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

Percentage Accurate: 99.5% → 99.5%

Time: 11.0s

Alternatives: 9

Speedup: 1.0×

• 44.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\frac{t \cdot t}{2}}\\ \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{+82} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;t_1 \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(y \cdot \left(-\sqrt{z + z}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (exp (/ (* t t) 2.0))))
   (if (or (<= (* x 0.5) -2e+82) (not (<= (* x 0.5) 2e-8)))
     (* t_1 (* x (sqrt (* 0.5 z))))
     (* t_1 (* y (- (sqrt (+ z z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = exp(((t * t) / 2.0));
	double tmp;
	if (((x * 0.5) <= -2e+82) || !((x * 0.5) <= 2e-8)) {
		tmp = t_1 * (x * sqrt((0.5 * z)));
	} else {
		tmp = t_1 * (y * -sqrt((z + z)));
	}
	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 = exp(((t * t) / 2.0d0))
    if (((x * 0.5d0) <= (-2d+82)) .or. (.not. ((x * 0.5d0) <= 2d-8))) then
        tmp = t_1 * (x * sqrt((0.5d0 * z)))
    else
        tmp = t_1 * (y * -sqrt((z + z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.exp(((t * t) / 2.0));
	double tmp;
	if (((x * 0.5) <= -2e+82) || !((x * 0.5) <= 2e-8)) {
		tmp = t_1 * (x * Math.sqrt((0.5 * z)));
	} else {
		tmp = t_1 * (y * -Math.sqrt((z + z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.exp(((t * t) / 2.0))
	tmp = 0
	if ((x * 0.5) <= -2e+82) or not ((x * 0.5) <= 2e-8):
		tmp = t_1 * (x * math.sqrt((0.5 * z)))
	else:
		tmp = t_1 * (y * -math.sqrt((z + z)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = exp(Float64(Float64(t * t) / 2.0))
	tmp = 0.0
	if ((Float64(x * 0.5) <= -2e+82) || !(Float64(x * 0.5) <= 2e-8))
		tmp = Float64(t_1 * Float64(x * sqrt(Float64(0.5 * z))));
	else
		tmp = Float64(t_1 * Float64(y * Float64(-sqrt(Float64(z + z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = exp(((t * t) / 2.0));
	tmp = 0.0;
	if (((x * 0.5) <= -2e+82) || ~(((x * 0.5) <= 2e-8)))
		tmp = t_1 * (x * sqrt((0.5 * z)));
	else
		tmp = t_1 * (y * -sqrt((z + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -2e+82], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-8]], $MachinePrecision]], N[(t$95$1 * N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(y * (-N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -1.9999999999999999e82 or 2e-8 < (*.f64 x 1/2)

   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. Step-by-step derivation

      1. add-sqr-sqrt 57.9%

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

      2. sqrt-unprod 48.2%

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

      3. *-commutative 48.2%

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

      4. *-commutative 48.2%

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

      5. swap-sqr 45.1%

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

      6. add-sqr-sqrt 45.2%

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

      7. add-log-exp 23.9%

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

      8. exp-lft-sqr 23.9%

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

      9. log-prod 23.9%

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

      10. add-log-exp 36.8%

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

      11. add-log-exp 45.2%

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

      12. pow2 45.2%

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

   3. Applied egg-rr 45.2%

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

   4. Taylor expanded in x around inf 45.1%

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

      1. *-commutative 45.1%

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

      2. unpow2 45.1%

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

   6. Simplified 45.1%

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

      1. associate-*r* 45.1%

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

      2. sqrt-prod 46.6%

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

      3. sqrt-prod 53.3%

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

      4. add-sqr-sqrt 86.2%

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

   8. Applied egg-rr 86.2%

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

   if -1.9999999999999999e82 < (*.f64 x 1/2) < 2e-8

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 84.5%

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

      1. mul-1-neg 84.5%

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

      2. associate-*l* 84.6%

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

      3. distribute-rgt-neg-in 84.6%

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

      4. *-commutative 84.6%

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

      5. distribute-rgt-neg-in 84.6%

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

   4. Simplified 84.6%

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

      1. distribute-rgt-neg-out 84.6%

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

      2. add-sqr-sqrt 84.6%

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

      3. sqr-neg 84.6%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 3.2%

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

      6. *-commutative 3.2%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 84.6%

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

      9. sqr-neg 84.6%

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

      10. add-sqr-sqrt 84.6%

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

      11. sqrt-prod 84.7%

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

      12. count-2 84.7%

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

   6. Applied egg-rr 84.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{+82} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{z + z}\right)\right)\\ \end{array} \]

