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

Percentage Accurate: 99.4% → 99.8%

Time: 15.1s

Alternatives: 15

Speedup: 1.0×

• 45.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

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. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg N/A

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

   7. pow-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. div-inv N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left({\mathsf{E}\left(\right)}^{\left(\left(t \cdot t\right) \cdot -0.5\right)}\right)}^{-1} \]
6. Add Preprocessing

Alternative 3: 85.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (exp (/ (* t t) 2.0)) 2.0)
     (* (* (- (* x 0.5) y) t_1) 1.0)
     (* (* (* (* (- (* 0.5 x) y) 0.5) 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 (exp(((t * t) / 2.0)) <= 2.0) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else {
		tmp = (((((0.5 * x) - y) * 0.5) * 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 (exp(((t * t) / 2.0d0)) <= 2.0d0) then
        tmp = (((x * 0.5d0) - y) * t_1) * 1.0d0
    else
        tmp = (((((0.5d0 * x) - y) * 0.5d0) * 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 (Math.exp(((t * t) / 2.0)) <= 2.0) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else {
		tmp = (((((0.5 * x) - y) * 0.5) * t) * t) * t_1;
	}
	return tmp;
}

Python

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

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (exp(Float64(Float64(t * t) / 2.0)) <= 2.0)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.5 * x) - y) * 0.5) * 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 (exp(((t * t) / 2.0)) <= 2.0)
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	else
		tmp = (((((0.5 * x) - y) * 0.5) * 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[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * 0.5), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (/.f64 (*.f64 t t) #s(literal 2 binary64))) < 2

   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 t around 0

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

   5. Applied rewrites 98.2%

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

   if 2 < (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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 79.5

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

   7. Applied rewrites 79.5%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 77.3%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 5: 74.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 73.2

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

7. Applied rewrites 73.2%

   \[\leadsto \left(\left(0.5 \cdot x - y\right) \cdot {\color{blue}{\left(\mathsf{fma}\left(0.5, t, 1\right)\right)}}^{t}\right) \cdot \sqrt{z \cdot 2} \]
8. Add Preprocessing

Alternative 6: 95.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right) \cdot t, t, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right) \cdot \sqrt{2 \cdot z} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (*
   (fma (* (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5) t) t 1.0)
   (fma x 0.5 (- y)))
  (sqrt (* 2.0 z))))

C

double code(double x, double y, double z, double t) {
	return (fma((fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5) * t), t, 1.0) * fma(x, 0.5, -y)) * sqrt((2.0 * z));
}

Julia

function code(x, y, z, t)
	return Float64(Float64(fma(Float64(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5) * t), t, 1.0) * fma(x, 0.5, Float64(-y))) * sqrt(Float64(2.0 * z)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision] * N[(x * 0.5 + (-y)), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2}} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), {t}^{2}, 1\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{{t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}}, {t}^{2}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2}} + \frac{1}{2}, {t}^{2}, 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right)}, {t}^{2}, 1\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right)}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   13. unpow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), \color{blue}{t \cdot t}, 1\right) \]

   14. lower-*.f64 94.9

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), \color{blue}{t \cdot t}, 1\right) \]

5. Applied rewrites 94.9%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 94.9%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), t \cdot t, 0.5\right) \cdot t, \color{blue}{t}, 1\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \left(\sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \left(\sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \left(\sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \color{blue}{\left(\sqrt{z \cdot 2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

   9. remove-double-div N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \color{blue}{\frac{1}{\frac{1}{\sqrt{z \cdot 2} \cdot \left(\frac{1}{2} \cdot x - y\right)}}} \]

   10. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot \frac{1}{\color{blue}{\frac{1}{\sqrt{z \cdot 2} \cdot \left(\frac{1}{2} \cdot x - y\right)}}} \]

   11. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot 1}{\frac{1}{\sqrt{z \cdot 2} \cdot \left(\frac{1}{2} \cdot x - y\right)}}} \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot 1}{\color{blue}{\frac{1}{\sqrt{z \cdot 2} \cdot \left(\frac{1}{2} \cdot x - y\right)}}} \]

   13. inv-pow N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \cdot 1}{\color{blue}{{\left(\sqrt{z \cdot 2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)}^{-1}}} \]

9. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right) \cdot t, t, 1\right) \cdot \mathsf{fma}\left(x, 0.5, -y\right)\right) \cdot \sqrt{2 \cdot z}} \]
10. Add Preprocessing

