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

Percentage Accurate: 99.4% → 99.8%

Time: 14.1s

Alternatives: 15

Speedup: 1.0×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64((exp(1) ^ Float64(t * t)) * Float64(2.0 * z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * 0.5) - y) * sqrt(((2.71828182845904523536 ^ (t * t)) * (2.0 * z)));
end

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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. 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 98.7%

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

   2. expm1-udef 74.9%

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

   3. pow-exp 74.9%

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

   4. sqrt-unprod 74.9%

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

5. Applied egg-rr 74.9%

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

   1. expm1-def 98.7%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. exp-prod 99.8%

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

7. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   2. sqrt-unprod 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right)\\ t_2 := \sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}}\\ t_3 := \frac{t_2}{\frac{-1}{y}}\\ \mathbf{if}\;t \cdot t \leq 50:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \cdot t \leq 4 \cdot 10^{+109}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+159}:\\ \;\;\;\;\frac{t_2}{\frac{2}{x}}\\ \mathbf{elif}\;t \cdot t \leq 10^{+277}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* (- (* x 0.5) y) (sqrt (* 2.0 z))) (+ 1.0 (* 0.5 (* t t)))))
        (t_2 (sqrt (* (* 2.0 z) (exp (* t t)))))
        (t_3 (/ t_2 (/ -1.0 y))))
   (if (<= (* t t) 50.0)
     t_1
     (if (<= (* t t) 4e+109)
       t_3
       (if (<= (* t t) 5e+159)
         (/ t_2 (/ 2.0 x))
         (if (<= (* t t) 1e+277) t_3 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((x * 0.5) - y) * sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	double t_2 = sqrt(((2.0 * z) * exp((t * t))));
	double t_3 = t_2 / (-1.0 / y);
	double tmp;
	if ((t * t) <= 50.0) {
		tmp = t_1;
	} else if ((t * t) <= 4e+109) {
		tmp = t_3;
	} else if ((t * t) <= 5e+159) {
		tmp = t_2 / (2.0 / x);
	} else if ((t * t) <= 1e+277) {
		tmp = t_3;
	} else {
		tmp = 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (((x * 0.5d0) - y) * sqrt((2.0d0 * z))) * (1.0d0 + (0.5d0 * (t * t)))
    t_2 = sqrt(((2.0d0 * z) * exp((t * t))))
    t_3 = t_2 / ((-1.0d0) / y)
    if ((t * t) <= 50.0d0) then
        tmp = t_1
    else if ((t * t) <= 4d+109) then
        tmp = t_3
    else if ((t * t) <= 5d+159) then
        tmp = t_2 / (2.0d0 / x)
    else if ((t * t) <= 1d+277) then
        tmp = t_3
    else
        tmp = t_1
    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) * Math.sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	double t_2 = Math.sqrt(((2.0 * z) * Math.exp((t * t))));
	double t_3 = t_2 / (-1.0 / y);
	double tmp;
	if ((t * t) <= 50.0) {
		tmp = t_1;
	} else if ((t * t) <= 4e+109) {
		tmp = t_3;
	} else if ((t * t) <= 5e+159) {
		tmp = t_2 / (2.0 / x);
	} else if ((t * t) <= 1e+277) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (((x * 0.5) - y) * math.sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)))
	t_2 = math.sqrt(((2.0 * z) * math.exp((t * t))))
	t_3 = t_2 / (-1.0 / y)
	tmp = 0
	if (t * t) <= 50.0:
		tmp = t_1
	elif (t * t) <= 4e+109:
		tmp = t_3
	elif (t * t) <= 5e+159:
		tmp = t_2 / (2.0 / x)
	elif (t * t) <= 1e+277:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z))) * Float64(1.0 + Float64(0.5 * Float64(t * t))))
	t_2 = sqrt(Float64(Float64(2.0 * z) * exp(Float64(t * t))))
	t_3 = Float64(t_2 / Float64(-1.0 / y))
	tmp = 0.0
	if (Float64(t * t) <= 50.0)
		tmp = t_1;
	elseif (Float64(t * t) <= 4e+109)
		tmp = t_3;
	elseif (Float64(t * t) <= 5e+159)
		tmp = Float64(t_2 / Float64(2.0 / x));
	elseif (Float64(t * t) <= 1e+277)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (((x * 0.5) - y) * sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	t_2 = sqrt(((2.0 * z) * exp((t * t))));
	t_3 = t_2 / (-1.0 / y);
	tmp = 0.0;
	if ((t * t) <= 50.0)
		tmp = t_1;
	elseif ((t * t) <= 4e+109)
		tmp = t_3;
	elseif ((t * t) <= 5e+159)
		tmp = t_2 / (2.0 / x);
	elseif ((t * t) <= 1e+277)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 50.0], t$95$1, If[LessEqual[N[(t * t), $MachinePrecision], 4e+109], t$95$3, If[LessEqual[N[(t * t), $MachinePrecision], 5e+159], N[(t$95$2 / N[(2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+277], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 50 or 1e277 < (*.f64 t t)

