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

Percentage Accurate: 99.3% → 99.3%

Time: 23.8s

Alternatives: 15

Speedup: 1.0×

• 44.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-sqrt 99.4%

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

   2. pow-exp 99.4%

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

   3. pow1/2 99.4%

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

   4. pow-pow 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{\left(z \cdot 2\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ t_3 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;t\_1 \cdot t\_3\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+123}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+163}:\\ \;\;\;\;\left(t \cdot \left(y \cdot \sqrt{2}\right)\right) \cdot \left(-\sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(t \cdot t\_3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y))
        (t_2 (sqrt (* (* z 2.0) (* t_1 t_1))))
        (t_3 (sqrt (* z 2.0))))
   (if (<= t 0.095)
     (* t_1 t_3)
     (if (<= t 2.8e+34)
       t_2
       (if (<= t 4.5e+86)
         (* 0.5 (* x (cbrt (pow (* z 2.0) 1.5))))
         (if (<= t 9.5e+123)
           t_2
           (if (<= t 7.4e+163)
             (* (* t (* y (sqrt 2.0))) (- (sqrt z)))
             (* 0.5 (* x (* t t_3))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt(((z * 2.0) * (t_1 * t_1)));
	double t_3 = sqrt((z * 2.0));
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * t_3;
	} else if (t <= 2.8e+34) {
		tmp = t_2;
	} else if (t <= 4.5e+86) {
		tmp = 0.5 * (x * cbrt(pow((z * 2.0), 1.5)));
	} else if (t <= 9.5e+123) {
		tmp = t_2;
	} else if (t <= 7.4e+163) {
		tmp = (t * (y * sqrt(2.0))) * -sqrt(z);
	} else {
		tmp = 0.5 * (x * (t * t_3));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt(((z * 2.0) * (t_1 * t_1)));
	double t_3 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * t_3;
	} else if (t <= 2.8e+34) {
		tmp = t_2;
	} else if (t <= 4.5e+86) {
		tmp = 0.5 * (x * Math.cbrt(Math.pow((z * 2.0), 1.5)));
	} else if (t <= 9.5e+123) {
		tmp = t_2;
	} else if (t <= 7.4e+163) {
		tmp = (t * (y * Math.sqrt(2.0))) * -Math.sqrt(z);
	} else {
		tmp = 0.5 * (x * (t * t_3));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(Float64(z * 2.0) * Float64(t_1 * t_1)))
	t_3 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 0.095)
		tmp = Float64(t_1 * t_3);
	elseif (t <= 2.8e+34)
		tmp = t_2;
	elseif (t <= 4.5e+86)
		tmp = Float64(0.5 * Float64(x * cbrt((Float64(z * 2.0) ^ 1.5))));
	elseif (t <= 9.5e+123)
		tmp = t_2;
	elseif (t <= 7.4e+163)
		tmp = Float64(Float64(t * Float64(y * sqrt(2.0))) * Float64(-sqrt(z)));
	else
		tmp = Float64(0.5 * Float64(x * Float64(t * t_3)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.095], N[(t$95$1 * t$95$3), $MachinePrecision], If[LessEqual[t, 2.8e+34], t$95$2, If[LessEqual[t, 4.5e+86], N[(0.5 * N[(x * N[Power[N[Power[N[(z * 2.0), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e+123], t$95$2, If[LessEqual[t, 7.4e+163], N[(N[(t * N[(y * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[z], $MachinePrecision])), $MachinePrecision], N[(0.5 * N[(x * N[(t * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < 0.095000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.7%

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

      2. pow-exp 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. sqrt-prod 72.0%

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

      3. sub-neg 72.0%

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

      4. distribute-rgt-in 70.4%

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

      5. *-commutative 70.4%

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

      6. *-commutative 70.4%

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

   7. Applied egg-rr 70.4%

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

      1. distribute-rgt-out 72.0%

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

      2. sub-neg 72.0%

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

      3. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   if 0.095000000000000001 < t < 2.80000000000000008e34 or 4.49999999999999993e86 < t < 9.4999999999999996e123