Alternative 3: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\frac{t \cdot t}{2}}\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{+39}:\\ \;\;\;\;t_1 \cdot \left(y \cdot \left(z \cdot -2\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+74}:\\ \;\;\;\;t_1 \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(y \cdot \left(y \cdot z\right)\right) \cdot e^{t \cdot t}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (exp (/ (* t t) 2.0))))
   (if (<= y -2.45e+39)
     (* t_1 (* y (* z -2.0)))
     (if (<= y 1.55e+74)
       (* t_1 (* x (sqrt (* 0.5 z))))
       (- (sqrt (* 2.0 (* (* y (* y z)) (exp (* t t))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = exp(((t * t) / 2.0));
	double tmp;
	if (y <= -2.45e+39) {
		tmp = t_1 * (y * (z * -2.0));
	} else if (y <= 1.55e+74) {
		tmp = t_1 * (x * sqrt((0.5 * z)));
	} else {
		tmp = -sqrt((2.0 * ((y * (y * z)) * exp((t * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp(((t * t) / 2.0d0))
    if (y <= (-2.45d+39)) then
        tmp = t_1 * (y * (z * (-2.0d0)))
    else if (y <= 1.55d+74) then
        tmp = t_1 * (x * sqrt((0.5d0 * z)))
    else
        tmp = -sqrt((2.0d0 * ((y * (y * z)) * exp((t * t)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.exp(((t * t) / 2.0));
	double tmp;
	if (y <= -2.45e+39) {
		tmp = t_1 * (y * (z * -2.0));
	} else if (y <= 1.55e+74) {
		tmp = t_1 * (x * Math.sqrt((0.5 * z)));
	} else {
		tmp = -Math.sqrt((2.0 * ((y * (y * z)) * Math.exp((t * t)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.exp(((t * t) / 2.0))
	tmp = 0
	if y <= -2.45e+39:
		tmp = t_1 * (y * (z * -2.0))
	elif y <= 1.55e+74:
		tmp = t_1 * (x * math.sqrt((0.5 * z)))
	else:
		tmp = -math.sqrt((2.0 * ((y * (y * z)) * math.exp((t * t)))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = exp(Float64(Float64(t * t) / 2.0))
	tmp = 0.0
	if (y <= -2.45e+39)
		tmp = Float64(t_1 * Float64(y * Float64(z * -2.0)));
	elseif (y <= 1.55e+74)
		tmp = Float64(t_1 * Float64(x * sqrt(Float64(0.5 * z))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(y * Float64(y * z)) * exp(Float64(t * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = exp(((t * t) / 2.0));
	tmp = 0.0;
	if (y <= -2.45e+39)
		tmp = t_1 * (y * (z * -2.0));
	elseif (y <= 1.55e+74)
		tmp = t_1 * (x * sqrt((0.5 * z)));
	else
		tmp = -sqrt((2.0 * ((y * (y * z)) * exp((t * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -2.45e+39], N[(t$95$1 * N[(y * N[(z * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+74], N[(t$95$1 * N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(2.0 * N[(N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.44999999999999994e39

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 92.4%

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

      1. mul-1-neg 92.4%

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

      2. associate-*l* 92.6%

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

      3. distribute-rgt-neg-in 92.6%

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

      4. *-commutative 92.6%

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

      5. distribute-rgt-neg-in 92.6%

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

   4. Simplified 92.6%

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

      1. *-commutative 92.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.4%

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

      4. sqr-neg 0.4%

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

      5. add-sqr-sqrt 0.4%

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

      6. sqrt-prod 0.4%

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

      7. count-2 0.4%

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

      8. expm1-log1p-u 0.4%

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. expm1-udef 0.6%

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

   6. Applied egg-rr 0.6%

      \[\leadsto \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   7. Step-by-step derivation

      1. expm1-def 0.4%

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. expm1-log1p 0.4%

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

   8. Simplified 0.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 70.3%

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

      3. sqr-neg 70.3%

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

      4. distribute-rgt-neg-out 70.3%

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

      5. distribute-rgt-neg-out 70.3%

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

      6. sqrt-unprod 92.5%

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

      7. add-sqr-sqrt 92.8%

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

      8. distribute-rgt-neg-out 92.8%

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

      9. neg-sub0 92.8%

         \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{z + z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \sqrt{z + z}} \cdot \sqrt{y \cdot \sqrt{z + z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqrt-unprod 0.3%

          \[\leadsto \left(0 - \color{blue}{\sqrt{\left(y \cdot \sqrt{z + z}\right) \cdot \left(y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. sqr-neg 0.3%

          \[\leadsto \left(0 - \sqrt{\color{blue}{\left(-y \cdot \sqrt{z + z}\right) \cdot \left(-y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. distribute-rgt-neg-out 0.3%

          \[\leadsto \left(0 - \sqrt{\color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)} \cdot \left(-y \cdot \sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      14. distribute-rgt-neg-out 0.3%

          \[\leadsto \left(0 - \sqrt{\left(y \cdot \left(-\sqrt{z + z}\right)\right) \cdot \color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      15. sqrt-unprod 0.4%

          \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \left(-\sqrt{z + z}\right)} \cdot \sqrt{y \cdot \left(-\sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      16. add-sqr-sqrt 0.4%

          \[\leadsto \left(0 - \color{blue}{y \cdot \left(-\sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      17. add-sqr-sqrt 0.0%

          \[\leadsto \left(0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z + z}} \cdot \sqrt{-\sqrt{z + z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. Applied egg-rr 64.0%

       \[\leadsto \color{blue}{\left(0 - y \cdot \left(z + z\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   11. Step-by-step derivation

