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

Percentage Accurate: 99.4% → 98.9%

Time: 15.3s

Alternatives: 8

Speedup: 1.0×

• 44.2% 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: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	return (((0.5 * t_1) * x) - (t_1 * y)) * 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
    real(8) :: t_1
    t_1 = sqrt((2.0d0 * z))
    code = (((0.5d0 * t_1) * x) - (t_1 * y)) * exp(((t * t) / 2.0d0))
end function

Java

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

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	return (((0.5 * t_1) * x) - (t_1 * y)) * math.exp(((t * t) / 2.0))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = (((0.5 * t_1) * x) - (t_1 * y)) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(0.5 * t$95$1), $MachinePrecision] * x), $MachinePrecision] - N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 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. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. distribute-rgt-in 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 49.8%

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

   2. sqrt-unprod 71.1%

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

   3. sqr-neg 71.1%

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

   4. sqrt-unprod 27.8%

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

   5. add-sqr-sqrt 51.2%

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

   6. cancel-sign-sub 51.2%

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

   7. add-log-exp 25.5%

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

   8. add-log-exp 17.2%

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

   9. diff-log 17.2%

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

   10. associate-*l* 17.2%

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

   11. exp-prod 15.8%

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

   12. exp-prod 14.3%

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

6. Applied egg-rr 36.0%

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

   1. log-div 36.0%

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

   2. log-pow 37.0%

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

   3. *-commutative 37.0%

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

   4. rem-log-exp 67.4%

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

   5. log-pow 68.3%

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

   6. rem-log-exp 99.8%

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

   7. *-commutative 99.8%

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

   8. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := t_1 \cdot y\\ \mathbf{if}\;t \cdot t \leq 0.045:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot x\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-e^{\frac{t \cdot t}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))) (t_2 (* t_1 y)))
   (if (<= (* t t) 0.045)
     (- (* t_1 (* 0.5 x)) t_2)
     (* t_2 (- (exp (/ (* t t) 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = t_1 * y;
	double tmp;
	if ((t * t) <= 0.045) {
		tmp = (t_1 * (0.5 * x)) - t_2;
	} else {
		tmp = t_2 * -exp(((t * t) / 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    t_2 = t_1 * y
    if ((t * t) <= 0.045d0) then
        tmp = (t_1 * (0.5d0 * x)) - t_2
    else
        tmp = t_2 * -exp(((t * t) / 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double t_2 = t_1 * y;
	double tmp;
	if ((t * t) <= 0.045) {
		tmp = (t_1 * (0.5 * x)) - t_2;
	} else {
		tmp = t_2 * -Math.exp(((t * t) / 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	t_2 = t_1 * y
	tmp = 0
	if (t * t) <= 0.045:
		tmp = (t_1 * (0.5 * x)) - t_2
	else:
		tmp = t_2 * -math.exp(((t * t) / 2.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = Float64(t_1 * y)
	tmp = 0.0
	if (Float64(t * t) <= 0.045)
		tmp = Float64(Float64(t_1 * Float64(0.5 * x)) - t_2);
	else
		tmp = Float64(t_2 * Float64(-exp(Float64(Float64(t * t) / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	t_2 = t_1 * y;
	tmp = 0.0;
	if ((t * t) <= 0.045)
		tmp = (t_1 * (0.5 * x)) - t_2;
	else
		tmp = t_2 * -exp(((t * t) / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 0.045], N[(N[(t$95$1 * N[(0.5 * x), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(t$95$2 * (-N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 0.044999999999999998

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. associate-*r* 99.3%

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

      4. distribute-rgt-out 99.3%

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

      5. associate-*r* 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*r* 99.3%

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

      8. neg-mul-1 99.3%

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

      9. distribute-rgt-in 99.3%

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

      10. *-commutative 99.3%

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

      11. sub-neg 99.3%

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

      12. *-commutative 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-*l* 99.2%