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 94.9%

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

      2. pow-exp 94.9%

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

      3. pow1/2 94.9%

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

      4. pow-pow 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 11.0%

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

      1. associate-*r* 11.0%

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

      2. sqrt-prod 11.0%

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

      3. *-commutative 11.0%

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

      4. *-commutative 11.0%

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

      5. add-sqr-sqrt 10.0%

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

      6. sqrt-unprod 39.8%

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

      7. *-commutative 39.8%

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

      8. sub-neg 39.8%

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

      9. *-commutative 39.8%

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

      10. sub-neg 39.8%

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

      11. *-commutative 39.8%

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

      12. sub-neg 39.8%

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

      13. *-commutative 39.8%

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

      14. sub-neg 39.8%

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

   7. Applied egg-rr 54.9%

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

   9. Applied egg-rr 54.9%

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

   if 2.80000000000000008e34 < t < 4.49999999999999993e86

   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. Add Preprocessing

   3. Taylor expanded in x around inf 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*l* 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in t around 0 14.5%

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

      1. *-commutative 14.5%

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

      2. associate-*l* 14.5%

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

      3. *-commutative 14.5%

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

   8. Simplified 14.5%

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

      1. pow1 14.5%

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

      2. *-commutative 14.5%

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

      3. *-commutative 14.5%

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

      4. associate-*l* 14.5%

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

      5. sqrt-prod 14.5%

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

      6. *-commutative 14.5%

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

   10. Applied egg-rr 14.5%

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

       1. unpow1 14.5%

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

   12. Simplified 14.5%

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

   13. Taylor expanded in x around 0 14.5%

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

       1. associate-*l* 14.5%

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

   15. Simplified 14.5%

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

       1. sqrt-unprod 14.5%

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

       2. *-commutative 14.5%

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

       3. pow1/2 14.5%

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

       4. metadata-eval 14.5%

          \[\leadsto 0.5 \cdot \left(x \cdot {\left(z \cdot 2\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}\right) \]

       5. pow-prod-up 14.5%

          \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\left({\left(z \cdot 2\right)}^{0.25} \cdot {\left(z \cdot 2\right)}^{0.25}\right)}\right) \]

       6. pow2 14.5%

          \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{{\left({\left(z \cdot 2\right)}^{0.25}\right)}^{2}}\right) \]

   17. Applied egg-rr 14.5%

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

       1. pow-pow 14.5%

          \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{{\left(z \cdot 2\right)}^{\left(0.25 \cdot 2\right)}}\right) \]

       2. metadata-eval 14.5%

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

       3. pow1/2 14.5%

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

       4. add-cbrt-cube 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\sqrt[3]{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{z \cdot 2}}}\right) \]

       5. add-sqr-sqrt 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{\color{blue}{\left(z \cdot 2\right)} \cdot \sqrt{z \cdot 2}}\right) \]

       6. pow1 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{\color{blue}{{\left(z \cdot 2\right)}^{1}} \cdot \sqrt{z \cdot 2}}\right) \]

       7. pow1/2 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}}\right) \]

       8. pow-prod-up 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{\color{blue}{{\left(z \cdot 2\right)}^{\left(1 + 0.5\right)}}}\right) \]

       9. *-commutative 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{{\color{blue}{\left(2 \cdot z\right)}}^{\left(1 + 0.5\right)}}\right) \]

       10. metadata-eval 45.7%

           \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{\color{blue}{1.5}}}\right) \]

   19. Applied egg-rr 45.7%

       \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}}\right) \]

   if 9.4999999999999996e123 < t < 7.39999999999999986e163

   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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 85.1%

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

       1. +-commutative 85.1%

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

       2. unpow2 85.1%

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

       3. fma-define 85.1%

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

   11. Simplified 85.1%

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

   12. Taylor expanded in t around inf 85.1%

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

   13. Taylor expanded in x around 0 66.9%

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

   if 7.39999999999999986e163 < 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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 100.0%

