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

Percentage Accurate: 99.3% → 99.3%

Time: 13.1s

Alternatives: 12

Speedup: 1.0×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% 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: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-sqrt 99.8%

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

   2. pow-exp 99.8%

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

   3. pow1/2 99.8%

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

   4. pow-pow 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sqr-neg 99.8%

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

   2. associate-/l* 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. exp-neg 99.8%

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

   5. associate-*r/ 99.8%

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

   6. *-rgt-identity 99.8%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. associate-*r/ 99.8%

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

   10. exp-neg 99.8%

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

   11. distribute-frac-neg 99.8%

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

   12. associate-/l* 99.8%

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

   13. sqr-neg 99.8%

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

   14. 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. Simplified 99.8%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 75.6%

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

   3. pow-exp 75.6%

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

   4. sqrt-unprod 75.6%

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

   5. associate-*l* 75.6%

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

5. Applied egg-rr 75.6%

   \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{z \cdot \left(2 \cdot {\left(e^{t}\right)}^{t}\right)}\right)} - 1\right)} \]
6. Step-by-step derivation

   1. expm1-def 98.6%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 4: 86.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sqr-neg 99.8%

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

   2. associate-/l* 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. exp-neg 99.8%

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

   5. associate-*r/ 99.8%

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

   6. *-rgt-identity 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*r/ 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. associate-*r/ 99.8%

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

   11. exp-neg 99.8%

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

   12. distribute-frac-neg 99.8%

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

   13. associate-/l* 99.8%

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

   14. sqr-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 91.0%

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

   1. distribute-lft-out 91.0%

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

   2. unpow2 91.0%

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

   3. associate-*l* 89.5%

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

6. Simplified 89.5%

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

7. Final simplification 89.5%

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

Alternative 5: 87.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sqr-neg 99.8%

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

   2. associate-/l* 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. exp-neg 99.8%

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

   5. associate-*r/ 99.8%

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

   6. *-rgt-identity 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*r/ 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. associate-*r/ 99.8%

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

   11. exp-neg 99.8%

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

   12. distribute-frac-neg 99.8%

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

   13. associate-/l* 99.8%

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

   14. sqr-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 91.0%

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

   1. distribute-lft-out 91.0%

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

   2. unpow2 91.0%

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

6. Simplified 91.0%

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

7. Final simplification 91.0%

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

Alternative 6: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t \cdot \left(0.5 \cdot t\right)\right) \cdot \sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 7.8e+170)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (* (+ 1.0 (* t (* 0.5 t))) (sqrt (* 2.0 (* y (* y z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 7.8e+170) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = (1.0 + (t * (0.5 * t))) * sqrt((2.0 * (y * (y * 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 <= 7.8d+170) then
        tmp = sqrt((z * 2.0d0)) * ((x * 0.5d0) - y)
    else
        tmp = (1.0d0 + (t * (0.5d0 * t))) * sqrt((2.0d0 * (y * (y * z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 7.8e+170) {
		tmp = Math.sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = (1.0 + (t * (0.5 * t))) * Math.sqrt((2.0 * (y * (y * z))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 7.8e+170:
		tmp = math.sqrt((z * 2.0)) * ((x * 0.5) - y)
	else:
		tmp = (1.0 + (t * (0.5 * t))) * math.sqrt((2.0 * (y * (y * z))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 7.8e+170)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(Float64(1.0 + Float64(t * Float64(0.5 * t))) * sqrt(Float64(2.0 * Float64(y * Float64(y * z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 7.8e+170)
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	else
		tmp = (1.0 + (t * (0.5 * t))) * sqrt((2.0 * (y * (y * z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 7.8e+170], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t * N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.8000000000000005e170

   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. sqr-neg 99.8%

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

      2. associate-/l* 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. exp-neg 99.8%

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

      5. associate-*r/ 99.8%

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

      6. *-rgt-identity 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. associate-*r/ 99.8%

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

      11. exp-neg 99.8%

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

      12. distribute-frac-neg 99.8%

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

      13. associate-/l* 99.8%

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

      14. sqr-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 63.0%

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

   if 7.8000000000000005e170 < 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. Step-by-step derivation