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

      15. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in t around 0 97.7%

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

      1. associate-*r* 97.7%

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

      2. sqrt-prod 98.1%

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

      3. *-commutative 98.1%

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

      4. sub-neg 98.1%

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

      5. distribute-rgt-in 98.1%

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

      6. *-commutative 98.1%

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

      7. associate-*r* 98.1%

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

      8. *-commutative 98.1%

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

      9. cancel-sign-sub-inv 98.1%

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

      10. *-commutative 98.1%

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

      11. associate-*r* 98.1%

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

   8. Applied egg-rr 98.1%

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

   if 0.044999999999999998 < (*.f64 t t)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add-sqr-sqrt 53.0%

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

      2. sqrt-unprod 76.1%

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

      3. sqr-neg 76.1%

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

      4. sqrt-unprod 24.8%

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

      5. add-sqr-sqrt 48.7%

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

      6. cancel-sign-sub 48.7%

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

      7. add-log-exp 47.9%

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

      8. add-log-exp 31.6%

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

      9. diff-log 31.6%

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

      10. associate-*l* 31.6%

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

      11. exp-prod 29.9%

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

      12. exp-prod 26.5%

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

   6. Applied egg-rr 65.8%

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

      1. log-div 65.8%

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

      2. log-pow 65.8%

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

      3. *-commutative 65.8%

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

      4. rem-log-exp 85.5%

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

      5. log-pow 86.3%

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

      6. rem-log-exp 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. sub-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. sqrt-prod 100.0%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-*r* 100.0%

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

      13. sqrt-prod 100.0%

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

      14. pow1/2 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      16. pow-prod-up 100.0%

          \[\leadsto \left(\left(0.5 \cdot x - y\right) \cdot \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      17. associate-*r* 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   11. Step-by-step derivation

       1. associate-*l* 100.0%

          \[\leadsto \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   13. Taylor expanded in x around 0 80.3%

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

       1. mul-1-neg 80.3%

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

       2. associate-*l* 80.3%

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

       3. distribute-lft-neg-in 80.3%

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

       4. unpow1/2 80.3%

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

       5. metadata-eval 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left({2}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot \sqrt{z}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       6. pow-sqr 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left(\color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)} \cdot \sqrt{z}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       7. unpow1/2 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{{z}^{0.5}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       8. metadata-eval 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot {z}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       9. pow-sqr 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       10. swap-sqr 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \color{blue}{\left(\left({2}^{0.25} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

       11. exp-to-pow 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(\left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       12. exp-to-pow 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(\left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       13. exp-sum 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       14. distribute-rgt-in 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       15. log-prod 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       16. exp-to-pow 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       17. exp-to-pow 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       18. exp-sum 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       19. distribute-rgt-in 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       20. log-prod 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   15. Simplified 80.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.045:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x\right) - \sqrt{2 \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2 \cdot z} \cdot y\right) \cdot \left(-e^{\frac{t \cdot t}{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 0.045:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot y\right) \cdot \left(-e^{\frac{t \cdot t}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= (* t t) 0.045)
     (* t_1 (- (* 0.5 x) y))
     (* (* t_1 y) (- (exp (/ (* t t) 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 0.045) {
		tmp = t_1 * ((0.5 * x) - y);
	} else {
		tmp = (t_1 * y) * -exp(((t * t) / 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if ((t * t) <= 0.045d0) then
        tmp = t_1 * ((0.5d0 * x) - y)
    else
        tmp = (t_1 * y) * -exp(((t * t) / 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if ((t * t) <= 0.045) {
		tmp = t_1 * ((0.5 * x) - y);
	} else {
		tmp = (t_1 * y) * -Math.exp(((t * t) / 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (t * t) <= 0.045:
		tmp = t_1 * ((0.5 * x) - y)
	else:
		tmp = (t_1 * y) * -math.exp(((t * t) / 2.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t * t) <= 0.045)
		tmp = Float64(t_1 * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(Float64(t_1 * y) * Float64(-exp(Float64(Float64(t * t) / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((t * t) <= 0.045)
		tmp = t_1 * ((0.5 * x) - y);
	else
		tmp = (t_1 * y) * -exp(((t * t) / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 0.045], N[(t$95$1 * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * y), $MachinePrecision] * (-N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 0.044999999999999998