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

   13. Taylor expanded in x around inf 54.6%

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

       1. associate-*l* 54.6%

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

       2. *-commutative 54.6%

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

   15. Simplified 54.6%

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

       1. pow1 54.6%

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

       2. *-commutative 54.6%

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

       3. *-commutative 54.6%

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

       4. associate-*r* 54.6%

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

       5. *-commutative 54.6%

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

       6. associate-*l* 54.6%

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

       7. sqrt-unprod 54.6%

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

       8. *-commutative 54.6%

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

   17. Applied egg-rr 54.6%

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

       1. unpow1 54.6%

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

       2. associate-*r* 54.6%

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

       3. *-commutative 54.6%

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

       4. associate-*l* 58.1%

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

       5. *-commutative 58.1%

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

   19. Simplified 58.1%

       \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(\sqrt{2 \cdot z} \cdot t\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+163}:\\ \;\;\;\;\left(t \cdot \left(y \cdot \sqrt{2}\right)\right) \cdot \left(-\sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(t \cdot \sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{hypot}\left(1, t\right) \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{+33} \lor \neg \left(t \leq 7.4 \cdot 10^{+89}\right) \land t \leq 1.5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 0.095)
     (* t_1 (* (hypot 1.0 t) (sqrt (* z 2.0))))
     (if (or (<= t 9.4e+33) (and (not (<= t 7.4e+89)) (<= t 1.5e+111)))
       (sqrt (* (* z 2.0) (* t_1 t_1)))
       (* t_1 (sqrt (* 2.0 (* z (pow t 2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * (hypot(1.0, t) * sqrt((z * 2.0)));
	} else if ((t <= 9.4e+33) || (!(t <= 7.4e+89) && (t <= 1.5e+111))) {
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	} else {
		tmp = t_1 * sqrt((2.0 * (z * pow(t, 2.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * (Math.hypot(1.0, t) * Math.sqrt((z * 2.0)));
	} else if ((t <= 9.4e+33) || (!(t <= 7.4e+89) && (t <= 1.5e+111))) {
		tmp = Math.sqrt(((z * 2.0) * (t_1 * t_1)));
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z * Math.pow(t, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 0.095:
		tmp = t_1 * (math.hypot(1.0, t) * math.sqrt((z * 2.0)))
	elif (t <= 9.4e+33) or (not (t <= 7.4e+89) and (t <= 1.5e+111)):
		tmp = math.sqrt(((z * 2.0) * (t_1 * t_1)))
	else:
		tmp = t_1 * math.sqrt((2.0 * (z * math.pow(t, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 0.095)
		tmp = Float64(t_1 * Float64(hypot(1.0, t) * sqrt(Float64(z * 2.0))));
	elseif ((t <= 9.4e+33) || (!(t <= 7.4e+89) && (t <= 1.5e+111)))
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(t_1 * t_1)));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * (t ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 0.095)
		tmp = t_1 * (hypot(1.0, t) * sqrt((z * 2.0)));
	elseif ((t <= 9.4e+33) || (~((t <= 7.4e+89)) && (t <= 1.5e+111)))
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	else
		tmp = t_1 * sqrt((2.0 * (z * (t ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 0.095], N[(t$95$1 * N[(N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 9.4e+33], And[N[Not[LessEqual[t, 7.4e+89]], $MachinePrecision], LessEqual[t, 1.5e+111]]], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.095000000000000001

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. exp-sqrt 99.7%

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

      3. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 89.5%

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

       1. +-commutative 89.5%

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

       2. unpow2 89.5%

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

       3. fma-define 89.5%

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

   11. Simplified 89.5%

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

       1. associate-*r* 89.5%

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

       2. *-commutative 89.5%

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

       3. sqrt-prod 86.1%

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

       4. *-commutative 86.1%

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

   13. Applied egg-rr 86.1%

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

       1. fma-undefine 86.1%

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

       2. unpow2 86.1%

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

       3. +-commutative 86.1%

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

       4. unpow2 86.1%

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

       5. hypot-1-def 79.3%

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

   15. Simplified 79.3%

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

   if 0.095000000000000001 < t < 9.3999999999999996e33 or 7.3999999999999996e89 < t < 1.5e111

