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

Percentage Accurate: 99.4% → 99.4%

Time: 15.9s

Alternatives: 9

Speedup: 1.0×

• 44.7% 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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.8%

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

Alternative 2: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 6200:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 6200.0)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* 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 <= 6200.0) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * 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 <= 6200.0d0) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * (0.5d0 * (x * 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 <= 6200.0) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (0.5 * (x * t_1));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 6200.0:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * (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 (t <= 6200.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(0.5 * Float64(x * 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 <= 6200.0)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * 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[t, 6200.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 6200

   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 t around 0 69.2%

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

      1. associate-*r* 69.1%

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

      2. sqrt-prod 69.3%

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

      3. *-commutative 69.3%

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

      4. add-sqr-sqrt 40.9%

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

      5. sqrt-unprod 46.5%

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

      6. sqr-neg 46.5%

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

      7. sqrt-unprod 11.3%

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

      8. add-sqr-sqrt 34.3%

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

      9. distribute-rgt-out-- 33.7%

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

      10. *-commutative 33.7%

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

      11. cancel-sign-sub-inv 33.7%

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

      12. *-commutative 33.7%

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

      13. associate-*r* 33.7%

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

      14. add-sqr-sqrt 10.8%

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

      15. sqrt-unprod 46.5%

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

      16. sqr-neg 46.5%

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

      17. sqrt-unprod 40.9%

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

      18. add-sqr-sqrt 69.3%

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

   5. Applied egg-rr 69.3%

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

      1. distribute-lft-neg-out 69.3%

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

      2. unsub-neg 69.3%

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

      3. associate-*r* 69.3%

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

      4. *-commutative 69.3%

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

      5. add-sqr-sqrt 69.3%

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

      6. sqr-neg 69.3%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 33.7%

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

      9. *-commutative 33.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 69.3%

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

      12. sqr-neg 69.3%

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

      13. add-sqr-sqrt 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. distribute-rgt-out-- 69.3%

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

      2. *-commutative 69.3%

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

   9. Simplified 69.3%

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

   if 6200 < 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 inf 75.4%

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

      1. associate-*l* 75.4%

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

   5. Simplified 75.4%

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

      1. *-commutative 75.4%

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

      2. sqrt-prod 75.4%

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

   7. Applied egg-rr 75.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6200:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+244} \lor \neg \left(y \leq -6.2 \cdot 10^{+112}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.06e+244) (not (<= y -6.2e+112)))
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (sqrt (* (* z 2.0) (pow y 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.06e+244) || !(y <= -6.2e+112)) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt(((z * 2.0) * pow(y, 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 ((y <= (-1.06d+244)) .or. (.not. (y <= (-6.2d+112)))) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = sqrt(((z * 2.0d0) * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.06e+244) || !(y <= -6.2e+112)) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.06e+244) or not (y <= -6.2e+112):
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(((z * 2.0) * math.pow(y, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.06e+244) || !(y <= -6.2e+112))
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.06e+244) || ~((y <= -6.2e+112)))
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = sqrt(((z * 2.0) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.06e+244], N[Not[LessEqual[y, -6.2e+112]], $MachinePrecision]], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0599999999999999e244 or -6.19999999999999965e112 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 60.0%