       1. neg-sub0 64.0%

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

       2. distribute-rgt-neg-in 64.0%

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

       3. count-2 64.0%

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

       4. distribute-lft-neg-in 64.0%

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

       5. metadata-eval 64.0%

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

   12. Simplified 64.0%

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

   if -2.44999999999999994e39 < y < 1.55000000000000011e74

   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. Step-by-step derivation

      1. add-sqr-sqrt 54.3%

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

      2. sqrt-unprod 46.1%

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

      3. *-commutative 46.1%

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

      4. *-commutative 46.1%

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

      5. swap-sqr 42.8%

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

      6. add-sqr-sqrt 42.9%

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

      7. add-log-exp 18.4%

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

      8. exp-lft-sqr 18.4%

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

      9. log-prod 18.4%

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

      10. add-log-exp 27.9%

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

      11. add-log-exp 42.9%

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

      12. pow2 42.9%

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

   3. Applied egg-rr 42.9%

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

   4. Taylor expanded in x around inf 33.3%

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

      1. *-commutative 33.3%

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

      2. unpow2 33.3%

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

   6. Simplified 33.3%

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

      1. associate-*r* 33.3%

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

      2. sqrt-prod 35.8%

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

      3. sqrt-prod 42.1%

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

      4. add-sqr-sqrt 77.5%

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

   8. Applied egg-rr 77.5%

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

   if 1.55000000000000011e74 < y

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-- 67.0%

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

      3. associate-*r/ 67.0%

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

      4. add-log-exp 35.0%

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

      5. exp-lft-sqr 35.0%

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

      6. log-prod 35.0%

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

      7. add-log-exp 59.7%

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

      8. add-log-exp 67.0%

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

      9. swap-sqr 67.0%

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

      10. metadata-eval 67.0%

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

      11. fma-def 67.0%

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

   3. Applied egg-rr 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-/l* 54.7%

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

   5. Simplified 54.7%

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

   6. Taylor expanded in x around 0 89.9%

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

      1. mul-1-neg 89.9%

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

      2. distribute-rgt-neg-in 89.9%

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

   8. Simplified 89.9%

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

      1. distribute-rgt-neg-out 89.9%

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

      2. *-commutative 89.9%

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

      3. associate-*r* 89.9%

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

      4. distribute-lft-neg-out 89.9%

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

      5. add-sqr-sqrt 89.9%

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

      6. sqr-neg 89.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 6.3%

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

      9. associate-*l* 6.3%

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

      10. *-commutative 6.3%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 76.9%

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

   10. Applied egg-rr 71.3%

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

       1. *-commutative 71.3%

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

       2. associate-*l* 71.3%

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

       3. *-commutative 71.3%

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

       4. associate-*l* 77.0%

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

       5. exp-prod 77.0%

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

   12. Simplified 77.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+39}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(z \cdot -2\right)\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+74}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \left(\left(y \cdot \left(y \cdot z\right)\right) \cdot e^{t \cdot t}\right)}\\ \end{array} \]

Alternative 4: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\frac{t \cdot t}{2}}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+39} \lor \neg \left(y \leq 9 \cdot 10^{+113}\right):\\ \;\;\;\;t_1 \cdot \left(y \cdot \left(z \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (exp (/ (* t t) 2.0))))
   (if (or (<= y -2.6e+39) (not (<= y 9e+113)))
     (* t_1 (* y (* z -2.0)))
     (* t_1 (* x (sqrt (* 0.5 z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = exp(((t * t) / 2.0));
	double tmp;
	if ((y <= -2.6e+39) || !(y <= 9e+113)) {
		tmp = t_1 * (y * (z * -2.0));
	} else {
		tmp = t_1 * (x * sqrt((0.5 * z)));
	}
	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 = exp(((t * t) / 2.0d0))
    if ((y <= (-2.6d+39)) .or. (.not. (y <= 9d+113))) then
        tmp = t_1 * (y * (z * (-2.0d0)))
    else
        tmp = t_1 * (x * sqrt((0.5d0 * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.exp(((t * t) / 2.0));
	double tmp;
	if ((y <= -2.6e+39) || !(y <= 9e+113)) {
		tmp = t_1 * (y * (z * -2.0));
	} else {
		tmp = t_1 * (x * Math.sqrt((0.5 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.exp(((t * t) / 2.0))
	tmp = 0
	if (y <= -2.6e+39) or not (y <= 9e+113):
		tmp = t_1 * (y * (z * -2.0))
	else:
		tmp = t_1 * (x * math.sqrt((0.5 * z)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = exp(Float64(Float64(t * t) / 2.0))
	tmp = 0.0
	if ((y <= -2.6e+39) || !(y <= 9e+113))
		tmp = Float64(t_1 * Float64(y * Float64(z * -2.0)));
	else
		tmp = Float64(t_1 * Float64(x * sqrt(Float64(0.5 * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = exp(((t * t) / 2.0));
	tmp = 0.0;
	if ((y <= -2.6e+39) || ~((y <= 9e+113)))
		tmp = t_1 * (y * (z * -2.0));
	else
		tmp = t_1 * (x * sqrt((0.5 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -2.6e+39], N[Not[LessEqual[y, 9e+113]], $MachinePrecision]], N[(t$95$1 * N[(y * N[(z * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e39 or 9.0000000000000001e113 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 93.6%

