   5. sqr-neg 16.6%

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

   6. mul-1-neg 16.6%

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

   7. mul-1-neg 16.6%

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

   8. sqrt-unprod 1.3%

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

   9. add-sqr-sqrt 2.7%

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

   10. *-commutative 2.7%

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

   11. add-sqr-sqrt 1.3%

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

   12. sqrt-unprod 16.6%

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

   13. mul-1-neg 16.6%

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

   14. mul-1-neg 16.6%

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

   15. sqr-neg 16.6%

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

   16. sqrt-unprod 16.5%

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

   17. add-sqr-sqrt 32.4%

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

8. Applied egg-rr 32.4%

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

   1. neg-mul-1 32.4%

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

   2. *-commutative 32.4%

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

   3. distribute-rgt-neg-in 32.4%

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

   4. *-commutative 32.4%

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

10. Simplified 32.4%

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

11. Final simplification 32.4%

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

Alternative 10: 2.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Sqrt[N[(z * 2.0), $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. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. associate-*l* 99.8%

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

   3. fma-neg 99.8%

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

   4. associate-*l/ 99.8%

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

   5. exp-prod 99.8%

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

   6. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 66.8%

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

6. Taylor expanded in t around 0 32.4%

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

   1. add-sqr-sqrt 15.8%

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

   2. sqrt-unprod 18.5%

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

   3. mul-1-neg 18.5%

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

   4. mul-1-neg 18.5%

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

   5. sqr-neg 18.5%

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

   6. sqrt-unprod 1.5%

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

   7. add-sqr-sqrt 2.7%

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

8. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{\sqrt{z \cdot 2} \cdot y} \]
9. Add Preprocessing

Developer target: 99.6% 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 2024096 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))