   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. Taylor expanded in t around 0 99.1%

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

      1. unpow2 99.1%

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

   4. Simplified 99.1%

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

   if 50 < (*.f64 t t) < 3.99999999999999993e109 or 5.00000000000000003e159 < (*.f64 t t) < 1e277

   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. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. *-commutative 100.0%

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

      4. flip-- 93.8%

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

      5. associate-*r/ 93.8%

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

      6. pow-exp 93.8%

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

      7. sqrt-unprod 93.8%

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

      8. swap-sqr 93.8%

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

      9. metadata-eval 93.8%

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

      10. fma-def 93.8%

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

   3. Applied egg-rr 93.8%

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

      1. associate-/l* 93.8%

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

      2. *-commutative 93.8%

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

      3. exp-prod 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in x around 0 91.7%

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

   if 3.99999999999999993e109 < (*.f64 t t) < 5.00000000000000003e159

   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. associate-*r* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. *-commutative 100.0%

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

      4. flip-- 81.8%

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

      5. associate-*r/ 81.8%

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

      6. pow-exp 81.8%

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

      7. sqrt-unprod 81.8%

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

      8. swap-sqr 81.8%

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

      9. metadata-eval 81.8%

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

      10. fma-def 81.8%

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

   3. Applied egg-rr 81.8%

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

      1. associate-/l* 81.8%

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

      2. *-commutative 81.8%

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

      3. exp-prod 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in x around inf 90.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 50:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 4 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}}}{\frac{-1}{y}}\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}}}{\frac{2}{x}}\\ \mathbf{elif}\;t \cdot t \leq 10^{+277}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}}}{\frac{-1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right)\\ \end{array} \]

Alternative 4: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 50 \lor \neg \left(t \cdot t \leq 10^{+277}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}}}{\frac{-1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* t t) 50.0) (not (<= (* t t) 1e+277)))
   (* (* (- (* x 0.5) y) (sqrt (* 2.0 z))) (+ 1.0 (* 0.5 (* t t))))
   (/ (sqrt (* (* 2.0 z) (exp (* t t)))) (/ -1.0 y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((t * t) <= 50.0) || !((t * t) <= 1e+277)) {
		tmp = (((x * 0.5) - y) * sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	} else {
		tmp = sqrt(((2.0 * z) * exp((t * t)))) / (-1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((t * t) <= 50.0d0) .or. (.not. ((t * t) <= 1d+277))) then
        tmp = (((x * 0.5d0) - y) * sqrt((2.0d0 * z))) * (1.0d0 + (0.5d0 * (t * t)))
    else
        tmp = sqrt(((2.0d0 * z) * exp((t * t)))) / ((-1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((t * t) <= 50.0) || !((t * t) <= 1e+277)) {
		tmp = (((x * 0.5) - y) * Math.sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	} else {
		tmp = Math.sqrt(((2.0 * z) * Math.exp((t * t)))) / (-1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((t * t) <= 50.0) or not ((t * t) <= 1e+277):
		tmp = (((x * 0.5) - y) * math.sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)))
	else:
		tmp = math.sqrt(((2.0 * z) * math.exp((t * t)))) / (-1.0 / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(t * t) <= 50.0) || !(Float64(t * t) <= 1e+277))
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z))) * Float64(1.0 + Float64(0.5 * Float64(t * t))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * exp(Float64(t * t)))) / Float64(-1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((t * t) <= 50.0) || ~(((t * t) <= 1e+277)))
		tmp = (((x * 0.5) - y) * sqrt((2.0 * z))) * (1.0 + (0.5 * (t * t)));
	else
		tmp = sqrt(((2.0 * z) * exp((t * t)))) / (-1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(t * t), $MachinePrecision], 50.0], N[Not[LessEqual[N[(t * t), $MachinePrecision], 1e+277]], $MachinePrecision]], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 50 or 1e277 < (*.f64 t t)

   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. Taylor expanded in t around 0 99.1%

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

      1. unpow2 99.1%

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

   4. Simplified 99.1%

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

   if 50 < (*.f64 t t) < 1e277

   1. Initial program 98.3%

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

      1. associate-*r* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. *-commutative 100.0%

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

      4. flip-- 91.5%

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

      5. associate-*r/ 91.5%

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

      6. pow-exp 91.5%

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

      7. sqrt-unprod 91.5%

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

      8. swap-sqr 91.5%

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

      9. metadata-eval 91.5%

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

      10. fma-def 91.5%

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

   3. Applied egg-rr 91.5%

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

      1. associate-/l* 91.5%

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

      2. *-commutative 91.5%

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

      3. exp-prod 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in x around 0 83.1%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 50 \lor \neg \left(t \cdot t \leq 10^{+277}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}}}{\frac{-1}{y}}\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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. 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 98.7%