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

      1. mul-1-neg 93.6%

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

      2. associate-*l* 93.7%

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

      3. distribute-rgt-neg-in 93.7%

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

      4. *-commutative 93.7%

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

      5. distribute-rgt-neg-in 93.7%

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

   4. Simplified 93.7%

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

      1. *-commutative 93.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 1.3%

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

      4. sqr-neg 1.3%

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

      5. add-sqr-sqrt 1.3%

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

      6. sqrt-prod 1.3%

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

      7. count-2 1.3%

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

      8. expm1-log1p-u 1.3%

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. expm1-udef 1.5%

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

   6. Applied egg-rr 1.5%

      \[\leadsto \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   7. Step-by-step derivation

      1. expm1-def 1.3%

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. expm1-log1p 1.3%

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

   8. Simplified 1.3%

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

      1. add-sqr-sqrt 1.1%

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

      2. sqrt-unprod 41.1%

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

      3. sqr-neg 41.1%

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

      4. distribute-rgt-neg-out 41.1%

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

      5. distribute-rgt-neg-out 41.1%

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

      6. sqrt-unprod 52.6%

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

      7. add-sqr-sqrt 93.9%

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

      8. distribute-rgt-neg-out 93.9%

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

      9. neg-sub0 93.9%

         \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{z + z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      10. add-sqr-sqrt 41.1%

          \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \sqrt{z + z}} \cdot \sqrt{y \cdot \sqrt{z + z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqrt-unprod 35.4%

          \[\leadsto \left(0 - \color{blue}{\sqrt{\left(y \cdot \sqrt{z + z}\right) \cdot \left(y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. sqr-neg 35.4%

          \[\leadsto \left(0 - \sqrt{\color{blue}{\left(-y \cdot \sqrt{z + z}\right) \cdot \left(-y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. distribute-rgt-neg-out 35.4%

          \[\leadsto \left(0 - \sqrt{\color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)} \cdot \left(-y \cdot \sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      14. distribute-rgt-neg-out 35.4%

          \[\leadsto \left(0 - \sqrt{\left(y \cdot \left(-\sqrt{z + z}\right)\right) \cdot \color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      15. sqrt-unprod 0.2%

          \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \left(-\sqrt{z + z}\right)} \cdot \sqrt{y \cdot \left(-\sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      16. add-sqr-sqrt 1.3%

          \[\leadsto \left(0 - \color{blue}{y \cdot \left(-\sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      17. add-sqr-sqrt 0.0%

          \[\leadsto \left(0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z + z}} \cdot \sqrt{-\sqrt{z + z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. Applied egg-rr 68.8%

       \[\leadsto \color{blue}{\left(0 - y \cdot \left(z + z\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   11. Step-by-step derivation

       1. neg-sub0 68.8%

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

       2. distribute-rgt-neg-in 68.8%

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

       3. count-2 68.8%

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

       4. distribute-lft-neg-in 68.8%

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

       5. metadata-eval 68.8%

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

   12. Simplified 68.8%

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

   if -2.6e39 < y < 9.0000000000000001e113

   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. Step-by-step derivation

      1. add-sqr-sqrt 53.1%

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

      2. sqrt-unprod 44.8%

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

      3. *-commutative 44.8%

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

      4. *-commutative 44.8%

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

      5. swap-sqr 41.7%

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

      6. add-sqr-sqrt 41.8%

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

      7. add-log-exp 18.1%

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

      8. exp-lft-sqr 18.1%

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

      9. log-prod 18.1%

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

      10. add-log-exp 27.7%

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

      11. add-log-exp 41.8%

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

      12. pow2 41.8%

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

   3. Applied egg-rr 41.8%

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

   4. Taylor expanded in x around inf 32.7%

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

      1. *-commutative 32.7%

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

      2. unpow2 32.7%

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

   6. Simplified 32.7%

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

      1. associate-*r* 32.7%

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

      2. sqrt-prod 35.1%

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

      3. sqrt-prod 41.6%

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

      4. add-sqr-sqrt 75.1%

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

   8. Applied egg-rr 75.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+39} \lor \neg \left(y \leq 9 \cdot 10^{+113}\right):\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(z \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \end{array} \]

Alternative 5: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 17500000:\\ \;\;\;\;y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-2 \cdot \left(z \cdot \sqrt{e^{t \cdot t}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 17500000.0)
   (* y (* (sqrt 2.0) (- (sqrt z))))
   (* y (* -2.0 (* z (sqrt (exp (* t t))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 17500000.0) {
		tmp = y * (sqrt(2.0) * -sqrt(z));
	} else {
		tmp = y * (-2.0 * (z * sqrt(exp((t * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t * t) <= 17500000.0d0) then
        tmp = y * (sqrt(2.0d0) * -sqrt(z))
    else
        tmp = y * ((-2.0d0) * (z * sqrt(exp((t * t)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 17500000.0) {
		tmp = y * (Math.sqrt(2.0) * -Math.sqrt(z));
	} else {
		tmp = y * (-2.0 * (z * Math.sqrt(Math.exp((t * t)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t * t) <= 17500000.0:
		tmp = y * (math.sqrt(2.0) * -math.sqrt(z))
	else:
		tmp = y * (-2.0 * (z * math.sqrt(math.exp((t * t)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 17500000.0)
		tmp = Float64(y * Float64(sqrt(2.0) * Float64(-sqrt(z))));
	else
		tmp = Float64(y * Float64(-2.0 * Float64(z * sqrt(exp(Float64(t * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t * t) <= 17500000.0)
		tmp = y * (sqrt(2.0) * -sqrt(z));
	else
		tmp = y * (-2.0 * (z * sqrt(exp((t * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 17500000.0], N[(y * N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[z], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(y * N[(-2.0 * N[(z * N[Sqrt[N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.75e7