Alternative 7: 94.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(0.020833333333333332 \cdot t\right) \cdot t, t \cdot t, 0.5\right) \cdot t, t, 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma (* (fma (* (* 0.020833333333333332 t) t) (* t t) 0.5) t) t 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma((fma(((0.020833333333333332 * t) * t), (t * t), 0.5) * t), t, 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(Float64(fma(Float64(Float64(0.020833333333333332 * t) * t), Float64(t * t), 0.5) * t), t, 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(0.020833333333333332 * t), $MachinePrecision] * t), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2}} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), {t}^{2}, 1\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{{t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}}, {t}^{2}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2}} + \frac{1}{2}, {t}^{2}, 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right)}, {t}^{2}, 1\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right)}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   13. unpow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), \color{blue}{t \cdot t}, 1\right) \]

   14. lower-*.f64 94.9

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), \color{blue}{t \cdot t}, 1\right) \]

5. Applied rewrites 94.9%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 94.9%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), t \cdot t, 0.5\right) \cdot t, \color{blue}{t}, 1\right) \]

8. Taylor expanded in t around inf

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2}, t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \]
9. Step-by-step derivation

10. Applied rewrites 94.7%

    \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(0.020833333333333332 \cdot t\right) \cdot t, t \cdot t, 0.5\right) \cdot t, t, 1\right) \]
11. Add Preprocessing

Alternative 8: 93.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathsf{fma}\left(t\_1 \cdot \mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, t\_1\right) \cdot \sqrt{z \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (* (fma (* t_1 (fma 0.125 (* t t) 0.5)) (* t t) t_1) (sqrt (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	return fma((t_1 * fma(0.125, (t * t), 0.5)), (t * t), t_1) * sqrt((z * 2.0));
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	return Float64(fma(Float64(t_1 * fma(0.125, Float64(t * t), 0.5)), Float64(t * t), t_1) * sqrt(Float64(z * 2.0)))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, N[(N[(N[(t$95$1 * N[(0.125 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + t$95$1), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({t}^{2} \cdot \left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot x\right)} - y\right) \cdot \sqrt{z \cdot 2} \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \left(\frac{1}{2} \cdot x - y\right)\right)} \cdot \sqrt{z \cdot 2} \]

   3. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot {t}^{2}} + \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z \cdot 2} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right), {t}^{2}, \frac{1}{2} \cdot x - y\right)} \cdot \sqrt{z \cdot 2} \]

7. Applied rewrites 93.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.5 \cdot x - y\right) \cdot \mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 0.5 \cdot x - y\right)} \cdot \sqrt{z \cdot 2} \]
8. Add Preprocessing

Alternative 9: 92.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.5 \cdot x - y\\ \mathsf{fma}\left(\left(\left(0.125 \cdot t\_1\right) \cdot t\right) \cdot t, t \cdot t, t\_1\right) \cdot \sqrt{z \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* 0.5 x) y)))
   (* (fma (* (* (* 0.125 t_1) t) t) (* t t) t_1) (sqrt (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (0.5 * x) - y;
	return fma((((0.125 * t_1) * t) * t), (t * t), t_1) * sqrt((z * 2.0));
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(0.5 * x) - y)
	return Float64(fma(Float64(Float64(Float64(0.125 * t_1) * t) * t), Float64(t * t), t_1) * sqrt(Float64(z * 2.0)))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]}, N[(N[(N[(N[(N[(0.125 * t$95$1), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * N[(t * t), $MachinePrecision] + t$95$1), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({t}^{2} \cdot \left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot x\right)} - y\right) \cdot \sqrt{z \cdot 2} \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \left(\frac{1}{2} \cdot x - y\right)\right)} \cdot \sqrt{z \cdot 2} \]

   3. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot {t}^{2}} + \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \sqrt{z \cdot 2} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot x - y\right), {t}^{2}, \frac{1}{2} \cdot x - y\right)} \cdot \sqrt{z \cdot 2} \]

7. Applied rewrites 93.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.5 \cdot x - y\right) \cdot \mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 0.5 \cdot x - y\right)} \cdot \sqrt{z \cdot 2} \]

8. Taylor expanded in t around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{8} \cdot \left({t}^{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right), \color{blue}{t} \cdot t, \frac{1}{2} \cdot x - y\right) \cdot \sqrt{z \cdot 2} \]
9. Step-by-step derivation