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

   2. expm1-udef 74.9%

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

   3. pow-exp 74.9%

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

   4. sqrt-unprod 74.9%

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

5. Applied egg-rr 74.9%

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

   1. expm1-def 98.7%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. exp-prod 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 6: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \cdot t \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(y \cdot y\right) \cdot \left(z + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot z\right) \cdot \left(1 + t \cdot t\right)}}{\frac{2}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 4.2e+16)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (if (<= (* t t) 1.7e+154)
     (sqrt (* (* y y) (+ z z)))
     (/ (sqrt (* (* 2.0 z) (+ 1.0 (* t t)))) (/ 2.0 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 4.2e+16) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else if ((t * t) <= 1.7e+154) {
		tmp = sqrt(((y * y) * (z + z)));
	} else {
		tmp = sqrt(((2.0 * z) * (1.0 + (t * t)))) / (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t * t) <= 4.2d+16) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else if ((t * t) <= 1.7d+154) then
        tmp = sqrt(((y * y) * (z + z)))
    else
        tmp = sqrt(((2.0d0 * z) * (1.0d0 + (t * t)))) / (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 4.2e+16) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else if ((t * t) <= 1.7e+154) {
		tmp = Math.sqrt(((y * y) * (z + z)));
	} else {
		tmp = Math.sqrt(((2.0 * z) * (1.0 + (t * t)))) / (2.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t * t) <= 4.2e+16:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	elif (t * t) <= 1.7e+154:
		tmp = math.sqrt(((y * y) * (z + z)))
	else:
		tmp = math.sqrt(((2.0 * z) * (1.0 + (t * t)))) / (2.0 / x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 4.2e+16)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	elseif (Float64(t * t) <= 1.7e+154)
		tmp = sqrt(Float64(Float64(y * y) * Float64(z + z)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * Float64(1.0 + Float64(t * t)))) / Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t * t) <= 4.2e+16)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	elseif ((t * t) <= 1.7e+154)
		tmp = sqrt(((y * y) * (z + z)));
	else
		tmp = sqrt(((2.0 * z) * (1.0 + (t * t)))) / (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 4.2e+16], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1.7e+154], N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(1.0 + N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 4.2e16

   1. Initial program 99.6%

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

   2. Taylor expanded in t around 0 96.2%

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

      1. associate-*l* 96.1%

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

   4. Simplified 96.1%

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

      1. expm1-log1p-u 60.7%

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

      2. expm1-udef 37.1%

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

      3. *-commutative 37.1%

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

      4. *-commutative 37.1%

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

      5. associate-*r* 37.1%

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

      6. sqrt-prod 37.1%

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

      7. *-commutative 37.1%

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

   6. Applied egg-rr 37.1%

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

      1. expm1-def 60.8%

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

      2. expm1-log1p 96.3%

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

      3. *-commutative 96.3%

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

      4. *-commutative 96.3%

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

      5. *-commutative 96.3%

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

   8. Simplified 96.3%

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

   if 4.2e16 < (*.f64 t t) < 1.69999999999999987e154

   1. Initial program 96.4%

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

   2. Taylor expanded in t around 0 15.2%

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

      1. associate-*l* 15.2%

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

   4. Simplified 15.2%

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

      1. add-sqr-sqrt 10.3%

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

      2. sqrt-unprod 40.3%

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

      3. swap-sqr 40.3%

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

      4. add-sqr-sqrt 40.3%

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

      5. *-commutative 40.3%

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

      6. *-commutative 40.3%

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

      7. *-commutative 40.3%

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

      8. *-commutative 40.3%

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

      9. swap-sqr 50.6%

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

      10. add-sqr-sqrt 50.6%

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

      11. pow2 50.6%

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

   6. Applied egg-rr 50.6%

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

      1. *-commutative 50.6%

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

      2. *-commutative 50.6%

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

      3. associate-*l* 50.6%

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

      4. *-commutative 50.6%

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

      5. *-commutative 50.6%

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

   8. Simplified 50.6%

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

   9. Taylor expanded in x around 0 43.8%

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

       1. *-commutative 43.8%

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

       2. associate-*l* 43.8%

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

       3. unpow2 43.8%

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

       4. *-commutative 43.8%

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

       5. rem-log-exp 43.3%

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

       6. log-pow 43.3%

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

       7. unpow2 43.3%

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

       8. prod-exp 43.3%

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

       9. rem-log-exp 43.8%

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

   11. Simplified 43.8%

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

   if 1.69999999999999987e154 < (*.f64 t t)

   1. Initial program 100.0%

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

      1. associate-*r* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. *-commutative 100.0%