      1. add-sqr-sqrt 53.3%

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

      2. sqrt-unprod 46.7%

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

      3. *-commutative 46.7%

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

      4. *-commutative 46.7%

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

      5. swap-sqr 46.7%

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

      6. add-sqr-sqrt 46.7%

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

      7. pow2 46.7%

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

   3. Applied egg-rr 46.7%

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

   4. Taylor expanded in x around 0 40.0%

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

      1. unpow2 40.0%

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

      2. associate-*l* 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in t around 0 40.0%

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

      1. *-commutative 40.0%

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

      2. unpow2 40.0%

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

      3. associate-*r* 40.0%

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

   9. Simplified 40.0%

      \[\leadsto \sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)} \cdot \color{blue}{\left(1 + t \cdot \left(t \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t \cdot \left(0.5 \cdot t\right)\right) \cdot \sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 7: 85.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (sqrt((z * 2.0)) * ((x * 0.5) - y)) * (1.0 + (t * (0.5 * 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 = (sqrt((z * 2.0d0)) * ((x * 0.5d0) - y)) * (1.0d0 + (t * (0.5d0 * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 88.5%

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

   1. *-commutative 21.7%

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

   2. unpow2 21.7%

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

   3. associate-*r* 21.7%

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

4. Simplified 88.5%

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

5. Final simplification 88.5%

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

Alternative 8: 31.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 5.6e+26) (* y (- (sqrt (* z 2.0)))) (sqrt (* 2.0 (* y (* y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.6e+26) {
		tmp = y * -sqrt((z * 2.0));
	} else {
		tmp = sqrt((2.0 * (y * (y * 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 <= 5.6d+26) then
        tmp = y * -sqrt((z * 2.0d0))
    else
        tmp = sqrt((2.0d0 * (y * (y * z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.6e+26) {
		tmp = y * -Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt((2.0 * (y * (y * z))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 5.6e+26:
		tmp = y * -math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt((2.0 * (y * (y * z))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 5.6e+26)
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	else
		tmp = sqrt(Float64(2.0 * Float64(y * Float64(y * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 5.6e+26)
		tmp = y * -sqrt((z * 2.0));
	else
		tmp = sqrt((2.0 * (y * (y * z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 5.6e+26], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(2.0 * N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.59999999999999999e26

   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. sqr-neg 99.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-frac-neg 99.7%

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

      4. exp-neg 99.8%

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

      5. associate-*r/ 99.8%

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

      6. *-rgt-identity 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. associate-*r/ 99.8%

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

      11. exp-neg 99.7%

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

      12. distribute-frac-neg 99.7%

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

      13. associate-/l* 99.7%

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

      14. sqr-neg 99.7%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 69.5%

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

      1. associate-*r* 70.0%

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

      2. *-commutative 70.0%

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

      3. associate-*l* 70.0%

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

   6. Simplified 70.0%

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

   7. Taylor expanded in x around 0 35.6%

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

      1. mul-1-neg 35.6%

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

      2. distribute-rgt-neg-in 35.6%

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

   9. Simplified 35.6%

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

       1. associate-*r* 35.1%

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

       2. *-commutative 35.1%

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

       3. distribute-rgt-neg-out 35.1%

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

       4. neg-sub0 35.1%

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

       5. add-sqr-sqrt 35.1%

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

       6. sqr-neg 35.1%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 3.5%

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

       9. associate-*l* 3.5%

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

       10. add-sqr-sqrt 0.0%

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

       11. sqrt-unprod 35.6%

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

       12. sqr-neg 35.6%

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

       13. add-sqr-sqrt 35.6%

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

       14. sqrt-prod 35.7%

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

       15. *-commutative 35.7%

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

   11. Applied egg-rr 35.7%

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

       1. neg-sub0 35.7%

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

       2. *-commutative 35.7%

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

       3. distribute-rgt-neg-in 35.7%

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

   13. Simplified 35.7%

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

   if 5.59999999999999999e26 < 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. Step-by-step derivation

      1. sqr-neg 100.0%

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

      2. associate-/l* 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r/ 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. associate-*r/ 100.0%