   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. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. flip-- 61.0%

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

      3. associate-*r/ 55.3%

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

      4. add-log-exp 15.6%

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

      5. exp-lft-sqr 15.6%

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

      6. log-prod 15.6%

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

      7. add-log-exp 19.5%

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

      8. add-log-exp 55.3%

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

      9. swap-sqr 55.3%

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

      10. metadata-eval 55.3%

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

      11. fma-def 55.3%

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

   3. Applied egg-rr 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/l* 60.4%

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

   5. Simplified 60.4%

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

   6. Taylor expanded in x around 0 46.8%

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

      1. mul-1-neg 46.8%

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

      2. distribute-rgt-neg-in 46.8%

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

   8. Simplified 46.8%

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

   9. Taylor expanded in t around 0 46.8%

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

       1. mul-1-neg 46.8%

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

       2. associate-*r* 46.8%

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

       3. distribute-lft-neg-in 46.8%

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

   11. Simplified 46.8%

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

   if 1.75e7 < (*.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. Taylor expanded in x around 0 81.1%

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

      1. mul-1-neg 81.1%

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

      2. associate-*l* 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

      4. *-commutative 81.1%

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

      5. distribute-rgt-neg-in 81.1%

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

   4. Simplified 81.1%

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

      1. *-commutative 81.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 16.4%

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

      4. sqr-neg 16.4%

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

      5. add-sqr-sqrt 16.4%

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

      6. sqrt-prod 16.4%

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

      7. count-2 16.4%

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

      8. expm1-log1p-u 16.4%

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. expm1-udef 10.7%

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

   6. Applied egg-rr 10.7%

      \[\leadsto \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   7. Step-by-step derivation

      1. expm1-def 16.4%

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. expm1-log1p 16.4%

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

   8. Simplified 16.4%

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

      1. add-sqr-sqrt 12.3%

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

      2. sqrt-unprod 49.2%

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

      3. sqr-neg 49.2%

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

      4. distribute-rgt-neg-out 49.2%

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

      5. distribute-rgt-neg-out 49.2%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 81.1%

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

      8. distribute-rgt-neg-out 81.1%

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

      9. neg-sub0 81.1%

         \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{z + z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      10. add-sqr-sqrt 37.7%

          \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \sqrt{z + z}} \cdot \sqrt{y \cdot \sqrt{z + z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqrt-unprod 34.4%

          \[\leadsto \left(0 - \color{blue}{\sqrt{\left(y \cdot \sqrt{z + z}\right) \cdot \left(y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. sqr-neg 34.4%

          \[\leadsto \left(0 - \sqrt{\color{blue}{\left(-y \cdot \sqrt{z + z}\right) \cdot \left(-y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. distribute-rgt-neg-out 34.4%

          \[\leadsto \left(0 - \sqrt{\color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)} \cdot \left(-y \cdot \sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      14. distribute-rgt-neg-out 34.4%

          \[\leadsto \left(0 - \sqrt{\left(y \cdot \left(-\sqrt{z + z}\right)\right) \cdot \color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      15. sqrt-unprod 4.1%

          \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \left(-\sqrt{z + z}\right)} \cdot \sqrt{y \cdot \left(-\sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      16. add-sqr-sqrt 16.4%

          \[\leadsto \left(0 - \color{blue}{y \cdot \left(-\sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      17. add-sqr-sqrt 0.0%

          \[\leadsto \left(0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z + z}} \cdot \sqrt{-\sqrt{z + z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. Applied egg-rr 78.7%

       \[\leadsto \color{blue}{\left(0 - y \cdot \left(z + z\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   11. Step-by-step derivation

       1. neg-sub0 78.7%

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

       2. distribute-rgt-neg-in 78.7%

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

       3. count-2 78.7%

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

       4. distribute-lft-neg-in 78.7%

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

       5. metadata-eval 78.7%

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

   12. Simplified 78.7%

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

   13. Taylor expanded in y around 0 82.8%

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

       1. *-commutative 82.8%

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

       2. associate-*l* 82.8%

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

       3. *-commutative 82.8%

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

       4. exp-prod 82.8%

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

       5. unpow1/2 82.8%

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

       6. unpow2 82.8%

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

   15. Simplified 82.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 17500000:\\ \;\;\;\;y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-2 \cdot \left(z \cdot \sqrt{e^{t \cdot t}}\right)\right)\\ \end{array} \]

