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

Percentage Accurate: 99.3% → 99.8%

Time: 24.1s

Alternatives: 8

Speedup: 1.0×

• 43.8% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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. Final simplification 99.8%

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

Alternative 2: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1.45 \cdot 10^{+32}:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {t\_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.45e+32)
     (* t_1 (sqrt (* z 2.0)))
     (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 <= 1.45e+32) {
		tmp = t_1 * sqrt((z * 2.0));
	} 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 <= 1.45d+32) then
        tmp = t_1 * sqrt((z * 2.0d0))
    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 <= 1.45e+32) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} 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 <= 1.45e+32:
		tmp = t_1 * math.sqrt((z * 2.0))
	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 <= 1.45e+32)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(z * 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 <= 1.45e+32)
		tmp = t_1 * sqrt((z * 2.0));
	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, 1.45e+32], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.45000000000000001e32

   1. Initial program 98.8%

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

      1. associate-*r* 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. *-commutative 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-lft-in 83.3%

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

      5. exp-sqrt 83.3%

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

      6. pow-exp 83.3%

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

      7. sqrt-unprod 83.3%

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

      8. associate-*l* 83.3%

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

      9. pow-exp 83.3%

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

      10. pow2 83.3%

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

   4. Applied egg-rr 83.2%

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

      1. distribute-lft-out 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in t around 0 69.7%

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

   if 1.45000000000000001e32 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0 14.1%

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

      1. associate-*r* 14.1%

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

      2. sqrt-prod 14.1%

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

      3. pow1/2 14.1%

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

      4. pow1 14.1%

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

      5. metadata-eval 14.1%

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

      6. sqrt-pow1 20.8%

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

      7. *-commutative 20.8%

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

      8. pow1/2 20.8%

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

      9. unpow-prod-down 24.2%

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

      10. *-commutative 24.2%

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

      11. *-commutative 24.2%

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

      12. fma-neg 24.2%

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

   5. Applied egg-rr 24.2%

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

      1. unpow1/2 24.2%

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

      2. fma-neg 24.2%

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

   7. Simplified 24.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{+32}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r* 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. *-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 77.1%

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

   5. exp-sqrt 77.1%

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

   6. pow-exp 77.1%

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

   7. sqrt-unprod 77.1%

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

   8. associate-*l* 77.1%

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

   9. pow-exp 77.1%

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

   10. pow2 77.1%

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

4. Applied egg-rr 77.1%

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

   1. distribute-lft-out 99.8%

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

   2. *-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*l* 99.8%

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

6. Simplified 99.8%

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

   1. pow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+32}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot {x}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.2e+32)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (sqrt (* 0.5 (* z (pow x 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.2e+32) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt((0.5 * (z * pow(x, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.2d+32) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = sqrt((0.5d0 * (z * (x ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.2e+32) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt((0.5 * (z * Math.pow(x, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.2e+32:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt((0.5 * (z * math.pow(x, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.2e+32)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(0.5 * Float64(z * (x ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.2e+32)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = sqrt((0.5 * (z * (x ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.2e+32], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(z * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.19999999999999996e32

   1. Initial program 98.8%

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

      1. associate-*r* 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. *-commutative 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-lft-in 83.3%

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

      5. exp-sqrt 83.3%

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

      6. pow-exp 83.3%

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

      7. sqrt-unprod 83.3%

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

      8. associate-*l* 83.3%

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

      9. pow-exp 83.3%

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

      10. pow2 83.3%

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

   4. Applied egg-rr 83.2%

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

      1. distribute-lft-out 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in t around 0 69.7%

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

   if 1.19999999999999996e32 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0 14.1%

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

      1. associate-*r* 14.1%

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

      2. sqrt-prod 14.1%

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

      3. pow1/2 14.1%

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

      4. pow1 14.1%

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

      5. metadata-eval 14.1%

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

      6. sqrt-pow1 20.8%

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

      7. *-commutative 20.8%

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

      8. pow1/2 20.8%

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

      9. unpow-prod-down 24.2%

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

      10. *-commutative 24.2%

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

      11. *-commutative 24.2%

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

      12. fma-neg 24.2%

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

   5. Applied egg-rr 24.2%

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

      1. unpow1/2 24.2%

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

      2. fma-neg 24.2%

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

   7. Simplified 24.2%

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

   8. Taylor expanded in x around inf 22.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+32}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(z \cdot {x}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+87} \lor \neg \left(x \leq 1.02 \cdot 10^{-10} \lor \neg \left(x \leq 2.55 \cdot 10^{+32}\right) \land x \leq 2.5 \cdot 10^{+99}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= x -1.7e+87)
           (not
            (or (<= x 1.02e-10) (and (not (<= x 2.55e+32)) (<= x 2.5e+99)))))
     (* 0.5 (* x t_1))
     (* y (- t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((x <= -1.7e+87) || !((x <= 1.02e-10) || (!(x <= 2.55e+32) && (x <= 2.5e+99)))) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((x <= (-1.7d+87)) .or. (.not. (x <= 1.02d-10) .or. (.not. (x <= 2.55d+32)) .and. (x <= 2.5d+99))) then
        tmp = 0.5d0 * (x * t_1)
    else
        tmp = y * -t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((x <= -1.7e+87) || !((x <= 1.02e-10) || (!(x <= 2.55e+32) && (x <= 2.5e+99)))) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (x <= -1.7e+87) or not ((x <= 1.02e-10) or (not (x <= 2.55e+32) and (x <= 2.5e+99))):
		tmp = 0.5 * (x * t_1)
	else:
		tmp = y * -t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((x <= -1.7e+87) || !((x <= 1.02e-10) || (!(x <= 2.55e+32) && (x <= 2.5e+99))))
		tmp = Float64(0.5 * Float64(x * t_1));
	else
		tmp = Float64(y * Float64(-t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((x <= -1.7e+87) || ~(((x <= 1.02e-10) || (~((x <= 2.55e+32)) && (x <= 2.5e+99)))))
		tmp = 0.5 * (x * t_1);
	else
		tmp = y * -t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -1.7e+87], N[Not[Or[LessEqual[x, 1.02e-10], And[N[Not[LessEqual[x, 2.55e+32]], $MachinePrecision], LessEqual[x, 2.5e+99]]]], $MachinePrecision]], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7000000000000001e87 or 1.01999999999999997e-10 < x < 2.55000000000000002e32 or 2.50000000000000004e99 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 63.4%