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

      1. associate-*r* 59.6%

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

      2. sqrt-prod 59.8%

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

      3. *-commutative 59.8%

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

      4. add-sqr-sqrt 38.2%

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

      5. sqrt-unprod 46.4%

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

      6. sqr-neg 46.4%

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

      7. sqrt-unprod 10.6%

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

      8. add-sqr-sqrt 31.7%

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

      9. distribute-rgt-out-- 31.7%

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

      10. *-commutative 31.7%

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

      11. cancel-sign-sub-inv 31.7%

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

      12. *-commutative 31.7%

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

      13. associate-*r* 31.6%

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

      14. add-sqr-sqrt 10.6%

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

      15. sqrt-unprod 46.4%

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

      16. sqr-neg 46.4%

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

      17. sqrt-unprod 38.2%

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

      18. add-sqr-sqrt 59.8%

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

   5. Applied egg-rr 59.8%

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

      1. distribute-lft-neg-out 59.8%

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

      2. unsub-neg 59.8%

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

      3. associate-*r* 59.8%

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

      4. *-commutative 59.8%

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

      5. add-sqr-sqrt 59.8%

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

      6. sqr-neg 59.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 31.7%

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

      9. *-commutative 31.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 59.8%

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

      12. sqr-neg 59.8%

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

      13. add-sqr-sqrt 59.8%

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

   7. Applied egg-rr 59.8%

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

      1. distribute-rgt-out-- 59.8%

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

      2. *-commutative 59.8%

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

   9. Simplified 59.8%

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

   if -1.0599999999999999e244 < y < -6.19999999999999965e112

   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 t around 0 45.3%

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

   4. Taylor expanded in x around 0 45.3%

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

      1. mul-1-neg 45.3%

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

      2. *-commutative 45.3%

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

      3. distribute-rgt-neg-in 45.3%

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

   6. Simplified 45.3%

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

      1. associate-*r* 45.3%

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

      2. sqrt-prod 45.4%

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

      3. distribute-rgt-neg-in 45.4%

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

      4. distribute-lft-neg-out 45.4%

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

      5. add-sqr-sqrt 45.3%

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

      6. sqrt-unprod 59.8%

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

      7. swap-sqr 78.2%

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

      8. sqr-neg 78.2%

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

      9. add-sqr-sqrt 78.2%

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

      10. pow2 78.2%

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

   8. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. *-commutative 78.2%

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

   10. Simplified 78.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+244} \lor \neg \left(y \leq -6.2 \cdot 10^{+112}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+242}:\\ \;\;\;\;\sqrt{z} \cdot \left(y \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.7e+242)
   (* (sqrt z) (* y (- (sqrt 2.0))))
   (if (<= y -5.1e+110)
     (sqrt (* (* z 2.0) (pow y 2.0)))
     (* (- (* x 0.5) y) (sqrt (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.7e+242) {
		tmp = sqrt(z) * (y * -sqrt(2.0));
	} else if (y <= -5.1e+110) {
		tmp = sqrt(((z * 2.0) * pow(y, 2.0)));
	} else {
		tmp = ((x * 0.5) - y) * sqrt((z * 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 (y <= (-3.7d+242)) then
        tmp = sqrt(z) * (y * -sqrt(2.0d0))
    else if (y <= (-5.1d+110)) then
        tmp = sqrt(((z * 2.0d0) * (y ** 2.0d0)))
    else
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.7e+242) {
		tmp = Math.sqrt(z) * (y * -Math.sqrt(2.0));
	} else if (y <= -5.1e+110) {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	} else {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3.7e+242:
		tmp = math.sqrt(z) * (y * -math.sqrt(2.0))
	elif y <= -5.1e+110:
		tmp = math.sqrt(((z * 2.0) * math.pow(y, 2.0)))
	else:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.7e+242)
		tmp = Float64(sqrt(z) * Float64(y * Float64(-sqrt(2.0))));
	elseif (y <= -5.1e+110)
		tmp = sqrt(Float64(Float64(z * 2.0) * (y ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -3.7e+242)
		tmp = sqrt(z) * (y * -sqrt(2.0));
	elseif (y <= -5.1e+110)
		tmp = sqrt(((z * 2.0) * (y ^ 2.0)));
	else
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.7e+242], N[(N[Sqrt[z], $MachinePrecision] * N[(y * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.1e+110], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7e242

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0 82.1%

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

   4. Taylor expanded in x around 0 79.7%

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

      1. mul-1-neg 79.7%

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

      2. *-commutative 79.7%

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

      3. distribute-rgt-neg-in 79.7%

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

   6. Simplified 79.7%

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

   if -3.7e242 < y < -5.1000000000000002e110

   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 t around 0 45.3%

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

   4. Taylor expanded in x around 0 45.3%

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

      1. mul-1-neg 45.3%

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

      2. *-commutative 45.3%

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

      3. distribute-rgt-neg-in 45.3%

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

   6. Simplified 45.3%

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

      1. associate-*r* 45.3%

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

      2. sqrt-prod 45.4%

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

      3. distribute-rgt-neg-in 45.4%

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

      4. distribute-lft-neg-out 45.4%

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

      5. add-sqr-sqrt 45.3%

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

      6. sqrt-unprod 59.8%

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

      7. swap-sqr 78.2%

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

      8. sqr-neg 78.2%

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

      9. add-sqr-sqrt 78.2%

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

      10. pow2 78.2%

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

   8. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. *-commutative 78.2%

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

   10. Simplified 78.2%

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

   if -5.1000000000000002e110 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 58.4%