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. associate-*r* 99.3%

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

      4. distribute-rgt-out 99.3%

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

      5. associate-*r* 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*r* 99.3%

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

      8. neg-mul-1 99.3%

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

      9. distribute-rgt-in 99.3%

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

      10. *-commutative 99.3%

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

      11. sub-neg 99.3%

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

      12. *-commutative 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-*l* 99.2%

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

      15. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in t around 0 97.7%

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

      1. associate-*r* 97.7%

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

      2. sqrt-prod 98.1%

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

      3. *-commutative 98.1%

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

      4. *-commutative 98.1%

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

      5. expm1-log1p-u 56.8%

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

      6. expm1-udef 33.5%

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

      7. log1p-udef 33.5%

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

      8. rem-exp-log 74.8%

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

      9. *-commutative 74.8%

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

      10. *-commutative 74.8%

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

      11. sub-neg 74.8%

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

      12. *-commutative 74.8%

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

      13. sub-neg 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. add-exp-log 33.5%

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

      2. log1p-udef 33.5%

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

      3. *-commutative 33.5%

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

      4. pow1/2 33.5%

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

      5. metadata-eval 33.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}\right)} - 1 \]

      6. pow-prod-up 33.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)}\right)} - 1 \]

      7. associate-*r* 33.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}}\right)} - 1 \]

      8. expm1-udef 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right)\right)} \]

      9. associate-*r* 56.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot x - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)}\right)\right) \]

      10. pow-prod-up 56.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot \color{blue}{{\left(2 \cdot z\right)}^{\left(0.25 + 0.25\right)}}\right)\right) \]

      11. metadata-eval 56.8%

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

   10. Applied egg-rr 98.1%

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

   if 0.044999999999999998 < (*.f64 t t)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add-sqr-sqrt 53.0%

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

      2. sqrt-unprod 76.1%

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

      3. sqr-neg 76.1%

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

      4. sqrt-unprod 24.8%

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

      5. add-sqr-sqrt 48.7%

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

      6. cancel-sign-sub 48.7%

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

      7. add-log-exp 47.9%

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

      8. add-log-exp 31.6%

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

      9. diff-log 31.6%

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

      10. associate-*l* 31.6%

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

      11. exp-prod 29.9%

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

      12. exp-prod 26.5%

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

   6. Applied egg-rr 65.8%

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

      1. log-div 65.8%

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

      2. log-pow 65.8%

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

      3. *-commutative 65.8%

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

      4. rem-log-exp 85.5%

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

      5. log-pow 86.3%

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

      6. rem-log-exp 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. sub-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. sqrt-prod 100.0%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-*r* 100.0%

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

      13. sqrt-prod 100.0%

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

      14. pow1/2 100.0%

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

      15. metadata-eval 100.0%

          \[\leadsto \left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      16. pow-prod-up 100.0%

          \[\leadsto \left(\left(0.5 \cdot x - y\right) \cdot \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      17. associate-*r* 100.0%

          \[\leadsto \color{blue}{\left(\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   11. Step-by-step derivation

       1. associate-*l* 100.0%

          \[\leadsto \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   13. Taylor expanded in x around 0 80.3%

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

       1. mul-1-neg 80.3%

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

       2. associate-*l* 80.3%

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

       3. distribute-lft-neg-in 80.3%

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

       4. unpow1/2 80.3%

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

       5. metadata-eval 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left({2}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot \sqrt{z}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       6. pow-sqr 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left(\color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)} \cdot \sqrt{z}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       7. unpow1/2 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{{z}^{0.5}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       8. metadata-eval 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot {z}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       9. pow-sqr 80.3%

          \[\leadsto \left(\left(-y\right) \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       10. swap-sqr 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \color{blue}{\left(\left({2}^{0.25} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

       11. exp-to-pow 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(\left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       12. exp-to-pow 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(\left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       13. exp-sum 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       14. distribute-rgt-in 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       15. log-prod 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       16. exp-to-pow 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       17. exp-to-pow 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       18. exp-sum 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       19. distribute-rgt-in 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