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

      11. exp-neg 100.0%

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

      12. distribute-frac-neg 100.0%

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

      13. associate-/l* 100.0%

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

      14. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 17.4%

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

      1. associate-*r* 17.4%

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

      2. *-commutative 17.4%

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

      3. associate-*l* 17.4%

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

   6. Simplified 17.4%

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

   7. Taylor expanded in x around 0 4.7%

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

      1. mul-1-neg 4.7%

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

      2. distribute-rgt-neg-in 4.7%

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

   9. Simplified 4.7%

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

       1. add-sqr-sqrt 3.3%

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

       2. sqrt-unprod 14.8%

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

       3. swap-sqr 16.4%

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

       4. sqr-neg 16.4%

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

       5. add-sqr-sqrt 16.4%

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

       6. associate-*r* 14.8%

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

       7. sqrt-prod 14.8%

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

       8. pow1/2 14.8%

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

   11. Applied egg-rr 16.4%

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

       1. unpow1/2 16.4%

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

       2. *-commutative 16.4%

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

       3. unpow2 16.4%

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

       4. associate-*r* 16.4%

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

       5. *-commutative 16.4%

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

       6. unpow2 16.4%

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

       7. associate-*r* 14.8%

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

   13. Simplified 14.8%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(y \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 9: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.1e+31) (* y (- (sqrt (* z 2.0)))) (sqrt (* z (* 0.5 (* x x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.1e+31) {
		tmp = y * -sqrt((z * 2.0));
	} else {
		tmp = sqrt((z * (0.5 * (x * 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 (x <= 1.1d+31) then
        tmp = y * -sqrt((z * 2.0d0))
    else
        tmp = sqrt((z * (0.5d0 * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.1e+31) {
		tmp = y * -Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt((z * (0.5 * (x * x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 1.1e+31:
		tmp = y * -math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt((z * (0.5 * (x * x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.1e+31)
		tmp = Float64(y * Float64(-sqrt(Float64(z * 2.0))));
	else
		tmp = sqrt(Float64(z * Float64(0.5 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.1e+31)
		tmp = y * -sqrt((z * 2.0));
	else
		tmp = sqrt((z * (0.5 * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.1e+31], N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(z * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.10000000000000005e31

   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. sqr-neg 99.8%

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

      2. associate-/l* 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. exp-neg 99.8%

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

      5. associate-*r/ 99.8%

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

      6. *-rgt-identity 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. *-rgt-identity 99.8%

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

      10. associate-*r/ 99.8%

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

      11. exp-neg 99.8%

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

      12. distribute-frac-neg 99.8%

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

      13. associate-/l* 99.8%

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

      14. sqr-neg 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 54.4%

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

      1. associate-*r* 54.9%

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

      2. *-commutative 54.9%

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

      3. associate-*l* 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in x around 0 32.0%

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

      1. mul-1-neg 32.0%

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

      2. distribute-rgt-neg-in 32.0%

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

   9. Simplified 32.0%

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

       1. associate-*r* 31.6%

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

       2. *-commutative 31.6%

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

       3. distribute-rgt-neg-out 31.6%

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

       4. neg-sub0 31.6%

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

       5. add-sqr-sqrt 31.6%

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

       6. sqr-neg 31.6%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 2.7%

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

       9. associate-*l* 2.7%

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

       10. add-sqr-sqrt 0.0%

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

       11. sqrt-unprod 32.0%

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

       12. sqr-neg 32.0%

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

       13. add-sqr-sqrt 32.0%

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

       14. sqrt-prod 32.1%

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

       15. *-commutative 32.1%

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

   11. Applied egg-rr 32.1%

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

       1. neg-sub0 32.1%

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

       2. *-commutative 32.1%

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

       3. distribute-rgt-neg-in 32.1%

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

   13. Simplified 32.1%

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

   if 1.10000000000000005e31 < x

   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. sqr-neg 99.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-frac-neg 99.7%

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

      4. exp-neg 99.7%

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

      5. associate-*r/ 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. *-rgt-identity 99.7%