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

      1. associate-*r* 58.4%

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

      2. sqrt-prod 58.5%

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

      3. *-commutative 58.5%

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

      4. add-sqr-sqrt 41.1%

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

      5. sqrt-unprod 49.9%

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

      6. sqr-neg 49.9%

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

      7. sqrt-unprod 11.3%

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

      8. add-sqr-sqrt 34.0%

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

      9. distribute-rgt-out-- 34.0%

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

      10. *-commutative 34.0%

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

      11. cancel-sign-sub-inv 34.0%

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

      12. *-commutative 34.0%

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

      13. associate-*r* 34.0%

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

      14. add-sqr-sqrt 11.3%

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

      15. sqrt-unprod 49.9%

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

      16. sqr-neg 49.9%

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

      17. sqrt-unprod 41.1%

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

      18. add-sqr-sqrt 58.5%

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

   5. Applied egg-rr 58.5%

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

      1. distribute-lft-neg-out 58.5%

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

      2. unsub-neg 58.5%

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

      3. associate-*r* 58.5%

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

      4. *-commutative 58.5%

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

      5. add-sqr-sqrt 58.5%

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

      6. sqr-neg 58.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 34.0%

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

      9. *-commutative 34.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 58.5%

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

      12. sqr-neg 58.5%

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

      13. add-sqr-sqrt 58.5%

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

   7. Applied egg-rr 58.5%

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

      1. distribute-rgt-out-- 58.5%

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

      2. *-commutative 58.5%

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

   9. Simplified 58.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+242}:\\ \;\;\;\;\sqrt{z} \cdot \left(y \cdot \left(-\sqrt{2}\right)\right)\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{+110}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;y \leq -1.06 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{z} \cdot \left(t\_1 \cdot \sqrt{2}\right)\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= y -1.06e+244)
     (* (sqrt z) (* t_1 (sqrt 2.0)))
     (if (<= y -4.5e+111)
       (sqrt (* (* z 2.0) (pow y 2.0)))
       (* t_1 (sqrt (* z 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (y <= -1.06e+244) {
		tmp = sqrt(z) * (t_1 * sqrt(2.0));
	} else if (y <= -4.5e+111) {
		tmp = sqrt(((z * 2.0) * pow(y, 2.0)));
	} else {
		tmp = t_1 * sqrt((z * 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 (y <= (-1.06d+244)) then
        tmp = sqrt(z) * (t_1 * sqrt(2.0d0))
    else if (y <= (-4.5d+111)) then
        tmp = sqrt(((z * 2.0d0) * (y ** 2.0d0)))
    else
        tmp = t_1 * sqrt((z * 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 (y <= -1.06e+244) {
		tmp = Math.sqrt(z) * (t_1 * Math.sqrt(2.0));
	} else if (y <= -4.5e+111) {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	} else {
		tmp = t_1 * Math.sqrt((z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if y <= -1.06e+244:
		tmp = math.sqrt(z) * (t_1 * math.sqrt(2.0))
	elif y <= -4.5e+111:
		tmp = math.sqrt(((z * 2.0) * math.pow(y, 2.0)))
	else:
		tmp = t_1 * math.sqrt((z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (y <= -1.06e+244)
		tmp = Float64(sqrt(z) * Float64(t_1 * sqrt(2.0)));
	elseif (y <= -4.5e+111)
		tmp = sqrt(Float64(Float64(z * 2.0) * (y ^ 2.0)));
	else
		tmp = Float64(t_1 * sqrt(Float64(z * 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 (y <= -1.06e+244)
		tmp = sqrt(z) * (t_1 * sqrt(2.0));
	elseif (y <= -4.5e+111)
		tmp = sqrt(((z * 2.0) * (y ^ 2.0)));
	else
		tmp = t_1 * sqrt((z * 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[y, -1.06e+244], N[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.5e+111], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0599999999999999e244

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0 82.1%

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

   if -1.0599999999999999e244 < y < -4.50000000000000001e111

   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 t around 0 45.3%

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

   4. Taylor expanded in x around 0 45.3%

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

      1. mul-1-neg 45.3%

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

      2. *-commutative 45.3%

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

      3. distribute-rgt-neg-in 45.3%

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

   6. Simplified 45.3%

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

      1. associate-*r* 45.3%

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

      2. sqrt-prod 45.4%

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

      3. distribute-rgt-neg-in 45.4%

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

      4. distribute-lft-neg-out 45.4%

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

      5. add-sqr-sqrt 45.3%

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

      6. sqrt-unprod 59.8%

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

      7. swap-sqr 78.2%

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

      8. sqr-neg 78.2%

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

      9. add-sqr-sqrt 78.2%

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

      10. pow2 78.2%

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

   8. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. *-commutative 78.2%

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

   10. Simplified 78.2%

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

   if -4.50000000000000001e111 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 58.4%