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

      4. flip-- 86.8%

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

      5. associate-*r/ 86.8%

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

      6. pow-exp 86.8%

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

      7. sqrt-unprod 86.8%

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

      8. swap-sqr 86.8%

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

      9. metadata-eval 86.8%

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

      10. fma-def 86.8%

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

   3. Applied egg-rr 86.8%

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

      1. associate-/l* 86.8%

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

      2. *-commutative 86.8%

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

      3. exp-prod 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in x around inf 74.7%

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

   7. Taylor expanded in t around 0 61.1%

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

      1. unpow2 61.1%

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

   9. Simplified 61.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \cdot t \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(y \cdot y\right) \cdot \left(z + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot z\right) \cdot \left(1 + t \cdot t\right)}}{\frac{2}{x}}\\ \end{array} \]

Alternative 7: 85.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0 84.6%

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

   1. unpow2 84.6%

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

4. Simplified 84.6%

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

5. Final simplification 84.6%

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

Alternative 8: 84.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   2. sqrt-unprod 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in t around 0 84.5%

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

   1. +-commutative 84.5%

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

   2. unpow2 84.5%

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

8. Simplified 84.5%

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

9. Final simplification 84.5%

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

Alternative 9: 44.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;y \leq -1.22 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(y \cdot y\right) \cdot \left(z + z\right)}\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \left(0.5 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= y -1.22e+82)
     (sqrt (* (* y y) (+ z z)))
     (if (<= y 3.55e+30) (* x (* 0.5 t_1)) (* y (- t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (y <= -1.22e+82) {
		tmp = sqrt(((y * y) * (z + z)));
	} else if (y <= 3.55e+30) {
		tmp = x * (0.5 * 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 + z))
    if (y <= (-1.22d+82)) then
        tmp = sqrt(((y * y) * (z + z)))
    else if (y <= 3.55d+30) then
        tmp = x * (0.5d0 * 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 + z));
	double tmp;
	if (y <= -1.22e+82) {
		tmp = Math.sqrt(((y * y) * (z + z)));
	} else if (y <= 3.55e+30) {
		tmp = x * (0.5 * t_1);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if y <= -1.22e+82:
		tmp = math.sqrt(((y * y) * (z + z)))
	elif y <= 3.55e+30:
		tmp = x * (0.5 * t_1)
	else:
		tmp = y * -t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (y <= -1.22e+82)
		tmp = sqrt(Float64(Float64(y * y) * Float64(z + z)));
	elseif (y <= 3.55e+30)
		tmp = Float64(x * Float64(0.5 * 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 + z));
	tmp = 0.0;
	if (y <= -1.22e+82)
		tmp = sqrt(((y * y) * (z + z)));
	elseif (y <= 3.55e+30)
		tmp = x * (0.5 * 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 + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.22e+82], N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 3.55e+30], N[(x * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.22000000000000008e82

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 65.0%

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

      1. associate-*l* 65.0%

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

   4. Simplified 65.0%

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

      1. add-sqr-sqrt 60.5%

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

      2. sqrt-unprod 66.5%

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

      3. swap-sqr 66.4%

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

      4. add-sqr-sqrt 66.5%

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

      5. *-commutative 66.5%

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

      6. *-commutative 66.5%

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

      7. *-commutative 66.5%

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

      8. *-commutative 66.5%

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

      9. swap-sqr 64.4%

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

      10. add-sqr-sqrt 64.5%

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

      11. pow2 64.5%

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

   6. Applied egg-rr 64.5%

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

      1. *-commutative 64.5%

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

      2. *-commutative 64.5%

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

      3. associate-*l* 64.5%

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

      4. *-commutative 64.5%

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

      5. *-commutative 64.5%

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

   8. Simplified 64.5%

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

   9. Taylor expanded in x around 0 62.5%

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

       1. *-commutative 62.5%

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

       2. associate-*l* 62.5%

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

       3. unpow2 62.5%

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

       4. *-commutative 62.5%

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

       5. rem-log-exp 42.5%

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

       6. log-pow 42.5%

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

       7. unpow2 42.5%

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

       8. prod-exp 42.6%

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

       9. rem-log-exp 62.5%

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

   11. Simplified 62.5%

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

   if -1.22000000000000008e82 < y < 3.54999999999999992e30

   1. Initial program 99.1%

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

   2. Taylor expanded in t around 0 58.4%

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

      1. associate-*l* 58.3%

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

   4. Simplified 58.3%

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

   5. Taylor expanded in x around inf 47.5%

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

      1. expm1-log1p-u 31.4%

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

      2. expm1-udef 19.9%

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

      3. *-commutative 19.9%

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

      4. associate-*l* 19.9%

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

      5. sqrt-prod 19.9%

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

   7. Applied egg-rr 19.9%

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

      1. expm1-def 31.4%

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

      2. expm1-log1p 47.5%

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

      3. *-commutative 47.5%

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

      4. associate-*l* 47.5%

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

      5. *-commutative 47.5%

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

      6. rem-log-exp 29.2%

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

      7. log-pow 29.2%

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

      8. unpow2 29.2%

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

      9. prod-exp 29.2%

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

      10. rem-log-exp 47.5%

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

   9. Simplified 47.5%

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

   if 3.54999999999999992e30 < y

   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 62.0%

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

      1. associate-*l* 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in x around 0 44.6%