   1. Initial program 99.3%

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

      1. exp-sqrt 94.0%

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

      2. pow-exp 94.0%

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

      3. pow1/2 94.0%

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

      4. pow-pow 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 12.0%

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

      1. associate-*r* 12.0%

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

      2. sqrt-prod 12.0%

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

      3. *-commutative 12.0%

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

      4. *-commutative 12.0%

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

      5. add-sqr-sqrt 10.8%

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

      6. sqrt-unprod 34.6%

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

      7. *-commutative 34.6%

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

      8. sub-neg 34.6%

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

      9. *-commutative 34.6%

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

      10. sub-neg 34.6%

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

      11. *-commutative 34.6%

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

      12. sub-neg 34.6%

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

      13. *-commutative 34.6%

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

      14. sub-neg 34.6%

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

   7. Applied egg-rr 52.5%

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

   9. Applied egg-rr 52.5%

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

   if 9.3999999999999996e33 < t < 7.3999999999999996e89 or 1.5e111 < 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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 85.5%

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

       1. +-commutative 85.5%

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

       2. unpow2 85.5%

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

       3. fma-define 85.5%

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

   11. Simplified 85.5%

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

   12. Taylor expanded in t around inf 85.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\mathsf{hypot}\left(1, t\right) \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{+33} \lor \neg \left(t \leq 7.4 \cdot 10^{+89}\right) \land t \leq 1.5 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{\left(z \cdot 2\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ t_3 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;t\_1 \cdot t\_3\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+87}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+181}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(t \cdot t\_3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y))
        (t_2 (sqrt (* (* z 2.0) (* t_1 t_1))))
        (t_3 (sqrt (* z 2.0))))
   (if (<= t 0.095)
     (* t_1 t_3)
     (if (<= t 1.35e+34)
       t_2
       (if (<= t 1.1e+87)
         (* 0.5 (* x (cbrt (pow (* z 2.0) 1.5))))
         (if (<= t 1.8e+181) t_2 (* 0.5 (* x (* t t_3)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt(((z * 2.0) * (t_1 * t_1)));
	double t_3 = sqrt((z * 2.0));
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * t_3;
	} else if (t <= 1.35e+34) {
		tmp = t_2;
	} else if (t <= 1.1e+87) {
		tmp = 0.5 * (x * cbrt(pow((z * 2.0), 1.5)));
	} else if (t <= 1.8e+181) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (x * (t * t_3));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt(((z * 2.0) * (t_1 * t_1)));
	double t_3 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * t_3;
	} else if (t <= 1.35e+34) {
		tmp = t_2;
	} else if (t <= 1.1e+87) {
		tmp = 0.5 * (x * Math.cbrt(Math.pow((z * 2.0), 1.5)));
	} else if (t <= 1.8e+181) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (x * (t * t_3));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(Float64(z * 2.0) * Float64(t_1 * t_1)))
	t_3 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 0.095)
		tmp = Float64(t_1 * t_3);
	elseif (t <= 1.35e+34)
		tmp = t_2;
	elseif (t <= 1.1e+87)
		tmp = Float64(0.5 * Float64(x * cbrt((Float64(z * 2.0) ^ 1.5))));
	elseif (t <= 1.8e+181)
		tmp = t_2;
	else
		tmp = Float64(0.5 * Float64(x * Float64(t * t_3)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.095], N[(t$95$1 * t$95$3), $MachinePrecision], If[LessEqual[t, 1.35e+34], t$95$2, If[LessEqual[t, 1.1e+87], N[(0.5 * N[(x * N[Power[N[Power[N[(z * 2.0), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8e+181], t$95$2, N[(0.5 * N[(x * N[(t * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 0.095000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.7%

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

      2. pow-exp 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. sqrt-prod 72.0%