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

      1. associate-*r* 58.4%

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

      2. sqrt-prod 58.5%

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

      3. *-commutative 58.5%

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

      4. add-sqr-sqrt 41.1%

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

      5. sqrt-unprod 49.9%

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

      6. sqr-neg 49.9%

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

      7. sqrt-unprod 11.3%

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

      8. add-sqr-sqrt 34.0%

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

      9. distribute-rgt-out-- 34.0%

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

      10. *-commutative 34.0%

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

      11. cancel-sign-sub-inv 34.0%

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

      12. *-commutative 34.0%

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

      13. associate-*r* 34.0%

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

      14. add-sqr-sqrt 11.3%

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

      15. sqrt-unprod 49.9%

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

      16. sqr-neg 49.9%

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

      17. sqrt-unprod 41.1%

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

      18. add-sqr-sqrt 58.5%

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

   5. Applied egg-rr 58.5%

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

      1. distribute-lft-neg-out 58.5%

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

      2. unsub-neg 58.5%

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

      3. associate-*r* 58.5%

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

      4. *-commutative 58.5%

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

      5. add-sqr-sqrt 58.5%

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

      6. sqr-neg 58.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 34.0%

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

      9. *-commutative 34.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 58.5%

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

      12. sqr-neg 58.5%

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

      13. add-sqr-sqrt 58.5%

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

   7. Applied egg-rr 58.5%

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

      1. distribute-rgt-out-- 58.5%

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

      2. *-commutative 58.5%

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

   9. Simplified 58.5%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-43} \lor \neg \left(y \leq 1.9 \cdot 10^{+28}\right):\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5 + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= y -2e-43) (not (<= y 1.9e+28)))
     (* t_1 (- y))
     (* t_1 (+ (* x 0.5) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((y <= -2e-43) || !(y <= 1.9e+28)) {
		tmp = t_1 * -y;
	} else {
		tmp = t_1 * ((x * 0.5) + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((y <= (-2d-43)) .or. (.not. (y <= 1.9d+28))) then
        tmp = t_1 * -y
    else
        tmp = t_1 * ((x * 0.5d0) + y)
    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 ((y <= -2e-43) || !(y <= 1.9e+28)) {
		tmp = t_1 * -y;
	} else {
		tmp = t_1 * ((x * 0.5) + y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (y <= -2e-43) or not (y <= 1.9e+28):
		tmp = t_1 * -y
	else:
		tmp = t_1 * ((x * 0.5) + y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((y <= -2e-43) || !(y <= 1.9e+28))
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((y <= -2e-43) || ~((y <= 1.9e+28)))
		tmp = t_1 * -y;
	else
		tmp = t_1 * ((x * 0.5) + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -2e-43], N[Not[LessEqual[y, 1.9e+28]], $MachinePrecision]], N[(t$95$1 * (-y)), $MachinePrecision], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.00000000000000015e-43 or 1.8999999999999999e28 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 62.9%

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

   4. Taylor expanded in x around 0 51.7%

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

      1. mul-1-neg 51.7%

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

      2. *-commutative 51.7%

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

      3. distribute-rgt-neg-in 51.7%

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

   6. Simplified 51.7%

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

      1. associate-*r* 51.1%

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

      2. sqrt-prod 51.2%

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

      3. distribute-rgt-neg-in 51.2%

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

      4. neg-sub0 51.2%

         \[\leadsto \color{blue}{0 - \sqrt{z \cdot 2} \cdot y} \]

      5. add-sqr-sqrt 51.2%

         \[\leadsto 0 - \sqrt{\color{blue}{\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}}} \cdot y \]

      6. sqr-neg 51.2%

         \[\leadsto 0 - \sqrt{\color{blue}{\left(-\sqrt{z \cdot 2}\right) \cdot \left(-\sqrt{z \cdot 2}\right)}} \cdot y \]

      7. sqrt-unprod 0.0%

         \[\leadsto 0 - \color{blue}{\left(\sqrt{-\sqrt{z \cdot 2}} \cdot \sqrt{-\sqrt{z \cdot 2}}\right)} \cdot y \]

      8. add-sqr-sqrt 0.8%

         \[\leadsto 0 - \color{blue}{\left(-\sqrt{z \cdot 2}\right)} \cdot y \]