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

      1. mul-1-neg 44.6%

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

      2. distribute-lft-neg-out 44.6%

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

      3. *-commutative 44.6%

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

   7. Simplified 44.6%

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

      1. expm1-log1p-u 2.1%

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

      2. expm1-udef 1.7%

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

      3. associate-*r* 1.7%

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

      4. pow1/2 1.7%

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

      5. pow1/2 1.7%

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

      6. pow-prod-down 1.7%

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

      7. count-2 1.7%

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

      8. pow1/2 1.7%

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

   9. Applied egg-rr 1.7%

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

       1. expm1-def 2.1%

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

       2. expm1-log1p 44.6%

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

   11. Simplified 44.6%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{\left(y \cdot y\right) \cdot \left(z + z\right)}\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \sqrt{z + z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z + z}\right)\\ \end{array} \]

Alternative 10: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 350000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\left(y \cdot y\right) \cdot \left(z + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot z\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 350000.0)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (if (<= t 6e+60) (sqrt (* (* y y) (+ z z))) (sqrt (* (* 0.5 z) (* x x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 350000.0) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else if (t <= 6e+60) {
		tmp = sqrt(((y * y) * (z + z)));
	} else {
		tmp = sqrt(((0.5 * z) * (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 (t <= 350000.0d0) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else if (t <= 6d+60) then
        tmp = sqrt(((y * y) * (z + z)))
    else
        tmp = sqrt(((0.5d0 * z) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 350000.0) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else if (t <= 6e+60) {
		tmp = Math.sqrt(((y * y) * (z + z)));
	} else {
		tmp = Math.sqrt(((0.5 * z) * (x * x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 350000.0:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	elif t <= 6e+60:
		tmp = math.sqrt(((y * y) * (z + z)))
	else:
		tmp = math.sqrt(((0.5 * z) * (x * x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 350000.0)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	elseif (t <= 6e+60)
		tmp = sqrt(Float64(Float64(y * y) * Float64(z + z)));
	else
		tmp = sqrt(Float64(Float64(0.5 * z) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 350000.0)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	elseif (t <= 6e+60)
		tmp = sqrt(((y * y) * (z + z)));
	else
		tmp = sqrt(((0.5 * z) * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 350000.0], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e+60], N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(0.5 * z), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.5e5

   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. Taylor expanded in t around 0 69.2%

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

      1. associate-*l* 69.2%

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

   4. Simplified 69.2%

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

      1. expm1-log1p-u 43.4%

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

      2. expm1-udef 27.2%

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

      3. *-commutative 27.2%

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

      4. *-commutative 27.2%

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

      5. associate-*r* 27.2%

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

      6. sqrt-prod 27.2%

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

      7. *-commutative 27.2%

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

   6. Applied egg-rr 27.2%

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

      1. expm1-def 43.5%

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

      2. expm1-log1p 69.3%

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

      3. *-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. *-commutative 69.3%

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

   8. Simplified 69.3%

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

   if 3.5e5 < t < 5.9999999999999997e60

   1. Initial program 88.9%

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

   2. Taylor expanded in t around 0 15.8%

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

      1. associate-*l* 15.8%

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

   4. Simplified 15.8%

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

      1. add-sqr-sqrt 15.5%

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

      2. sqrt-unprod 67.3%

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

      3. swap-sqr 67.3%

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

      4. add-sqr-sqrt 67.3%

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

      5. *-commutative 67.3%

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

      6. *-commutative 67.3%

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

      7. *-commutative 67.3%

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

      8. *-commutative 67.3%

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

      9. swap-sqr 67.3%

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

      10. add-sqr-sqrt 67.3%

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

      11. pow2 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. *-commutative 67.3%

