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

Percentage Accurate: 99.5% → 99.1%

Time: 2.5s

Alternatives: 9

Speedup: 1.0×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% 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: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 290:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot e^{\frac{t \cdot t}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 290.0)
     (* t_1 (sqrt (* z 2.0)))
     (* t_1 (exp (/ (* t t) 2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 290.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = t_1 * exp(((t * t) / 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if ((t * t) <= 290.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else
        tmp = t_1 * exp(((t * t) / 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 290.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = t_1 * Math.exp(((t * t) / 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (t * t) <= 290.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	else:
		tmp = t_1 * math.exp(((t * t) / 2.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 290.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(t_1 * exp(Float64(Float64(t * t) / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((t * t) <= 290.0)
		tmp = t_1 * sqrt((z * 2.0));
	else
		tmp = t_1 * exp(((t * t) / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 290.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 290

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.0%

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

   if 290 < (*.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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right)} \cdot e^{\frac{t \cdot t}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 18.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;\left(t\_1 \cdot t\_2\right) \cdot e^{\frac{t \cdot t}{2}} \leq -5 \cdot 10^{-215}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot t\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= (* (* t_1 t_2) (exp (/ (* t t) 2.0))) -5e-215)
     t_1
     (* (* t t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (((t_1 * t_2) * exp(((t * t) / 2.0))) <= -5e-215) {
		tmp = t_1;
	} else {
		tmp = (t * t) * t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((z * 2.0d0))
    if (((t_1 * t_2) * exp(((t * t) / 2.0d0))) <= (-5d-215)) then
        tmp = t_1
    else
        tmp = (t * t) * t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (((t_1 * t_2) * Math.exp(((t * t) / 2.0))) <= -5e-215) {
		tmp = t_1;
	} else {
		tmp = (t * t) * t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if ((t_1 * t_2) * math.exp(((t * t) / 2.0))) <= -5e-215:
		tmp = t_1
	else:
		tmp = (t * t) * t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * t_2) * exp(Float64(Float64(t * t) / 2.0))) <= -5e-215)
		tmp = t_1;
	else
		tmp = Float64(Float64(t * t) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (((t_1 * t_2) * exp(((t * t) / 2.0))) <= -5e-215)
		tmp = t_1;
	else
		tmp = (t * t) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -5e-215], t$95$1, N[(N[(t * t), $MachinePrecision] * t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (-.f64 (*.f64 x #s(literal 1/2 binary64)) y) (sqrt.f64 (*.f64 z #s(literal 2 binary64)))) (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64)))) < -4.99999999999999956e-215

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.8%

      \[\leadsto x \cdot 0.5 - \color{blue}{y} \]

   if -4.99999999999999956e-215 < (*.f64 (*.f64 (-.f64 (*.f64 x #s(literal 1/2 binary64)) y) (sqrt.f64 (*.f64 z #s(literal 2 binary64)))) (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))))

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 30.1%

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

Alternative 3: 17.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;\left(t\_1 \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* (* t_1 (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))) 5e+292)
     t_1
     (* t t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (((t_1 * sqrt((z * 2.0))) * exp(((t * t) / 2.0))) <= 5e+292) {
		tmp = t_1;
	} else {
		tmp = t * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (((t_1 * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))) <= 5d+292) then
        tmp = t_1
    else
        tmp = t * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (((t_1 * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0))) <= 5e+292) {
		tmp = t_1;
	} else {
		tmp = t * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if ((t_1 * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))) <= 5e+292:
		tmp = t_1
	else:
		tmp = t * t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(Float64(t_1 * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0))) <= 5e+292)
		tmp = t_1;
	else
		tmp = Float64(t * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (((t_1 * sqrt((z * 2.0))) * exp(((t * t) / 2.0))) <= 5e+292)
		tmp = t_1;
	else
		tmp = t * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+292], t$95$1, N[(t * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (-.f64 (*.f64 x #s(literal 1/2 binary64)) y) (sqrt.f64 (*.f64 z #s(literal 2 binary64)))) (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64)))) < 4.9999999999999996e292