      9. *-commutative 0.8%

         \[\leadsto 0 - \color{blue}{y \cdot \left(-\sqrt{z \cdot 2}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto 0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z \cdot 2}} \cdot \sqrt{-\sqrt{z \cdot 2}}\right)} \]

      11. sqrt-unprod 51.2%

          \[\leadsto 0 - y \cdot \color{blue}{\sqrt{\left(-\sqrt{z \cdot 2}\right) \cdot \left(-\sqrt{z \cdot 2}\right)}} \]

      12. sqr-neg 51.2%

          \[\leadsto 0 - y \cdot \sqrt{\color{blue}{\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}}} \]

      13. add-sqr-sqrt 51.2%

          \[\leadsto 0 - y \cdot \sqrt{\color{blue}{z \cdot 2}} \]

   8. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0 - y \cdot \sqrt{z \cdot 2}} \]
   9. Step-by-step derivation

      1. neg-sub0 51.2%

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

      2. distribute-rgt-neg-in 51.2%

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

      3. *-commutative 51.2%

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

   10. Simplified 51.2%

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

   if -2.00000000000000015e-43 < y < 1.8999999999999999e28

   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 t around 0 54.2%

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

      1. associate-*r* 54.1%

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

      2. sqrt-prod 54.3%

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

      3. *-commutative 54.3%

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

      4. add-sqr-sqrt 36.5%

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

      5. sqrt-unprod 49.4%

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

      6. sqr-neg 49.4%

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

      7. sqrt-unprod 14.8%

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

      8. add-sqr-sqrt 45.9%

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

      9. distribute-rgt-out-- 45.9%

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

      10. *-commutative 45.9%

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

      11. cancel-sign-sub-inv 45.9%

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

      12. *-commutative 45.9%

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

      13. associate-*r* 45.8%

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

      14. add-sqr-sqrt 14.7%

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

      15. sqrt-unprod 49.4%

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

      16. sqr-neg 49.4%

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

      17. sqrt-unprod 36.5%

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

      18. add-sqr-sqrt 54.3%

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

   5. Applied egg-rr 54.3%

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

      1. +-commutative 54.3%

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

      2. *-commutative 54.3%

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

      3. fma-define 54.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 45.8%

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

      6. sqr-neg 45.8%

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

      7. add-sqr-sqrt 45.8%

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

      8. associate-*r* 45.9%

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

      9. *-commutative 45.9%

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

   7. Applied egg-rr 45.9%

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

      1. fma-undefine 45.9%

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

      2. distribute-rgt-out 45.9%

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

      3. *-commutative 45.9%

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

   9. Simplified 45.9%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-43} \lor \neg \left(y \leq 1.9 \cdot 10^{+28}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * 0.5) - y) * 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 = ((x * 0.5d0) - y) * sqrt((z * 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));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 t around 0 58.6%

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

   1. associate-*r* 58.2%

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

   2. sqrt-prod 58.3%

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

   3. *-commutative 58.3%

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

   4. add-sqr-sqrt 34.3%

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

   5. sqrt-unprod 41.7%

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

   6. sqr-neg 41.7%

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

   7. sqrt-unprod 9.9%

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

   8. add-sqr-sqrt 28.9%

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

   9. distribute-rgt-out-- 28.5%

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

   10. *-commutative 28.5%

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

   11. cancel-sign-sub-inv 28.5%

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

   12. *-commutative 28.5%

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

   13. associate-*r* 28.5%

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

   14. add-sqr-sqrt 9.5%

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

   15. sqrt-unprod 41.7%

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

   16. sqr-neg 41.7%

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

   17. sqrt-unprod 34.3%

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

   18. add-sqr-sqrt 58.3%

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

5. Applied egg-rr 58.3%

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

   1. distribute-lft-neg-out 58.3%

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

   2. unsub-neg 58.3%

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

   3. associate-*r* 58.3%

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

   4. *-commutative 58.3%

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

   5. add-sqr-sqrt 58.3%

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

   6. sqr-neg 58.3%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 28.5%

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

   9. *-commutative 28.5%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-unprod 58.3%

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

   12. sqr-neg 58.3%

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

   13. add-sqr-sqrt 58.3%

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

7. Applied egg-rr 58.3%

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

   1. distribute-rgt-out-- 58.3%

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

   2. *-commutative 58.3%

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

9. Simplified 58.3%

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

10. Final simplification 58.3%

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

Alternative 8: 29.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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((z * 2.0d0)) * -y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 t around 0 58.6%