Alternative 6: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 17500000:\\ \;\;\;\;y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(z \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 17500000.0)
   (* y (* (sqrt 2.0) (- (sqrt z))))
   (* (exp (/ (* t t) 2.0)) (* y (* z -2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 17500000.0) {
		tmp = y * (sqrt(2.0) * -sqrt(z));
	} else {
		tmp = exp(((t * t) / 2.0)) * (y * (z * -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t * t) <= 17500000.0d0) then
        tmp = y * (sqrt(2.0d0) * -sqrt(z))
    else
        tmp = exp(((t * t) / 2.0d0)) * (y * (z * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 17500000.0) {
		tmp = y * (Math.sqrt(2.0) * -Math.sqrt(z));
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (y * (z * -2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t * t) <= 17500000.0:
		tmp = y * (math.sqrt(2.0) * -math.sqrt(z))
	else:
		tmp = math.exp(((t * t) / 2.0)) * (y * (z * -2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 17500000.0)
		tmp = Float64(y * Float64(sqrt(2.0) * Float64(-sqrt(z))));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * Float64(z * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t * t) <= 17500000.0)
		tmp = y * (sqrt(2.0) * -sqrt(z));
	else
		tmp = exp(((t * t) / 2.0)) * (y * (z * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 17500000.0], N[(y * N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[z], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * N[(z * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.75e7

   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. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. flip-- 61.0%

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

      3. associate-*r/ 55.3%

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

      4. add-log-exp 15.6%

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

      5. exp-lft-sqr 15.6%

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

      6. log-prod 15.6%

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

      7. add-log-exp 19.5%

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

      8. add-log-exp 55.3%

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

      9. swap-sqr 55.3%

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

      10. metadata-eval 55.3%

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

      11. fma-def 55.3%

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

   3. Applied egg-rr 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/l* 60.4%

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

   5. Simplified 60.4%

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

   6. Taylor expanded in x around 0 46.8%

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

      1. mul-1-neg 46.8%

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

      2. distribute-rgt-neg-in 46.8%

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

   8. Simplified 46.8%

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

   9. Taylor expanded in t around 0 46.8%

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

       1. mul-1-neg 46.8%

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

       2. associate-*r* 46.8%

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

       3. distribute-lft-neg-in 46.8%

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

   11. Simplified 46.8%

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

   if 1.75e7 < (*.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. Taylor expanded in x around 0 81.1%

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

      1. mul-1-neg 81.1%

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

      2. associate-*l* 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

      4. *-commutative 81.1%

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

      5. distribute-rgt-neg-in 81.1%

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

   4. Simplified 81.1%

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

      1. *-commutative 81.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 16.4%

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

      4. sqr-neg 16.4%

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

      5. add-sqr-sqrt 16.4%

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

      6. sqrt-prod 16.4%

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

      7. count-2 16.4%

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

      8. expm1-log1p-u 16.4%

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. expm1-udef 10.7%

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

   6. Applied egg-rr 10.7%

      \[\leadsto \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   7. Step-by-step derivation

      1. expm1-def 16.4%

         \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. expm1-log1p 16.4%

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

   8. Simplified 16.4%

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

      1. add-sqr-sqrt 12.3%

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

      2. sqrt-unprod 49.2%

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

      3. sqr-neg 49.2%

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

      4. distribute-rgt-neg-out 49.2%

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

      5. distribute-rgt-neg-out 49.2%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 81.1%

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

      8. distribute-rgt-neg-out 81.1%

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

      9. neg-sub0 81.1%

         \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{z + z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      10. add-sqr-sqrt 37.7%

          \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \sqrt{z + z}} \cdot \sqrt{y \cdot \sqrt{z + z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. sqrt-unprod 34.4%

          \[\leadsto \left(0 - \color{blue}{\sqrt{\left(y \cdot \sqrt{z + z}\right) \cdot \left(y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. sqr-neg 34.4%

          \[\leadsto \left(0 - \sqrt{\color{blue}{\left(-y \cdot \sqrt{z + z}\right) \cdot \left(-y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. distribute-rgt-neg-out 34.4%

          \[\leadsto \left(0 - \sqrt{\color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)} \cdot \left(-y \cdot \sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      14. distribute-rgt-neg-out 34.4%

          \[\leadsto \left(0 - \sqrt{\left(y \cdot \left(-\sqrt{z + z}\right)\right) \cdot \color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      15. sqrt-unprod 4.1%

          \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \left(-\sqrt{z + z}\right)} \cdot \sqrt{y \cdot \left(-\sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      16. add-sqr-sqrt 16.4%

          \[\leadsto \left(0 - \color{blue}{y \cdot \left(-\sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      17. add-sqr-sqrt 0.0%

          \[\leadsto \left(0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z + z}} \cdot \sqrt{-\sqrt{z + z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   10. Applied egg-rr 78.7%

       \[\leadsto \color{blue}{\left(0 - y \cdot \left(z + z\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   11. Step-by-step derivation

       1. neg-sub0 78.7%

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

       2. distribute-rgt-neg-in 78.7%

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

       3. count-2 78.7%

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

       4. distribute-lft-neg-in 78.7%

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

       5. metadata-eval 78.7%

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

   12. Simplified 78.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 17500000:\\ \;\;\;\;y \cdot \left(\sqrt{2} \cdot \left(-\sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(z \cdot -2\right)\right)\\ \end{array} \]