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

      3. sub-neg 72.0%

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

      4. distribute-rgt-in 70.4%

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

      5. *-commutative 70.4%

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

      6. *-commutative 70.4%

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

   7. Applied egg-rr 70.4%

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

      1. distribute-rgt-out 72.0%

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

      2. sub-neg 72.0%

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

      3. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   if 0.095000000000000001 < t < 1.35e34 or 1.1e87 < t < 1.79999999999999992e181

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 96.8%

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

      2. pow-exp 96.8%

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

      3. pow1/2 96.8%

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

      4. pow-pow 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 15.3%

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

      1. associate-*r* 15.3%

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

      2. sqrt-prod 15.3%

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

      3. *-commutative 15.3%

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

      4. *-commutative 15.3%

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

      5. add-sqr-sqrt 13.7%

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

      6. sqrt-unprod 42.0%

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

      7. *-commutative 42.0%

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

      8. sub-neg 42.0%

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

      9. *-commutative 42.0%

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

      10. sub-neg 42.0%

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

      11. *-commutative 42.0%

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

      12. sub-neg 42.0%

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

      13. *-commutative 42.0%

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

      14. sub-neg 42.0%

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

   7. Applied egg-rr 51.6%

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

   9. Applied egg-rr 51.6%

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

   if 1.35e34 < t < 1.1e87

   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. Add Preprocessing

   3. Taylor expanded in x around inf 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*l* 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in t around 0 14.5%

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

      1. *-commutative 14.5%

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

      2. associate-*l* 14.5%

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

      3. *-commutative 14.5%

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

   8. Simplified 14.5%

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

      1. pow1 14.5%

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

      2. *-commutative 14.5%

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

      3. *-commutative 14.5%

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

      4. associate-*l* 14.5%

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

      5. sqrt-prod 14.5%

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

      6. *-commutative 14.5%

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

   10. Applied egg-rr 14.5%

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

       1. unpow1 14.5%

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

   12. Simplified 14.5%

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

   13. Taylor expanded in x around 0 14.5%

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

       1. associate-*l* 14.5%

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

   15. Simplified 14.5%

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

       1. sqrt-unprod 14.5%

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

       2. *-commutative 14.5%

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

       3. pow1/2 14.5%

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

       4. metadata-eval 14.5%

          \[\leadsto 0.5 \cdot \left(x \cdot {\left(z \cdot 2\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}\right) \]

       5. pow-prod-up 14.5%

          \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\left({\left(z \cdot 2\right)}^{0.25} \cdot {\left(z \cdot 2\right)}^{0.25}\right)}\right) \]

       6. pow2 14.5%

          \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{{\left({\left(z \cdot 2\right)}^{0.25}\right)}^{2}}\right) \]

   17. Applied egg-rr 14.5%

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

       1. pow-pow 14.5%

          \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{{\left(z \cdot 2\right)}^{\left(0.25 \cdot 2\right)}}\right) \]

       2. metadata-eval 14.5%

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

       3. pow1/2 14.5%

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

       4. add-cbrt-cube 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\sqrt[3]{\left(\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{z \cdot 2}}}\right) \]

       5. add-sqr-sqrt 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{\color{blue}{\left(z \cdot 2\right)} \cdot \sqrt{z \cdot 2}}\right) \]

       6. pow1 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{\color{blue}{{\left(z \cdot 2\right)}^{1}} \cdot \sqrt{z \cdot 2}}\right) \]

       7. pow1/2 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1} \cdot \color{blue}{{\left(z \cdot 2\right)}^{0.5}}}\right) \]

       8. pow-prod-up 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{\color{blue}{{\left(z \cdot 2\right)}^{\left(1 + 0.5\right)}}}\right) \]

       9. *-commutative 45.7%

          \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{{\color{blue}{\left(2 \cdot z\right)}}^{\left(1 + 0.5\right)}}\right) \]

       10. metadata-eval 45.7%

           \[\leadsto 0.5 \cdot \left(x \cdot \sqrt[3]{{\left(2 \cdot z\right)}^{\color{blue}{1.5}}}\right) \]