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

      2. *-commutative 67.3%

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

      3. associate-*l* 67.3%

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

      4. *-commutative 67.3%

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

      5. *-commutative 67.3%

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

   8. Simplified 67.3%

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

   9. Taylor expanded in x around 0 56.6%

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

       1. *-commutative 56.6%

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

       2. associate-*l* 56.6%

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

       3. unpow2 56.6%

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

       4. *-commutative 56.6%

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

       5. rem-log-exp 67.0%

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

       6. log-pow 67.0%

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

       7. unpow2 67.0%

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

       8. prod-exp 67.0%

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

       9. rem-log-exp 56.6%

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

   11. Simplified 56.6%

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

   if 5.9999999999999997e60 < t

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0 30.8%

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

      1. associate-*l* 30.8%

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

   4. Simplified 30.8%

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

      1. add-sqr-sqrt 18.5%

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

      2. sqrt-unprod 28.7%

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

      3. swap-sqr 28.7%

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

      4. add-sqr-sqrt 28.7%

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

      5. *-commutative 28.7%

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

      6. *-commutative 28.7%

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

      7. *-commutative 28.7%

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

      8. *-commutative 28.7%

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

      9. swap-sqr 32.7%

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

      10. add-sqr-sqrt 32.7%

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

      11. pow2 32.7%

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

   6. Applied egg-rr 32.7%

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

      1. *-commutative 32.7%

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

      2. *-commutative 32.7%

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

      3. associate-*l* 32.7%

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

      4. *-commutative 32.7%

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

      5. *-commutative 32.7%

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

   8. Simplified 32.7%

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

   9. Taylor expanded in x around inf 28.6%

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

       1. associate-*r* 28.6%

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

       2. *-commutative 28.6%

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

       3. unpow2 28.6%

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

   11. Simplified 28.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 350000:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\left(y \cdot y\right) \cdot \left(z + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot z\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 11: 31.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 950:\\ \;\;\;\;y \cdot \left(-\sqrt{z + z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(2 \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 950.0) (* y (- (sqrt (+ z z)))) (sqrt (* y (* 2.0 (* y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 950.0) {
		tmp = y * -sqrt((z + z));
	} else {
		tmp = sqrt((y * (2.0 * (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 <= 950.0d0) then
        tmp = y * -sqrt((z + z))
    else
        tmp = sqrt((y * (2.0d0 * (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 <= 950.0) {
		tmp = y * -Math.sqrt((z + z));
	} else {
		tmp = Math.sqrt((y * (2.0 * (y * z))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 950.0:
		tmp = y * -math.sqrt((z + z))
	else:
		tmp = math.sqrt((y * (2.0 * (y * z))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 950.0)
		tmp = Float64(y * Float64(-sqrt(Float64(z + z))));
	else
		tmp = sqrt(Float64(y * Float64(2.0 * Float64(y * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 950.0)
		tmp = y * -sqrt((z + z));
	else
		tmp = sqrt((y * (2.0 * (y * z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 950.0], N[(y * (-N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(y * N[(2.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 950

   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. Taylor expanded in t around 0 69.2%

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

      1. associate-*l* 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in x around 0 30.0%

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

      1. mul-1-neg 30.0%

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

      2. distribute-lft-neg-out 30.0%

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

      3. *-commutative 30.0%

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

   7. Simplified 30.0%

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

      1. expm1-log1p-u 19.4%

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

      2. expm1-udef 12.3%

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

      3. associate-*r* 12.3%

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

      4. pow1/2 12.3%

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

      5. pow1/2 12.3%

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

      6. pow-prod-down 12.3%

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

      7. count-2 12.3%

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

      8. pow1/2 12.3%

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

   9. Applied egg-rr 12.3%

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

       1. expm1-def 19.4%

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

       2. expm1-log1p 30.1%

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

   11. Simplified 30.1%

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

   if 950 < 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. Taylor expanded in t around 0 28.4%

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

      1. associate-*l* 28.4%

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

   4. Simplified 28.4%

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

      1. add-sqr-sqrt 18.0%

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

      2. sqrt-unprod 34.9%

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

      3. swap-sqr 34.9%

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

      4. add-sqr-sqrt 34.9%

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

      5. *-commutative 34.9%

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

      6. *-commutative 34.9%

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

      7. *-commutative 34.9%

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

      8. *-commutative 34.9%

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

      9. swap-sqr 38.3%

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

      10. add-sqr-sqrt 38.3%

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

      11. pow2 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. *-commutative 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-*l* 38.3%

