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

Percentage Accurate: 99.4% → 99.7%

Time: 18.1s

Alternatives: 8

Speedup: 1.0×

• 45.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(x \cdot \left(\left(-0.5 + \frac{y}{x}\right) \cdot \left(-\sqrt{z}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.8e-5)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* (sqrt 2.0) (* x (* (+ -0.5 (/ y x)) (- (sqrt z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.8e-5) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt(2.0) * (x * ((-0.5 + (y / x)) * -sqrt(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) :: tmp
    if (t <= 2.8d-5) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt(2.0d0) * (x * (((-0.5d0) + (y / x)) * -sqrt(z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 2.8e-5) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(2.0) * (x * ((-0.5 + (y / x)) * -Math.sqrt(z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 2.8e-5:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(2.0) * (x * ((-0.5 + (y / x)) * -math.sqrt(z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 2.8e-5)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(sqrt(2.0) * Float64(x * Float64(Float64(-0.5 + Float64(y / x)) * Float64(-sqrt(z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 2.8e-5)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt(2.0) * (x * ((-0.5 + (y / x)) * -sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.8e-5], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(x * N[(N[(-0.5 + N[(y / x), $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[z], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.79999999999999996e-5

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0 71.9%

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

      1. associate-*r* 71.9%

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

      2. sqrt-prod 72.1%

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

      3. sub-neg 72.1%

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

      4. distribute-lft-in 71.1%

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

      5. *-commutative 71.1%

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

      6. *-commutative 71.1%

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

      7. *-commutative 71.1%

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

   5. Applied egg-rr 71.1%

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

      1. distribute-lft-out 72.1%

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

      2. sub-neg 72.1%

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

      3. *-commutative 72.1%

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

      4. *-commutative 72.1%

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

   7. Simplified 72.1%

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

   if 2.79999999999999996e-5 < t

   1. Initial program 98.4%

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

      1. add-sqr-sqrt 48.4%

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

      2. sqrt-unprod 46.8%

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

      3. *-commutative 46.8%

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

      4. *-commutative 46.8%

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

      5. swap-sqr 46.8%

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

      6. add-sqr-sqrt 46.8%

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

      7. pow2 46.8%

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

   4. Applied egg-rr 46.8%

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

      1. *-commutative 46.8%

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

      2. *-commutative 46.8%

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

      3. *-commutative 46.8%

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

   6. Simplified 46.8%

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

      1. sqrt-prod 46.8%

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

      2. *-commutative 46.8%

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

      3. sqrt-pow1 98.4%

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

      4. metadata-eval 98.4%

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

      5. pow1 98.4%

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

      6. add-sqr-sqrt 98.4%

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

      7. associate-*r* 98.4%

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

      8. *-commutative 98.4%

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

      9. pow1/2 98.4%

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

      10. *-commutative 98.4%

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

      11. sqrt-pow1 98.4%

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

      12. metadata-eval 98.4%

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

      13. pow1/2 98.4%

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

      14. *-commutative 98.4%

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

      15. sqrt-pow1 98.4%

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

      16. metadata-eval 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. associate-*l* 98.4%

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

      2. *-commutative 98.4%

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

   10. Simplified 98.4%

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

   11. Taylor expanded in t around 0 10.8%

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

       1. *-commutative 10.8%

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

       2. fma-neg 10.8%

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

       3. associate-*l* 10.8%

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

       4. *-commutative 10.8%

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

       5. fma-neg 10.8%

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

   13. Simplified 10.8%

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

   14. Taylor expanded in x around -inf 29.0%

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

       1. mul-1-neg 29.0%

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

       2. *-commutative 29.0%

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

       3. distribute-rgt-neg-in 29.0%

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

       4. distribute-rgt-out 29.0%

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

   16. Simplified 29.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(x \cdot \left(\left(-0.5 + \frac{y}{x}\right) \cdot \left(-\sqrt{z}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1.22 \cdot 10^{+30}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot {t\_1}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.22e+30)
     (* t_1 (sqrt (* 2.0 z)))
     (sqrt (* z (* 2.0 (pow t_1 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.22e+30) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = sqrt((z * (2.0 * pow(t_1, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 1.22d+30) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = sqrt((z * (2.0d0 * (t_1 ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.22e+30) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt((z * (2.0 * Math.pow(t_1, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 1.22e+30:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt((z * (2.0 * math.pow(t_1, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.22e+30)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(z * Float64(2.0 * (t_1 ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 1.22e+30)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = sqrt((z * (2.0 * (t_1 ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1.22e+30], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(z * N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.22e30