   19. Applied egg-rr 45.7%

       \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\sqrt[3]{{\left(2 \cdot z\right)}^{1.5}}}\right) \]

   if 1.79999999999999992e181 < 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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 100.0%

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

   13. Taylor expanded in x around inf 57.6%

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

       1. associate-*l* 57.6%

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

       2. *-commutative 57.6%

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

   15. Simplified 57.6%

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

       1. pow1 57.6%

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

       2. *-commutative 57.6%

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

       3. *-commutative 57.6%

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

       4. associate-*r* 57.6%

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

       5. *-commutative 57.6%

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

       6. associate-*l* 57.6%

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

       7. sqrt-unprod 57.6%

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

       8. *-commutative 57.6%

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

   17. Applied egg-rr 57.6%

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

       1. unpow1 57.6%

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

       2. associate-*r* 57.6%

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

       3. *-commutative 57.6%

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

       4. associate-*l* 61.8%

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

       5. *-commutative 61.8%

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

   19. Simplified 61.8%

       \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(\sqrt{2 \cdot z} \cdot t\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+87}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+181}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(t \cdot \sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 0.095)
     (* t_1 (sqrt (* z 2.0)))
     (if (<= t 6.4e+33)
       (sqrt (* (* z 2.0) (* t_1 t_1)))
       (* t_1 (sqrt (* 2.0 (* z (pow t 2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * sqrt((z * 2.0));
	} else if (t <= 6.4e+33) {
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	} else {
		tmp = t_1 * sqrt((2.0 * (z * pow(t, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 0.095d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else if (t <= 6.4d+33) then
        tmp = sqrt(((z * 2.0d0) * (t_1 * t_1)))
    else
        tmp = t_1 * sqrt((2.0d0 * (z * (t ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else if (t <= 6.4e+33) {
		tmp = Math.sqrt(((z * 2.0) * (t_1 * t_1)));
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z * Math.pow(t, 2.0))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 0.095:
		tmp = t_1 * math.sqrt((z * 2.0))
	elif t <= 6.4e+33:
		tmp = math.sqrt(((z * 2.0) * (t_1 * t_1)))
	else:
		tmp = t_1 * math.sqrt((2.0 * (z * math.pow(t, 2.0))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 0.095)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	elseif (t <= 6.4e+33)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(t_1 * t_1)));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z * (t ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 0.095)
		tmp = t_1 * sqrt((z * 2.0));
	elseif (t <= 6.4e+33)
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	else
		tmp = t_1 * sqrt((2.0 * (z * (t ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 0.095], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.4e+33], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.095000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.7%

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

      2. pow-exp 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. sqrt-prod 72.0%

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

      3. sub-neg 72.0%

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

      4. distribute-rgt-in 70.4%

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

      5. *-commutative 70.4%

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

      6. *-commutative 70.4%

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

   7. Applied egg-rr 70.4%

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

      1. distribute-rgt-out 72.0%

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

      2. sub-neg 72.0%

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

      3. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   if 0.095000000000000001 < t < 6.40000000000000034e33

   1. Initial program 98.7%

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

      1. exp-sqrt 89.3%

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

      2. pow-exp 89.3%

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

      3. pow1/2 89.3%

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

      4. pow-pow 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 18.1%

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

      1. associate-*r* 18.1%

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

      2. sqrt-prod 18.1%

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

      3. *-commutative 18.1%

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

      4. *-commutative 18.1%

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

      5. add-sqr-sqrt 17.1%

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

      6. sqrt-unprod 38.1%

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

      7. *-commutative 38.1%

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

      8. sub-neg 38.1%

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

      9. *-commutative 38.1%

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

      10. sub-neg 38.1%

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

      11. *-commutative 38.1%

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

      12. sub-neg 38.1%

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

      13. *-commutative 38.1%

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

      14. sub-neg 38.1%

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

   7. Applied egg-rr 48.7%

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

   9. Applied egg-rr 48.7%

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

   if 6.40000000000000034e33 < 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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. +-commutative 76.8%