Alternative 7: 40.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 63.1%

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

   1. mul-1-neg 63.1%

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

   2. associate-*l* 63.2%

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

   3. distribute-rgt-neg-in 63.2%

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

   4. *-commutative 63.2%

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

   5. distribute-rgt-neg-in 63.2%

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

4. Simplified 63.2%

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

   1. *-commutative 63.2%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 9.3%

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

   4. sqr-neg 9.3%

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

   5. add-sqr-sqrt 9.3%

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

   6. sqrt-prod 9.3%

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

   7. count-2 9.3%

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

   8. expm1-log1p-u 9.3%

      \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. expm1-udef 6.7%

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

6. Applied egg-rr 6.7%

   \[\leadsto \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
7. Step-by-step derivation

   1. expm1-def 9.3%

      \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. expm1-log1p 9.3%

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

8. Simplified 9.3%

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

   1. add-sqr-sqrt 6.9%

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

   2. sqrt-unprod 31.5%

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

   3. sqr-neg 31.5%

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

   4. distribute-rgt-neg-out 31.5%

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

   5. distribute-rgt-neg-out 31.5%

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

   6. sqrt-unprod 33.9%

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

   7. add-sqr-sqrt 63.3%

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

   8. distribute-rgt-neg-out 63.3%

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

   9. neg-sub0 63.3%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{z + z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. add-sqr-sqrt 29.7%

       \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \sqrt{z + z}} \cdot \sqrt{y \cdot \sqrt{z + z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   11. sqrt-unprod 25.4%

       \[\leadsto \left(0 - \color{blue}{\sqrt{\left(y \cdot \sqrt{z + z}\right) \cdot \left(y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   12. sqr-neg 25.4%

       \[\leadsto \left(0 - \sqrt{\color{blue}{\left(-y \cdot \sqrt{z + z}\right) \cdot \left(-y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   13. distribute-rgt-neg-out 25.4%

       \[\leadsto \left(0 - \sqrt{\color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)} \cdot \left(-y \cdot \sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   14. distribute-rgt-neg-out 25.4%

       \[\leadsto \left(0 - \sqrt{\left(y \cdot \left(-\sqrt{z + z}\right)\right) \cdot \color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   15. sqrt-unprod 2.9%

       \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \left(-\sqrt{z + z}\right)} \cdot \sqrt{y \cdot \left(-\sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   16. add-sqr-sqrt 9.3%

       \[\leadsto \left(0 - \color{blue}{y \cdot \left(-\sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   17. add-sqr-sqrt 0.0%

       \[\leadsto \left(0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z + z}} \cdot \sqrt{-\sqrt{z + z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

10. Applied egg-rr 44.5%

    \[\leadsto \color{blue}{\left(0 - y \cdot \left(z + z\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
11. Step-by-step derivation

    1. neg-sub0 44.5%

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

    2. distribute-rgt-neg-in 44.5%

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

    3. count-2 44.5%

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

    4. distribute-lft-neg-in 44.5%

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

    5. metadata-eval 44.5%

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

12. Simplified 44.5%

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

13. Final simplification 44.5%

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

Alternative 8: 30.7% accurate, 23.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 63.1%

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

   1. mul-1-neg 63.1%

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

   2. associate-*l* 63.2%

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

   3. distribute-rgt-neg-in 63.2%

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

   4. *-commutative 63.2%

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

   5. distribute-rgt-neg-in 63.2%

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

4. Simplified 63.2%

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

   1. *-commutative 63.2%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 9.3%

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

   4. sqr-neg 9.3%

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

   5. add-sqr-sqrt 9.3%

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

   6. sqrt-prod 9.3%

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

   7. count-2 9.3%

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

   8. expm1-log1p-u 9.3%

      \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. expm1-udef 6.7%

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

6. Applied egg-rr 6.7%

   \[\leadsto \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
7. Step-by-step derivation

   1. expm1-def 9.3%

      \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. expm1-log1p 9.3%

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

8. Simplified 9.3%

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

   1. add-sqr-sqrt 6.9%

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

   2. sqrt-unprod 31.5%

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

   3. sqr-neg 31.5%

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

   4. distribute-rgt-neg-out 31.5%

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

   5. distribute-rgt-neg-out 31.5%

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

   6. sqrt-unprod 33.9%

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

   7. add-sqr-sqrt 63.3%

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

   8. distribute-rgt-neg-out 63.3%

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

   9. neg-sub0 63.3%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{z + z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. add-sqr-sqrt 29.7%

       \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \sqrt{z + z}} \cdot \sqrt{y \cdot \sqrt{z + z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   11. sqrt-unprod 25.4%

       \[\leadsto \left(0 - \color{blue}{\sqrt{\left(y \cdot \sqrt{z + z}\right) \cdot \left(y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   12. sqr-neg 25.4%

       \[\leadsto \left(0 - \sqrt{\color{blue}{\left(-y \cdot \sqrt{z + z}\right) \cdot \left(-y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   13. distribute-rgt-neg-out 25.4%

       \[\leadsto \left(0 - \sqrt{\color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)} \cdot \left(-y \cdot \sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   14. distribute-rgt-neg-out 25.4%

       \[\leadsto \left(0 - \sqrt{\left(y \cdot \left(-\sqrt{z + z}\right)\right) \cdot \color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   15. sqrt-unprod 2.9%