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

   4. Taylor expanded in x around inf 55.5%

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

      1. associate-*l* 55.4%

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

   6. Simplified 55.4%

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

      1. sqrt-prod 55.6%

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

      2. pow1/2 55.6%

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

   8. Applied egg-rr 55.6%

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

      1. unpow1/2 55.6%

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

   10. Simplified 55.6%

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

   if -1.7000000000000001e87 < x < 1.01999999999999997e-10 or 2.55000000000000002e32 < x < 2.50000000000000004e99

   1. Initial program 97.9%

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

   3. Taylor expanded in t around 0 53.6%

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

   4. Taylor expanded in x around 0 44.1%

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

      1. associate-*r* 44.1%

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

      2. mul-1-neg 44.1%

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

      3. distribute-lft-neg-out 44.1%

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

      4. *-commutative 44.1%

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

   6. Simplified 44.1%

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

      1. distribute-rgt-neg-out 44.1%

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

      2. distribute-lft-neg-out 44.1%

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

      3. add-sqr-sqrt 24.4%

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

      4. sqrt-unprod 19.9%

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

      5. sqr-neg 19.9%

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

      6. sqrt-unprod 1.9%

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

      7. add-sqr-sqrt 2.7%

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

      8. *-commutative 2.7%

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

      9. associate-*l* 2.7%

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

      10. add-sqr-sqrt 1.9%

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

      11. sqrt-unprod 19.9%

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

      12. sqr-neg 19.9%

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

      13. sqrt-unprod 24.4%

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

      14. add-sqr-sqrt 44.2%

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

      15. sqrt-prod 44.2%

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

   8. Applied egg-rr 44.2%

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

      1. *-commutative 44.2%

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

      2. distribute-rgt-neg-in 44.2%

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

   10. Simplified 44.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+87} \lor \neg \left(x \leq 1.02 \cdot 10^{-10} \lor \neg \left(x \leq 2.55 \cdot 10^{+32}\right) \land x \leq 2.5 \cdot 10^{+99}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r* 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. *-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 77.1%

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

   5. exp-sqrt 77.1%

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

   6. pow-exp 77.1%

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

   7. sqrt-unprod 77.1%

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

   8. associate-*l* 77.1%

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

   9. pow-exp 77.1%

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

   10. pow2 77.1%

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

4. Applied egg-rr 77.1%

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

   1. distribute-lft-out 99.8%

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

   2. *-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*l* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in t around 0 57.5%

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

8. Final simplification 57.5%

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

Alternative 7: 30.2% 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 98.6%

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

3. Taylor expanded in t around 0 57.4%

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

4. Taylor expanded in x around 0 30.9%

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

   1. associate-*r* 30.9%

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

   2. mul-1-neg 30.9%

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

   3. distribute-lft-neg-out 30.9%

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

   4. *-commutative 30.9%

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

6. Simplified 30.9%

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

   1. distribute-rgt-neg-out 30.9%

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

   2. distribute-lft-neg-out 30.9%

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

   3. add-sqr-sqrt 16.6%

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

   4. sqrt-unprod 15.0%

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

   5. sqr-neg 15.0%

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

   6. sqrt-unprod 1.6%

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

   7. add-sqr-sqrt 2.7%

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

   8. *-commutative 2.7%

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

   9. associate-*l* 2.7%

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

   10. add-sqr-sqrt 1.6%

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

   11. sqrt-unprod 15.1%

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

   12. sqr-neg 15.1%

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

   13. sqrt-unprod 16.7%

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

   14. add-sqr-sqrt 30.9%

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

   15. sqrt-prod 30.9%

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

8. Applied egg-rr 30.9%

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

   1. *-commutative 30.9%

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

   2. distribute-rgt-neg-in 30.9%

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

10. Simplified 30.9%

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

11. Final simplification 30.9%

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

Alternative 8: 2.6% 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 98.6%

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

3. Taylor expanded in t around 0 57.4%

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

4. Taylor expanded in x around 0 30.9%

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

   1. associate-*r* 30.9%

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

   2. mul-1-neg 30.9%

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

   3. distribute-lft-neg-out 30.9%

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

   4. *-commutative 30.9%

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

6. Simplified 30.9%

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

   1. pow1 30.9%

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

   2. *-commutative 30.9%

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

   3. associate-*l* 30.9%

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

   4. add-sqr-sqrt 14.2%

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

   5. sqrt-unprod 13.1%

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

   6. sqr-neg 13.1%

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

   7. sqrt-unprod 1.1%

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

   8. add-sqr-sqrt 2.7%

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

   9. sqrt-prod 2.7%

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

8. Applied egg-rr 2.7%

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

   1. unpow1 2.7%

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

10. Simplified 2.7%

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

11. Final simplification 2.7%

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

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 2024076 
(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))))