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

       2. unpow2 76.8%

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

       3. fma-define 76.8%

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

   11. Simplified 76.8%

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

   12. Taylor expanded in t around inf 76.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z \cdot {t}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 0.095)
     (* t_1 (sqrt (* z 2.0)))
     (if (<= t 1.75e+34)
       (sqrt (* (* z 2.0) (* t_1 t_1)))
       (* (* t (* t_1 (sqrt 2.0))) (sqrt z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * sqrt((z * 2.0));
	} else if (t <= 1.75e+34) {
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	} else {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 0.095d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else if (t <= 1.75d+34) then
        tmp = sqrt(((z * 2.0d0) * (t_1 * t_1)))
    else
        tmp = (t * (t_1 * sqrt(2.0d0))) * sqrt(z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else if (t <= 1.75e+34) {
		tmp = Math.sqrt(((z * 2.0) * (t_1 * t_1)));
	} else {
		tmp = (t * (t_1 * Math.sqrt(2.0))) * Math.sqrt(z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 0.095:
		tmp = t_1 * math.sqrt((z * 2.0))
	elif t <= 1.75e+34:
		tmp = math.sqrt(((z * 2.0) * (t_1 * t_1)))
	else:
		tmp = (t * (t_1 * math.sqrt(2.0))) * math.sqrt(z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 0.095)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	elseif (t <= 1.75e+34)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(t_1 * t_1)));
	else
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 0.095)
		tmp = t_1 * sqrt((z * 2.0));
	elseif (t <= 1.75e+34)
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	else
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 0.095], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.75e+34], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.095000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.7%

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

      2. pow-exp 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. sqrt-prod 72.0%

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

      3. sub-neg 72.0%

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

      4. distribute-rgt-in 70.4%

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

      5. *-commutative 70.4%

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

      6. *-commutative 70.4%

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

   7. Applied egg-rr 70.4%

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

      1. distribute-rgt-out 72.0%

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

      2. sub-neg 72.0%

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

      3. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   if 0.095000000000000001 < t < 1.74999999999999999e34

   1. Initial program 98.7%

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

      1. exp-sqrt 89.3%

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

      2. pow-exp 89.3%

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

      3. pow1/2 89.3%

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

      4. pow-pow 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 18.1%

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

      1. associate-*r* 18.1%

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

      2. sqrt-prod 18.1%

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

      3. *-commutative 18.1%

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

      4. *-commutative 18.1%

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

      5. add-sqr-sqrt 17.1%

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

      6. sqrt-unprod 38.1%

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

      7. *-commutative 38.1%

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

      8. sub-neg 38.1%

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

      9. *-commutative 38.1%

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

      10. sub-neg 38.1%

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

      11. *-commutative 38.1%

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

      12. sub-neg 38.1%

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

      13. *-commutative 38.1%

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

      14. sub-neg 38.1%

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

   7. Applied egg-rr 48.7%

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

   9. Applied egg-rr 48.7%

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

   if 1.74999999999999999e34 < 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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.8%

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

       1. +-commutative 76.8%

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

       2. unpow2 76.8%

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

       3. fma-define 76.8%

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

   11. Simplified 76.8%

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

   12. Taylor expanded in t around inf 69.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 8: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. 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 99.4%

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

   3. exp-prod 99.4%

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

3. Simplified 99.4%

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

   1. pow1 99.4%

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

   2. sqrt-unprod 99.4%

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

   3. associate-*l* 99.4%

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

   4. pow-exp 99.4%

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

   5. pow2 99.4%

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

6. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in t around 0 84.8%

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

    1. +-commutative 84.8%

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

    2. unpow2 84.8%

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

    3. fma-define 84.8%

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

11. Simplified 84.8%

    \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, 1\right)}\right)} \]
12. Add Preprocessing