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

      4. *-commutative 38.3%

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

      5. *-commutative 38.3%

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

   8. Simplified 38.3%

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

   9. Taylor expanded in x around 0 20.9%

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

       1. *-commutative 20.9%

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

       2. associate-*l* 20.9%

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

       3. unpow2 20.9%

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

       4. *-commutative 20.9%

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

       5. rem-log-exp 31.0%

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

       6. log-pow 31.0%

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

       7. unpow2 31.0%

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

       8. prod-exp 31.0%

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

       9. rem-log-exp 20.9%

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

   11. Simplified 20.9%

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

   12. Taylor expanded in y around 0 20.9%

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

       1. unpow2 20.9%

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

       2. count-2 20.9%

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

       3. distribute-lft-out 20.9%

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

       4. associate-*r* 20.9%

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

       5. distribute-lft-in 20.9%

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

       6. count-2 20.9%

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

   14. Simplified 20.9%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 950:\\ \;\;\;\;y \cdot \left(-\sqrt{z + z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot \left(2 \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 12: 32.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 900:\\ \;\;\;\;y \cdot \left(-\sqrt{z + z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(y \cdot y\right) \cdot \left(z + z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 900.0) (* y (- (sqrt (+ z z)))) (sqrt (* (* y y) (+ z z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 900.0) {
		tmp = y * -sqrt((z + z));
	} else {
		tmp = sqrt(((y * y) * (z + z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 900.0d0) then
        tmp = y * -sqrt((z + z))
    else
        tmp = sqrt(((y * y) * (z + z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 900.0) {
		tmp = y * -Math.sqrt((z + z));
	} else {
		tmp = Math.sqrt(((y * y) * (z + z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 900.0:
		tmp = y * -math.sqrt((z + z))
	else:
		tmp = math.sqrt(((y * y) * (z + z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 900.0)
		tmp = Float64(y * Float64(-sqrt(Float64(z + z))));
	else
		tmp = sqrt(Float64(Float64(y * y) * Float64(z + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 900.0)
		tmp = y * -sqrt((z + z));
	else
		tmp = sqrt(((y * y) * (z + z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 900.0], N[(y * (-N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 900

   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. Taylor expanded in t around 0 69.2%

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

      1. associate-*l* 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in x around 0 30.0%

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

      1. mul-1-neg 30.0%

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

      2. distribute-lft-neg-out 30.0%

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

      3. *-commutative 30.0%

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

   7. Simplified 30.0%

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

      1. expm1-log1p-u 19.4%

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

      2. expm1-udef 12.3%

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

      3. associate-*r* 12.3%

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

      4. pow1/2 12.3%

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

      5. pow1/2 12.3%

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

      6. pow-prod-down 12.3%

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

      7. count-2 12.3%

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

      8. pow1/2 12.3%

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

   9. Applied egg-rr 12.3%

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

       1. expm1-def 19.4%

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

       2. expm1-log1p 30.1%

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

   11. Simplified 30.1%

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

   if 900 < 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. Taylor expanded in t around 0 28.4%

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

      1. associate-*l* 28.4%

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

   4. Simplified 28.4%

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

      1. add-sqr-sqrt 18.0%

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

      2. sqrt-unprod 34.9%

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

      3. swap-sqr 34.9%

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

      4. add-sqr-sqrt 34.9%

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

      5. *-commutative 34.9%

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

      6. *-commutative 34.9%

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

      7. *-commutative 34.9%

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

      8. *-commutative 34.9%

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

      9. swap-sqr 38.3%

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

      10. add-sqr-sqrt 38.3%

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

      11. pow2 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. *-commutative 38.3%

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

      2. *-commutative 38.3%

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

      3. associate-*l* 38.3%

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

      4. *-commutative 38.3%

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

      5. *-commutative 38.3%

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

   8. Simplified 38.3%

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

   9. Taylor expanded in x around 0 20.9%

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

       1. *-commutative 20.9%

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

       2. associate-*l* 20.9%

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

       3. unpow2 20.9%

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

       4. *-commutative 20.9%

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

       5. rem-log-exp 31.0%

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

       6. log-pow 31.0%

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

       7. unpow2 31.0%

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

       8. prod-exp 31.0%

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

       9. rem-log-exp 20.9%

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

   11. Simplified 20.9%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 900:\\ \;\;\;\;y \cdot \left(-\sqrt{z + z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(y \cdot y\right) \cdot \left(z + z\right)}\\ \end{array} \]

Alternative 13: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \left(-\sqrt{z + z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot z\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.4e+27) (* y (- (sqrt (+ z z)))) (sqrt (* (* 0.5 z) (* x x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.4e+27) {
		tmp = y * -sqrt((z + z));
	} else {
		tmp = sqrt(((0.5 * z) * (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 <= 2.4d+27) then
        tmp = y * -sqrt((z + z))
    else
        tmp = sqrt(((0.5d0 * z) * (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 <= 2.4e+27) {
		tmp = y * -Math.sqrt((z + z));
	} else {
		tmp = Math.sqrt(((0.5 * z) * (x * x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.4e+27:
		tmp = y * -math.sqrt((z + z))
	else:
		tmp = math.sqrt(((0.5 * z) * (x * x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.4e+27)
		tmp = Float64(y * Float64(-sqrt(Float64(z + z))));
	else
		tmp = sqrt(Float64(Float64(0.5 * z) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.4e+27)
		tmp = y * -sqrt((z + z));
	else
		tmp = sqrt(((0.5 * z) * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.4e+27], N[(y * (-N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(N[(0.5 * z), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999998e27