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.3%

      \[\leadsto x \cdot 0.5 - \color{blue}{y} \]

   if 4.9999999999999996e292 < (*.f64 (*.f64 (-.f64 (*.f64 x #s(literal 1/2 binary64)) y) (sqrt.f64 (*.f64 z #s(literal 2 binary64)))) (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))))

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto t \cdot \color{blue}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 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]

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. Add Preprocessing

Alternative 5: 86.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 320:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot e^{\frac{t \cdot t}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 320.0)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (* (* x 0.5) (exp (/ (* t t) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 320.0) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = (x * 0.5) * exp(((t * t) / 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t * t) <= 320.0d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = (x * 0.5d0) * exp(((t * t) / 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 320.0) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = (x * 0.5) * Math.exp(((t * t) / 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t * t) <= 320.0:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = (x * 0.5) * math.exp(((t * t) / 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 320.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(Float64(x * 0.5) * exp(Float64(Float64(t * t) / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t * t) <= 320.0)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = (x * 0.5) * exp(((t * t) / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 320.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 320

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.0%

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

   if 320 < (*.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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\left(x \cdot 0.5\right)} \cdot e^{\frac{t \cdot t}{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 1.02 \cdot 10^{+192}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t \cdot t}{2} \cdot t\_1\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 1.02e+192)
     (* (- (* x 0.5) y) t_1)
     (* (* (/ (* t t) 2.0) t_1) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 1.02e+192) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = (((t * t) / 2.0) * t_1) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((t * t) <= 1.02d+192) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = (((t * t) / 2.0d0) * t_1) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 1.02e+192) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = (((t * t) / 2.0) * t_1) * t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 1.02e+192:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = (((t * t) / 2.0) * t_1) * t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 1.02e+192)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(t * t) / 2.0) * t_1) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 1.02e+192)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = (((t * t) / 2.0) * t_1) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1.02e+192], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.01999999999999996e192

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.3%

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

   if 1.01999999999999996e192 < (*.f64 t 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 42.6%

       \[\leadsto \left(\frac{t \cdot t}{2} \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{z \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 2.9 \cdot 10^{+265}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot t\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 2.9e+265) (* (- (* x 0.5) y) t_1) (* (* t t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 2.9e+265) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = (t * t) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((t * t) <= 2.9d+265) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = (t * t) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 2.9e+265) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = (t * t) * t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 2.9e+265:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = (t * t) * t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 2.9e+265)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(Float64(t * t) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 2.9e+265)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = (t * t) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2.9e+265], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t * t), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.89999999999999996e265

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.6%

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

   if 2.89999999999999996e265 < (*.f64 t 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 47.2%

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

Alternative 8: 40.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 2.3 \cdot 10^{+235}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot t\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 2.3e+235) (* (* x 0.5) t_1) (* (* t t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 2.3e+235) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = (t * t) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((t * t) <= 2.3d+235) then
        tmp = (x * 0.5d0) * t_1
    else
        tmp = (t * t) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 2.3e+235) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = (t * t) * t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 2.3e+235:
		tmp = (x * 0.5) * t_1
	else:
		tmp = (t * t) * t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 2.3e+235)
		tmp = Float64(Float64(x * 0.5) * t_1);
	else
		tmp = Float64(Float64(t * t) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((t * t) <= 2.3e+235)
		tmp = (x * 0.5) * t_1;
	else
		tmp = (t * t) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2.3e+235], N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t * t), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.3e235

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.6%

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

   if 2.3e235 < (*.f64 t 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 44.0%

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

Alternative 9: 14.4% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ t \cdot t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return t * t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t * t
end function

Java

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

Python

def code(x, y, z, t):
	return t * t

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(t * t), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 55.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 60.3%

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

9. Taylor expanded in y around inf

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

11. Applied rewrites 16.0%

    \[\leadsto t \cdot \color{blue}{t} \]
12. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.6× 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 2024313 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

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