       20. log-prod 80.3%

           \[\leadsto \left(\left(-y\right) \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   15. Simplified 80.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.045:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2 \cdot z} \cdot y\right) \cdot \left(-e^{\frac{t \cdot t}{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 5: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \sqrt{2 \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1e+40)
   (* (sqrt (* 2.0 z)) (- (* 0.5 x) y))
   (* (sqrt z) (sqrt (* 2.0 (* y (- y x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1e+40) {
		tmp = sqrt((2.0 * z)) * ((0.5 * x) - y);
	} else {
		tmp = sqrt(z) * sqrt((2.0 * (y * (y - x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1d+40) then
        tmp = sqrt((2.0d0 * z)) * ((0.5d0 * x) - y)
    else
        tmp = sqrt(z) * sqrt((2.0d0 * (y * (y - x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1e+40) {
		tmp = Math.sqrt((2.0 * z)) * ((0.5 * x) - y);
	} else {
		tmp = Math.sqrt(z) * Math.sqrt((2.0 * (y * (y - x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1e+40:
		tmp = math.sqrt((2.0 * z)) * ((0.5 * x) - y)
	else:
		tmp = math.sqrt(z) * math.sqrt((2.0 * (y * (y - x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1e+40)
		tmp = Float64(sqrt(Float64(2.0 * z)) * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(sqrt(z) * sqrt(Float64(2.0 * Float64(y * Float64(y - x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1e+40)
		tmp = sqrt((2.0 * z)) * ((0.5 * x) - y);
	else
		tmp = sqrt(z) * sqrt((2.0 * (y * (y - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1e+40], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000003e40

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. associate-*r* 99.5%

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

      4. distribute-rgt-out 99.5%

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

      5. associate-*r* 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 99.5%

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

      8. neg-mul-1 99.5%

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

      9. distribute-rgt-in 99.5%

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

      10. *-commutative 99.5%

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

      11. sub-neg 99.5%

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

      12. *-commutative 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-*l* 99.4%

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

      15. *-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in t around 0 71.9%

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

      1. associate-*r* 71.9%

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

      2. sqrt-prod 72.1%

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

      3. *-commutative 72.1%

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

      4. *-commutative 72.1%

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

      5. expm1-log1p-u 42.1%

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

      6. expm1-udef 26.0%

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

      7. log1p-udef 26.0%

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

      8. rem-exp-log 56.0%

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

      9. *-commutative 56.0%

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

      10. *-commutative 56.0%

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

      11. sub-neg 56.0%

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

      12. *-commutative 56.0%

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

      13. sub-neg 56.0%

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

   8. Applied egg-rr 56.0%

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

      1. add-exp-log 26.0%

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

      2. log1p-udef 26.0%

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

      3. *-commutative 26.0%

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

      4. pow1/2 26.0%

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

      5. metadata-eval 26.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}\right)} - 1 \]

      6. pow-prod-up 26.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)}\right)} - 1 \]

      7. associate-*r* 26.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}}\right)} - 1 \]

      8. expm1-udef 42.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right)\right)} \]

      9. associate-*r* 42.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot x - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)}\right)\right) \]

      10. pow-prod-up 42.1%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot \color{blue}{{\left(2 \cdot z\right)}^{\left(0.25 + 0.25\right)}}\right)\right) \]

      11. metadata-eval 42.1%

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

   10. Applied egg-rr 72.1%

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

   if 1.00000000000000003e40 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-rgt-in 100.0%