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0 71.5%

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

      1. associate-*r* 71.5%

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

      2. sqrt-prod 71.7%

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

      3. sub-neg 71.7%

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

      4. distribute-lft-in 70.7%

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

      5. *-commutative 70.7%

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

      6. *-commutative 70.7%

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

      7. *-commutative 70.7%

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

   5. Applied egg-rr 70.7%

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

      1. distribute-lft-out 71.7%

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

      2. sub-neg 71.7%

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

      3. *-commutative 71.7%

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

      4. *-commutative 71.7%

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

   7. Simplified 71.7%

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

   if 1.22e30 < t

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0 8.0%

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

      1. associate-*r* 8.0%

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

      2. sqrt-prod 8.0%

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

      3. *-commutative 8.0%

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

      4. *-commutative 8.0%

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

      5. add-sqr-sqrt 3.8%

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

      6. unpow2 3.8%

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

      7. add-sqr-sqrt 3.8%

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

      8. sqrt-prod 23.3%

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

      9. unpow2 23.3%

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

      10. add-sqr-sqrt 23.3%

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

      11. *-commutative 23.3%

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

      12. sqrt-prod 23.3%

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

      13. *-commutative 23.3%

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

      14. associate-*r* 23.3%

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

      15. unpow2 23.3%

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

   5. Applied egg-rr 25.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.22 \cdot 10^{+30}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((2.0 * z))) * 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((2.0d0 * z))) * 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((2.0 * z))) * Math.exp(((t * t) / 2.0));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((2.0 * z))) * 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[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 5: 31.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 39000000:\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot \left(y - x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 39000000.0)
   (* y (- (sqrt (* 2.0 z))))
   (sqrt (* z (* 2.0 (* y (- y x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 39000000.0) {
		tmp = y * -sqrt((2.0 * z));
	} else {
		tmp = sqrt((z * (2.0 * (y * (y - x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 39000000.0d0) then
        tmp = y * -sqrt((2.0d0 * z))
    else
        tmp = sqrt((z * (2.0d0 * (y * (y - x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 39000000.0) {
		tmp = y * -Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt((z * (2.0 * (y * (y - x)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 39000000.0:
		tmp = y * -math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt((z * (2.0 * (y * (y - x)))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 39000000.0)
		tmp = Float64(y * Float64(-sqrt(Float64(2.0 * z))));
	else
		tmp = sqrt(Float64(z * Float64(2.0 * Float64(y * Float64(y - x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 39000000.0)
		tmp = y * -sqrt((2.0 * z));
	else
		tmp = sqrt((z * (2.0 * (y * (y - x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 39000000.0], N[(y * (-N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(z * N[(2.0 * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.9e7