10. Applied rewrites 93.2%

    \[\leadsto \mathsf{fma}\left(\left(\left(0.125 \cdot \left(0.5 \cdot x - y\right)\right) \cdot t\right) \cdot t, \color{blue}{t} \cdot t, 0.5 \cdot x - y\right) \cdot \sqrt{z \cdot 2} \]
11. Add Preprocessing

Alternative 10: 75.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 1.45 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot 1\\ \mathbf{elif}\;t \cdot t \leq 3.9 \cdot 10^{+276}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, t \cdot t, -1\right) \cdot y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25, t \cdot t, 0.5\right) \cdot x\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 1.45e+129)
     (* (* (- (* x 0.5) y) t_1) 1.0)
     (if (<= (* t t) 3.9e+276)
       (* (* (fma -0.5 (* t t) -1.0) y) t_1)
       (* (* (fma 0.25 (* t t) 0.5) x) 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) <= 1.45e+129) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else if ((t * t) <= 3.9e+276) {
		tmp = (fma(-0.5, (t * t), -1.0) * y) * t_1;
	} else {
		tmp = (fma(0.25, (t * t), 0.5) * x) * 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) <= 1.45e+129)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * 1.0);
	elseif (Float64(t * t) <= 3.9e+276)
		tmp = Float64(Float64(fma(-0.5, Float64(t * t), -1.0) * y) * t_1);
	else
		tmp = Float64(Float64(fma(0.25, Float64(t * t), 0.5) * x) * t_1);
	end
	return 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], 1.45e+129], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 3.9e+276], N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + -1.0), $MachinePrecision] * y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(0.25 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 1.45000000000000001e129

   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 t around 0

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

   5. Applied rewrites 84.4%

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

   if 1.45000000000000001e129 < (*.f64 t t) < 3.9000000000000002e276

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 86.8

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 70.0%

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

   if 3.9000000000000002e276 < (*.f64 t t)

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 96.3

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 74.3%

       \[\leadsto \left(\mathsf{fma}\left(0.25, t \cdot t, 0.5\right) \cdot \color{blue}{x}\right) \cdot \sqrt{z \cdot 2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 92.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma (fma 0.125 (* t t) 0.5) (* t t) 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma(fma(0.125, (t * t), 0.5), (t * t), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(fma(0.125, Float64(t * t), 0.5), Float64(t * t), 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) \cdot {t}^{2}} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}, {t}^{2}, 1\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{8} \cdot {t}^{2} + \frac{1}{2}}, {t}^{2}, 1\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{8}, {t}^{2}, \frac{1}{2}\right)}, {t}^{2}, 1\right) \]

   6. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), \color{blue}{t \cdot t}, 1\right) \]

   9. lower-*.f64 91.9

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), \color{blue}{t \cdot t}, 1\right) \]

5. Applied rewrites 91.9%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
6. Add Preprocessing

Alternative 12: 75.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 1.45 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, t \cdot t, -1\right) \cdot y\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 1.45e+129)
     (* (* (- (* x 0.5) y) t_1) 1.0)
     (* (* (fma -0.5 (* t t) -1.0) y) 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) <= 1.45e+129) {
		tmp = (((x * 0.5) - y) * t_1) * 1.0;
	} else {
		tmp = (fma(-0.5, (t * t), -1.0) * y) * 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) <= 1.45e+129)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * 1.0);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(t * t), -1.0) * y) * t_1);
	end
	return 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], 1.45e+129], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + -1.0), $MachinePrecision] * y), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.45000000000000001e129

   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 t around 0

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

   5. Applied rewrites 84.4%

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

   if 1.45000000000000001e129 < (*.f64 t t)

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 93.7

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.6%

       \[\leadsto \left(\mathsf{fma}\left(-0.5, t \cdot t, -1\right) \cdot \color{blue}{y}\right) \cdot \sqrt{z \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 87.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft1-in N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 89.1

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

7. Applied rewrites 89.1%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \left(0.5 \cdot x - y\right)\right)} \cdot \sqrt{z \cdot 2} \]
8. Add Preprocessing

Alternative 14: 57.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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 t around 0

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

5. Applied rewrites 54.7%

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

Alternative 15: 30.0% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision] * 1.0), $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 t around 0

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

5. Applied rewrites 54.7%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lower-*.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-sqrt.f64 31.0

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

8. Applied rewrites 31.0%

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

10. Applied rewrites 31.0%

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

Developer Target 1: 99.4% 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 2024318 
(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))))