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

      10. *-commutative 100.0%

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

      11. sub-neg 100.0%

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

      12. *-commutative 100.0%

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

      13. *-commutative 100.0%

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

      14. associate-*l* 100.0%

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

      15. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around 0 14.6%

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

      1. add-sqr-sqrt 5.6%

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

      2. sqrt-unprod 21.2%

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

      3. swap-sqr 21.2%

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

      4. rem-square-sqrt 21.2%

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

      5. pow2 21.2%

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

   8. Applied egg-rr 21.2%

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

   9. Taylor expanded in x around 0 13.8%

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

       1. +-commutative 13.8%

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

       2. unpow2 13.8%

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

       3. associate-*r* 13.8%

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

       4. distribute-rgt-out 13.8%

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

       5. mul-1-neg 13.8%

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

       6. unsub-neg 13.8%

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

   11. Simplified 13.8%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \sqrt{2 \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $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 x around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. associate-*r* 99.6%

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

   4. distribute-rgt-out 99.6%

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

   5. associate-*r* 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*r* 99.6%

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

   8. neg-mul-1 99.6%

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

   9. distribute-rgt-in 99.6%

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

   10. *-commutative 99.6%

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

   11. sub-neg 99.6%

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

   12. *-commutative 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-*l* 99.6%

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

   15. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in t around 0 59.6%

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)} \]
7. 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. *-commutative 59.8%

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

   5. expm1-log1p-u 34.3%

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

   6. expm1-udef 21.6%

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

   7. log1p-udef 21.6%

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

   8. rem-exp-log 47.0%

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

   9. *-commutative 47.0%

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

   10. *-commutative 47.0%

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

   11. sub-neg 47.0%

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

   12. *-commutative 47.0%

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

   13. sub-neg 47.0%

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

8. Applied egg-rr 47.0%

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

   1. add-exp-log 21.6%

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

   2. log1p-udef 21.6%

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

   3. *-commutative 21.6%

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

   4. pow1/2 21.6%

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

   5. metadata-eval 21.6%

      \[\leadsto e^{\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}\right)} - 1 \]

   6. pow-prod-up 21.6%

      \[\leadsto e^{\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)}\right)} - 1 \]

   7. associate-*r* 21.6%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}}\right)} - 1 \]

   8. expm1-udef 34.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(0.5 \cdot x - y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right)\right)} \]

   9. associate-*r* 34.3%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot x - y\right) \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(2 \cdot z\right)}^{0.25}\right)}\right)\right) \]

   10. pow-prod-up 34.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot x - y\right) \cdot \color{blue}{{\left(2 \cdot z\right)}^{\left(0.25 + 0.25\right)}}\right)\right) \]

   11. metadata-eval 34.3%

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

10. Applied egg-rr 59.8%

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

11. Final simplification 59.8%

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

Alternative 7: 2.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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)) * x) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * -0.5), $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 x around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. associate-*r* 99.6%

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

   4. distribute-rgt-out 99.6%

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

   5. associate-*r* 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*r* 99.6%

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

   8. neg-mul-1 99.6%

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

   9. distribute-rgt-in 99.6%

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

   10. *-commutative 99.6%

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

   11. sub-neg 99.6%

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

   12. *-commutative 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-*l* 99.6%

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

   15. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in t around 0 59.6%

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

   1. add-sqr-sqrt 27.4%

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

   2. sqrt-unprod 29.0%

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

   3. swap-sqr 29.0%

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

   4. rem-square-sqrt 29.0%

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

   5. pow2 29.0%

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

8. Applied egg-rr 29.0%

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

9. Taylor expanded in x around -inf 2.1%

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

    1. *-commutative 2.1%

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

11. Simplified 2.1%

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

12. Final simplification 2.1%

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

Alternative 8: 2.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt((2.0 * z)) * 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((2.0d0 * z)) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(2.0 * z), $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. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. distribute-rgt-in 99.8%