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0 72.0%

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

   4. Taylor expanded in x around 0 37.9%

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

      1. associate-*r* 37.9%

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

      2. mul-1-neg 37.9%

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

      3. distribute-lft-neg-out 37.9%

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

      4. *-commutative 37.9%

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

   6. Simplified 37.9%

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

      1. distribute-rgt-neg-out 37.9%

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

      2. distribute-lft-neg-out 37.9%

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

      3. *-commutative 37.9%

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

      4. add-sqr-sqrt 21.2%

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

      5. sqrt-unprod 19.1%

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

      6. sqr-neg 19.1%

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

      7. sqrt-unprod 0.7%

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

      8. add-sqr-sqrt 2.2%

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

      9. associate-*l* 2.2%

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

      10. sqrt-prod 2.2%

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

      11. add-sqr-sqrt 0.7%

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

      12. sqrt-unprod 19.1%

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

      13. sqr-neg 19.1%

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

      14. sqrt-unprod 21.2%

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

      15. add-sqr-sqrt 38.1%

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

   8. Applied egg-rr 38.1%

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

      1. distribute-rgt-neg-in 38.1%

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

      2. *-commutative 38.1%

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

   10. Simplified 38.1%

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

   if 3.9e7 < t

   1. Initial program 98.4%

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

   3. Taylor expanded in t around 0 9.3%

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

      1. associate-*r* 9.3%

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

      2. sqrt-prod 9.3%

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

      3. *-commutative 9.3%

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

      4. *-commutative 9.3%

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

      5. add-sqr-sqrt 3.7%

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

      6. unpow2 3.7%

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

      7. add-sqr-sqrt 3.7%

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

      8. sqrt-prod 22.2%

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

      9. unpow2 22.2%

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

      10. add-sqr-sqrt 22.2%

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

      11. *-commutative 22.2%

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

      12. sqrt-prod 22.2%

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

      13. *-commutative 22.2%

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

      14. associate-*r* 22.2%

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

      15. unpow2 22.2%

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

   5. Applied egg-rr 24.0%

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

   6. Taylor expanded in x around 0 17.5%

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

      1. +-commutative 17.5%

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

      2. unpow2 17.5%

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

      3. associate-*r* 17.5%

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

      4. distribute-rgt-in 17.5%

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

      5. mul-1-neg 17.5%

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

      6. unsub-neg 17.5%

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

   8. Simplified 17.5%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 39000000:\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \left(y \cdot \left(y - x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0 57.1%

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

   1. associate-*r* 57.1%

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

   2. sqrt-prod 57.3%

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

   3. sub-neg 57.3%

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

   4. distribute-lft-in 56.5%

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

   5. *-commutative 56.5%

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

   6. *-commutative 56.5%

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

   7. *-commutative 56.5%

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

5. Applied egg-rr 56.5%

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

   1. distribute-lft-out 57.3%

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

   2. sub-neg 57.3%

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

   3. *-commutative 57.3%

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

   4. *-commutative 57.3%

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

7. Simplified 57.3%

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

8. Final simplification 57.3%

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

Alternative 7: 30.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0 57.1%

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

4. Taylor expanded in x around 0 30.5%

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

   1. associate-*r* 30.5%

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

   2. mul-1-neg 30.5%

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

   3. distribute-lft-neg-out 30.5%

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

   4. *-commutative 30.5%

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

6. Simplified 30.5%

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

   1. distribute-rgt-neg-out 30.5%

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

   2. distribute-lft-neg-out 30.5%

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

   3. *-commutative 30.5%

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

   4. add-sqr-sqrt 16.9%

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

   5. sqrt-unprod 18.3%

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

   6. sqr-neg 18.3%

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

   7. sqrt-unprod 0.7%

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

   8. add-sqr-sqrt 1.9%

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

   9. associate-*l* 1.9%

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

   10. sqrt-prod 1.9%

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

   11. add-sqr-sqrt 0.7%

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

   12. sqrt-unprod 18.4%

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

   13. sqr-neg 18.4%

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

   14. sqrt-unprod 17.0%

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

   15. add-sqr-sqrt 30.6%

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

8. Applied egg-rr 30.6%

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

   1. distribute-rgt-neg-in 30.6%

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

   2. *-commutative 30.6%

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

10. Simplified 30.6%

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

11. Final simplification 30.6%

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

Alternative 8: 2.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in t around 0 57.1%

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

4. Taylor expanded in x around 0 30.5%

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

   1. associate-*r* 30.5%

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

   2. mul-1-neg 30.5%

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

   3. distribute-lft-neg-out 30.5%

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

   4. *-commutative 30.5%

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

6. Simplified 30.5%

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

   1. pow1 30.5%

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

   2. *-commutative 30.5%

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

   3. associate-*l* 30.5%

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

   4. sqrt-prod 30.6%

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

   5. add-sqr-sqrt 13.5%

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

   6. sqrt-unprod 15.3%

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

   7. sqr-neg 15.3%

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

   8. sqrt-unprod 1.2%

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

   9. add-sqr-sqrt 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1 1.9%

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

   2. *-commutative 1.9%

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

10. Simplified 1.9%

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

11. Final simplification 1.9%

    \[\leadsto y \cdot \sqrt{2 \cdot z} \]
12. Add Preprocessing

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024055 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (* (* (- (* 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))))