       \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \left(-\sqrt{z + z}\right)} \cdot \sqrt{y \cdot \left(-\sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   16. add-sqr-sqrt 9.3%

       \[\leadsto \left(0 - \color{blue}{y \cdot \left(-\sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   17. add-sqr-sqrt 0.0%

       \[\leadsto \left(0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z + z}} \cdot \sqrt{-\sqrt{z + z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

10. Applied egg-rr 44.5%

    \[\leadsto \color{blue}{\left(0 - y \cdot \left(z + z\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
11. Step-by-step derivation

    1. neg-sub0 44.5%

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

    2. distribute-rgt-neg-in 44.5%

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

    3. count-2 44.5%

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

    4. distribute-lft-neg-in 44.5%

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

    5. metadata-eval 44.5%

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

12. Simplified 44.5%

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

13. Taylor expanded in t around 0 32.5%

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

    1. associate-*r* 32.5%

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

    2. distribute-rgt-out 33.8%

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

    3. mul-1-neg 33.8%

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

    4. unsub-neg 33.8%

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

    5. unpow2 33.8%

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

15. Simplified 33.8%

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

16. Final simplification 33.8%

    \[\leadsto \left(y \cdot z\right) \cdot \left(-2 - t \cdot t\right) \]

Alternative 9: 13.7% accurate, 43.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 63.1%

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

   1. mul-1-neg 63.1%

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

   2. associate-*l* 63.2%

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

   3. distribute-rgt-neg-in 63.2%

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

   4. *-commutative 63.2%

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

   5. distribute-rgt-neg-in 63.2%

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

4. Simplified 63.2%

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

   1. *-commutative 63.2%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 9.3%

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

   4. sqr-neg 9.3%

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

   5. add-sqr-sqrt 9.3%

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

   6. sqrt-prod 9.3%

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

   7. count-2 9.3%

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

   8. expm1-log1p-u 9.3%

      \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   9. expm1-udef 6.7%

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

6. Applied egg-rr 6.7%

   \[\leadsto \left(y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z + z}\right)} - 1\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]
7. Step-by-step derivation

   1. expm1-def 9.3%

      \[\leadsto \left(y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{z + z}\right)\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   2. expm1-log1p 9.3%

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

8. Simplified 9.3%

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

   1. add-sqr-sqrt 6.9%

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

   2. sqrt-unprod 31.5%

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

   3. sqr-neg 31.5%

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

   4. distribute-rgt-neg-out 31.5%

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

   5. distribute-rgt-neg-out 31.5%

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

   6. sqrt-unprod 33.9%

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

   7. add-sqr-sqrt 63.3%

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

   8. distribute-rgt-neg-out 63.3%

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

   9. neg-sub0 63.3%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{z + z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   10. add-sqr-sqrt 29.7%

       \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \sqrt{z + z}} \cdot \sqrt{y \cdot \sqrt{z + z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   11. sqrt-unprod 25.4%

       \[\leadsto \left(0 - \color{blue}{\sqrt{\left(y \cdot \sqrt{z + z}\right) \cdot \left(y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   12. sqr-neg 25.4%

       \[\leadsto \left(0 - \sqrt{\color{blue}{\left(-y \cdot \sqrt{z + z}\right) \cdot \left(-y \cdot \sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   13. distribute-rgt-neg-out 25.4%

       \[\leadsto \left(0 - \sqrt{\color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)} \cdot \left(-y \cdot \sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   14. distribute-rgt-neg-out 25.4%

       \[\leadsto \left(0 - \sqrt{\left(y \cdot \left(-\sqrt{z + z}\right)\right) \cdot \color{blue}{\left(y \cdot \left(-\sqrt{z + z}\right)\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   15. sqrt-unprod 2.9%

       \[\leadsto \left(0 - \color{blue}{\sqrt{y \cdot \left(-\sqrt{z + z}\right)} \cdot \sqrt{y \cdot \left(-\sqrt{z + z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   16. add-sqr-sqrt 9.3%

       \[\leadsto \left(0 - \color{blue}{y \cdot \left(-\sqrt{z + z}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   17. add-sqr-sqrt 0.0%

       \[\leadsto \left(0 - y \cdot \color{blue}{\left(\sqrt{-\sqrt{z + z}} \cdot \sqrt{-\sqrt{z + z}}\right)}\right) \cdot e^{\frac{t \cdot t}{2}} \]

10. Applied egg-rr 44.5%

    \[\leadsto \color{blue}{\left(0 - y \cdot \left(z + z\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]
11. Step-by-step derivation

    1. neg-sub0 44.5%

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

    2. distribute-rgt-neg-in 44.5%

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

    3. count-2 44.5%

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

    4. distribute-lft-neg-in 44.5%

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

    5. metadata-eval 44.5%

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

12. Simplified 44.5%

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

13. Taylor expanded in t around 0 14.6%

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

14. Final simplification 14.6%

    \[\leadsto -2 \cdot \left(y \cdot z\right) \]

Developer target: 99.5% 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 2023271 
(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))))