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

4. Applied egg-rr 99.8%

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

   1. distribute-rgt-out 99.8%

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

   2. sub-neg 99.8%

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

   3. *-commutative 99.8%

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

   4. add-sqr-sqrt 99.5%

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

   5. associate-*r* 99.5%

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

   6. fma-neg 99.5%

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

   7. add-sqr-sqrt 49.8%

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

   8. sqrt-unprod 70.9%

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

   9. sqr-neg 70.9%

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

   10. sqrt-unprod 27.8%

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

   11. add-sqr-sqrt 51.5%

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

   12. pow1/2 51.5%

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

   13. sqrt-pow1 51.5%

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

   14. metadata-eval 51.5%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\left(z \cdot 2\right)}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   15. pow1/2 51.5%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\left(z \cdot 2\right)}^{0.25}\right) \cdot \sqrt{\color{blue}{{\left(z \cdot 2\right)}^{0.5}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   16. sqrt-pow1 51.4%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\left(z \cdot 2\right)}^{0.25}\right) \cdot \color{blue}{{\left(z \cdot 2\right)}^{\left(\frac{0.5}{2}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   17. metadata-eval 51.4%

       \[\leadsto \left(\left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\left(z \cdot 2\right)}^{0.25}\right) \cdot {\left(z \cdot 2\right)}^{\color{blue}{0.25}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

6. Applied egg-rr 51.4%

   \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\left(z \cdot 2\right)}^{0.25}\right) \cdot {\left(z \cdot 2\right)}^{0.25}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
7. Step-by-step derivation

   1. *-commutative 51.4%

      \[\leadsto \color{blue}{\left({\left(z \cdot 2\right)}^{0.25} \cdot \left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\left(z \cdot 2\right)}^{0.25}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   2. *-commutative 51.4%

      \[\leadsto \left({\color{blue}{\left(2 \cdot z\right)}}^{0.25} \cdot \left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\left(z \cdot 2\right)}^{0.25}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   3. *-commutative 51.4%

      \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\color{blue}{\left(2 \cdot z\right)}}^{0.25}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

8. Simplified 51.4%

   \[\leadsto \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot \left(\mathsf{fma}\left(x, 0.5, y\right) \cdot {\left(2 \cdot z\right)}^{0.25}\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

9. Taylor expanded in t around 0 32.0%

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

10. Taylor expanded in y around inf 2.0%

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

    1. associate-*l* 2.0%

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

    2. unpow1/2 2.0%

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

    3. metadata-eval 2.0%

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

    4. pow-sqr 2.0%

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

    5. unpow1/2 2.0%

       \[\leadsto y \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{{z}^{0.5}}\right) \]

    6. metadata-eval 2.0%

       \[\leadsto y \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot {z}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \]

    7. pow-sqr 2.0%

       \[\leadsto y \cdot \left(\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)}\right) \]

    8. swap-sqr 2.0%

       \[\leadsto y \cdot \color{blue}{\left(\left({2}^{0.25} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right)} \]

    9. exp-to-pow 2.0%

       \[\leadsto y \cdot \left(\left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

    10. exp-to-pow 2.0%

        \[\leadsto y \cdot \left(\left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right) \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

    11. exp-sum 2.0%

        \[\leadsto y \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

    12. distribute-rgt-in 2.0%

        \[\leadsto y \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

    13. log-prod 2.0%

        \[\leadsto y \cdot \left(e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}} \cdot \left({2}^{0.25} \cdot {z}^{0.25}\right)\right) \]

    14. exp-to-pow 2.0%

        \[\leadsto y \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot {z}^{0.25}\right)\right) \]

    15. exp-to-pow 2.0%

        \[\leadsto y \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \left(e^{\log 2 \cdot 0.25} \cdot \color{blue}{e^{\log z \cdot 0.25}}\right)\right) \]

    16. exp-sum 2.0%

        \[\leadsto y \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot \color{blue}{e^{\log 2 \cdot 0.25 + \log z \cdot 0.25}}\right) \]

    17. distribute-rgt-in 2.0%

        \[\leadsto y \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot e^{\color{blue}{0.25 \cdot \left(\log 2 + \log z\right)}}\right) \]

    18. log-prod 2.0%

        \[\leadsto y \cdot \left(e^{0.25 \cdot \log \left(2 \cdot z\right)} \cdot e^{0.25 \cdot \color{blue}{\log \left(2 \cdot z\right)}}\right) \]

12. Simplified 2.0%

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

13. Final simplification 2.0%

    \[\leadsto \sqrt{2 \cdot z} \cdot y \]
14. 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 2024010 
(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))))