   1. Initial program 99.2%

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

   2. Taylor expanded in t around 0 56.9%

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

      1. associate-*l* 56.8%

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

   4. Simplified 56.8%

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

   5. Taylor expanded in x around 0 30.0%

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

      1. mul-1-neg 30.0%

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

      2. distribute-lft-neg-out 30.0%

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

      3. *-commutative 30.0%

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

   7. Simplified 30.0%

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

      1. expm1-log1p-u 20.5%

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

      2. expm1-udef 12.8%

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

      3. associate-*r* 12.8%

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

      4. pow1/2 12.8%

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

      5. pow1/2 12.8%

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

      6. pow-prod-down 12.8%

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

      7. count-2 12.8%

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

      8. pow1/2 12.8%

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

   9. Applied egg-rr 12.8%

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

       1. expm1-def 20.6%

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

       2. expm1-log1p 30.1%

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

   11. Simplified 30.1%

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

   if 2.39999999999999998e27 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in t around 0 68.7%

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

      1. associate-*l* 68.8%

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

   4. Simplified 68.8%

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

      1. add-sqr-sqrt 59.0%

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

      2. sqrt-unprod 60.7%

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

      3. swap-sqr 60.6%

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

      4. add-sqr-sqrt 60.7%

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

      5. *-commutative 60.7%

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

      6. *-commutative 60.7%

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

      7. *-commutative 60.7%

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

      8. *-commutative 60.7%

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

      9. swap-sqr 60.7%

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

      10. add-sqr-sqrt 60.7%

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

      11. pow2 60.7%

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

   6. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-*l* 60.7%

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

      4. *-commutative 60.7%

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

      5. *-commutative 60.7%

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

   8. Simplified 60.7%

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

   9. Taylor expanded in x around inf 59.4%

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

       1. associate-*r* 59.4%

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

       2. *-commutative 59.4%

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

       3. unpow2 59.4%

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

   11. Simplified 59.4%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \left(-\sqrt{z + z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot z\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 14: 30.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0 60.3%

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

   1. associate-*l* 60.2%

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

4. Simplified 60.2%

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

5. Taylor expanded in x around 0 26.0%

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

   1. mul-1-neg 26.0%

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

   2. distribute-lft-neg-out 26.0%

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

   3. *-commutative 26.0%

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

7. Simplified 26.0%

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

   1. expm1-log1p-u 16.8%

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

   2. expm1-udef 11.2%

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

   3. associate-*r* 11.2%

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

   4. pow1/2 11.2%

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

   5. pow1/2 11.2%

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

   6. pow-prod-down 11.2%

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

   7. count-2 11.2%

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

   8. pow1/2 11.2%

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

9. Applied egg-rr 11.2%

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

    1. expm1-def 16.8%

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

    2. expm1-log1p 26.1%

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

11. Simplified 26.1%

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

12. Final simplification 26.1%

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

Alternative 15: 2.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0 60.3%

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

   1. associate-*l* 60.2%

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

4. Simplified 60.2%

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

   1. add-sqr-sqrt 32.8%

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

   2. sqrt-unprod 32.5%

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

   3. swap-sqr 32.5%

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

   4. add-sqr-sqrt 32.5%

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

   5. *-commutative 32.5%

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

   6. *-commutative 32.5%

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

   7. *-commutative 32.5%

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

   8. *-commutative 32.5%

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

   9. swap-sqr 32.6%

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

   10. add-sqr-sqrt 32.6%

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

   11. pow2 32.6%

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

6. Applied egg-rr 32.6%

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

   1. *-commutative 32.6%

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

   2. *-commutative 32.6%

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

   3. associate-*l* 32.6%

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

   4. *-commutative 32.6%

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

   5. *-commutative 32.6%

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

8. Simplified 32.6%

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

9. Taylor expanded in x around 0 17.8%

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

    1. *-commutative 17.8%

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

    2. associate-*l* 17.8%

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

    3. unpow2 17.8%

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

    4. *-commutative 17.8%

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

    5. rem-log-exp 16.1%

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

    6. log-pow 16.1%

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

    7. unpow2 16.1%

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

    8. prod-exp 16.2%

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

    9. rem-log-exp 17.8%

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

11. Simplified 17.8%

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

    1. pow1/2 17.8%

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

    2. *-commutative 17.8%

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

    3. unpow-prod-down 16.8%

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

    4. pow1/2 16.8%

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

    5. pow1/2 16.8%

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

13. Applied egg-rr 16.8%

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

    1. *-commutative 16.8%

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

    2. rem-sqrt-square 15.7%

       \[\leadsto \color{blue}{\left|y\right|} \cdot \sqrt{z + z} \]

    3. unpow1 15.7%

       \[\leadsto \left|\color{blue}{{y}^{1}}\right| \cdot \sqrt{z + z} \]

    4. sqr-pow 1.3%

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

    5. fabs-sqr 1.3%

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

    6. sqr-pow 2.4%

       \[\leadsto \color{blue}{{y}^{1}} \cdot \sqrt{z + z} \]

    7. unpow1 2.4%

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

15. Simplified 2.4%

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

16. Final simplification 2.4%

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

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023230 
(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))))