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

      10. associate-*r/ 99.7%

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

      11. exp-neg 99.7%

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

      12. distribute-frac-neg 99.7%

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

      13. associate-/l* 99.7%

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

      14. sqr-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 70.2%

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

      1. *-commutative 70.2%

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

      2. pow1 70.2%

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

      3. metadata-eval 70.2%

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

      4. sqrt-pow1 55.0%

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

      5. sqrt-prod 56.9%

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

      6. pow1/2 56.9%

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

      7. *-commutative 56.9%

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

   6. Applied egg-rr 56.9%

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

      1. unpow1/2 56.9%

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

      2. associate-*r* 56.9%

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

   8. Simplified 56.9%

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

   9. Taylor expanded in x around inf 53.3%

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

       1. associate-*r* 53.3%

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

       2. unpow2 53.3%

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

   11. Simplified 53.3%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+31}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternative 10: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sqr-neg 99.8%

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

   2. associate-/l* 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. exp-neg 99.8%

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

   5. associate-*r/ 99.8%

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

   6. *-rgt-identity 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*r/ 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. associate-*r/ 99.8%

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

   11. exp-neg 99.8%

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

   12. distribute-frac-neg 99.8%

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

   13. associate-/l* 99.8%

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

   14. sqr-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 58.0%

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

5. Final simplification 58.0%

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

Alternative 11: 30.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sqr-neg 99.8%

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

   2. associate-/l* 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. exp-neg 99.8%

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

   5. associate-*r/ 99.8%

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

   6. *-rgt-identity 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*r/ 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. associate-*r/ 99.8%

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

   11. exp-neg 99.8%

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

   12. distribute-frac-neg 99.8%

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

   13. associate-/l* 99.8%

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

   14. sqr-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 57.5%

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

   1. associate-*r* 57.9%

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

   2. *-commutative 57.9%

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

   3. associate-*l* 57.8%

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

6. Simplified 57.8%

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

7. Taylor expanded in x around 0 28.5%

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

   1. mul-1-neg 28.5%

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

   2. distribute-rgt-neg-in 28.5%

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

9. Simplified 28.5%

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

    1. associate-*r* 28.1%

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

    2. *-commutative 28.1%

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

    3. distribute-rgt-neg-out 28.1%

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

    4. neg-sub0 28.1%

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

    5. add-sqr-sqrt 28.1%

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

    6. sqr-neg 28.1%

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

    7. sqrt-unprod 0.0%

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

    8. add-sqr-sqrt 2.9%

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

    9. associate-*l* 2.9%

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

    10. add-sqr-sqrt 0.0%

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

    11. sqrt-unprod 28.5%

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

    12. sqr-neg 28.5%

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

    13. add-sqr-sqrt 28.5%

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

    14. sqrt-prod 28.6%

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

    15. *-commutative 28.6%

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

11. Applied egg-rr 28.6%

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

    1. neg-sub0 28.6%

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

    2. *-commutative 28.6%

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

    3. distribute-rgt-neg-in 28.6%

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

13. Simplified 28.6%

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

14. Final simplification 28.6%

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

Alternative 12: 2.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sqr-neg 99.8%

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

   2. associate-/l* 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. exp-neg 99.8%

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

   5. associate-*r/ 99.8%

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

   6. *-rgt-identity 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*r/ 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. associate-*r/ 99.8%

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

   11. exp-neg 99.8%

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

   12. distribute-frac-neg 99.8%

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

   13. associate-/l* 99.8%

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

   14. sqr-neg 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 57.5%

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

   1. associate-*r* 57.9%

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

   2. *-commutative 57.9%

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

   3. associate-*l* 57.8%

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

6. Simplified 57.8%

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

7. Taylor expanded in x around 0 28.5%

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

   1. mul-1-neg 28.5%

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

   2. distribute-rgt-neg-in 28.5%

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

9. Simplified 28.5%

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

    1. associate-*r* 28.1%

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

    2. *-commutative 28.1%

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

    3. expm1-log1p-u 16.5%

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

    4. expm1-udef 12.0%

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

    5. associate-*l* 12.4%

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

    6. add-sqr-sqrt 0.0%

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

    7. sqrt-unprod 2.2%

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

    8. sqr-neg 2.2%

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

    9. add-sqr-sqrt 2.2%

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

    10. sqrt-prod 2.2%

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

    11. *-commutative 2.2%

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

11. Applied egg-rr 2.2%

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

    1. expm1-def 2.2%

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

    2. expm1-log1p 2.9%

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

13. Simplified 2.9%

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

14. Final simplification 2.9%

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

Developer target: 99.3% 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 2023297 
(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))))