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

4. Taylor expanded in x around 0 31.9%

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

   1. mul-1-neg 31.9%

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

   2. *-commutative 31.9%

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

   3. distribute-rgt-neg-in 31.9%

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

6. Simplified 31.9%

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

   1. associate-*r* 31.5%

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

   2. sqrt-prod 31.6%

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

   3. distribute-rgt-neg-in 31.6%

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

   4. neg-sub0 31.6%

      \[\leadsto \color{blue}{0 - \sqrt{z \cdot 2} \cdot y} \]

   5. add-sqr-sqrt 31.6%

      \[\leadsto 0 - \sqrt{\color{blue}{\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}}} \cdot y \]

   6. sqr-neg 31.6%

      \[\leadsto 0 - \sqrt{\color{blue}{\left(-\sqrt{z \cdot 2}\right) \cdot \left(-\sqrt{z \cdot 2}\right)}} \cdot y \]

   7. sqrt-unprod 0.0%

      \[\leadsto 0 - \color{blue}{\left(\sqrt{-\sqrt{z \cdot 2}} \cdot \sqrt{-\sqrt{z \cdot 2}}\right)} \cdot y \]

   8. add-sqr-sqrt 2.5%

      \[\leadsto 0 - \color{blue}{\left(-\sqrt{z \cdot 2}\right)} \cdot y \]

   9. *-commutative 2.5%

      \[\leadsto 0 - \color{blue}{y \cdot \left(-\sqrt{z \cdot 2}\right)} \]

   10. add-sqr-sqrt 0.0%

       \[\leadsto 0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z \cdot 2}} \cdot \sqrt{-\sqrt{z \cdot 2}}\right)} \]

   11. sqrt-unprod 31.6%

       \[\leadsto 0 - y \cdot \color{blue}{\sqrt{\left(-\sqrt{z \cdot 2}\right) \cdot \left(-\sqrt{z \cdot 2}\right)}} \]

   12. sqr-neg 31.6%

       \[\leadsto 0 - y \cdot \sqrt{\color{blue}{\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}}} \]

   13. add-sqr-sqrt 31.6%

       \[\leadsto 0 - y \cdot \sqrt{\color{blue}{z \cdot 2}} \]

8. Applied egg-rr 31.6%

   \[\leadsto \color{blue}{0 - y \cdot \sqrt{z \cdot 2}} \]
9. Step-by-step derivation

   1. neg-sub0 31.6%

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

   2. distribute-rgt-neg-in 31.6%

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

   3. *-commutative 31.6%

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

10. Simplified 31.6%

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

11. Final simplification 31.6%

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

Alternative 9: 2.6% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 0.0)

C

double code(double x, double y, double z, double t) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0;
}

Python

def code(x, y, z, t):
	return 0.0

Julia

function code(x, y, z, t)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_] := 0.0

Derivation

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 t around 0 58.6%

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

   1. associate-*r* 58.2%

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

   2. sqrt-prod 58.3%

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

   3. *-commutative 58.3%

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

   4. add-sqr-sqrt 34.3%

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

   5. sqrt-unprod 41.7%

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

   6. sqr-neg 41.7%

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

   7. sqrt-unprod 9.9%

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

   8. add-sqr-sqrt 28.9%

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

   9. distribute-rgt-out-- 28.5%

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

   10. *-commutative 28.5%

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

   11. cancel-sign-sub-inv 28.5%

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

   12. *-commutative 28.5%

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

   13. associate-*r* 28.5%

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

   14. add-sqr-sqrt 9.5%

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

   15. sqrt-unprod 41.7%

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

   16. sqr-neg 41.7%

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

   17. sqrt-unprod 34.3%

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

   18. add-sqr-sqrt 58.3%

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

5. Applied egg-rr 58.3%

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

   1. distribute-lft-neg-out 58.3%

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

   2. unsub-neg 58.3%

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

   3. associate-*r* 58.3%

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

   4. *-commutative 58.3%

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

   5. add-sqr-sqrt 58.3%

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

   6. sqr-neg 58.3%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 28.5%

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

   9. *-commutative 28.5%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-unprod 58.3%

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

   12. sqr-neg 58.3%

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

   13. add-sqr-sqrt 58.3%

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

7. Applied egg-rr 58.3%

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

8. Taylor expanded in z around 0 3.1%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 3.1%

   \[\leadsto 0 \]
10. Add Preprocessing

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024039 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

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