Alternative 9: 42.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{-50} \lor \neg \left(y \leq -3.1 \cdot 10^{-131} \lor \neg \left(y \leq -2.05 \cdot 10^{-140}\right) \land y \leq 1.15 \cdot 10^{-81}\right):\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= y -1.55e-50)
           (not
            (or (<= y -3.1e-131)
                (and (not (<= y -2.05e-140)) (<= y 1.15e-81)))))
     (* t_1 (- y))
     (* 0.5 (* x t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((y <= -1.55e-50) || !((y <= -3.1e-131) || (!(y <= -2.05e-140) && (y <= 1.15e-81)))) {
		tmp = t_1 * -y;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((y <= (-1.55d-50)) .or. (.not. (y <= (-3.1d-131)) .or. (.not. (y <= (-2.05d-140))) .and. (y <= 1.15d-81))) then
        tmp = t_1 * -y
    else
        tmp = 0.5d0 * (x * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((y <= -1.55e-50) || !((y <= -3.1e-131) || (!(y <= -2.05e-140) && (y <= 1.15e-81)))) {
		tmp = t_1 * -y;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (y <= -1.55e-50) or not ((y <= -3.1e-131) or (not (y <= -2.05e-140) and (y <= 1.15e-81))):
		tmp = t_1 * -y
	else:
		tmp = 0.5 * (x * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((y <= -1.55e-50) || !((y <= -3.1e-131) || (!(y <= -2.05e-140) && (y <= 1.15e-81))))
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = Float64(0.5 * Float64(x * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((y <= -1.55e-50) || ~(((y <= -3.1e-131) || (~((y <= -2.05e-140)) && (y <= 1.15e-81)))))
		tmp = t_1 * -y;
	else
		tmp = 0.5 * (x * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -1.55e-50], N[Not[Or[LessEqual[y, -3.1e-131], And[N[Not[LessEqual[y, -2.05e-140]], $MachinePrecision], LessEqual[y, 1.15e-81]]]], $MachinePrecision]], N[(t$95$1 * (-y)), $MachinePrecision], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5500000000000001e-50 or -3.10000000000000021e-131 < y < -2.0500000000000001e-140 or 1.14999999999999996e-81 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.8%

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

      2. pow-exp 99.8%

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

      3. pow1/2 99.8%

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

      4. pow-pow 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 59.0%

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

   6. Taylor expanded in x around 0 41.8%

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

      1. mul-1-neg 41.8%

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

      2. associate-*l* 41.8%

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

      3. *-commutative 41.8%

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

      4. distribute-rgt-neg-in 41.8%

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

      5. distribute-rgt-neg-in 41.8%

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

   8. Simplified 41.8%

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

   10. Applied egg-rr 41.9%

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

   if -1.5500000000000001e-50 < y < -3.10000000000000021e-131 or -2.0500000000000001e-140 < y < 1.14999999999999996e-81

   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. Add Preprocessing

   3. Taylor expanded in x around inf 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l* 88.8%

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

   5. Simplified 88.8%

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

   6. Taylor expanded in t around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l* 52.7%

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

      3. *-commutative 52.7%

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

   8. Simplified 52.7%

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

      1. pow1 52.7%

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

      2. *-commutative 52.7%

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

      3. *-commutative 52.7%

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

      4. associate-*l* 52.7%

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

      5. sqrt-prod 52.8%

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

      6. *-commutative 52.8%

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

   10. Applied egg-rr 52.8%

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

       1. unpow1 52.8%

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

   12. Simplified 52.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-50} \lor \neg \left(y \leq -3.1 \cdot 10^{-131} \lor \neg \left(y \leq -2.05 \cdot 10^{-140}\right) \land y \leq 1.15 \cdot 10^{-81}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-140} \lor \neg \left(y \leq 3.1 \cdot 10^{-44}\right):\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= y -1.3e-50)
     (sqrt (* (* z 2.0) (* y (- y x))))
     (if (<= y -5.2e-135)
       (* 0.5 (* x t_1))
       (if (or (<= y -2.05e-140) (not (<= y 3.1e-44)))
         (* t_1 (- y))
         (* (* x 0.5) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (y <= -1.3e-50) {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	} else if (y <= -5.2e-135) {
		tmp = 0.5 * (x * t_1);
	} else if ((y <= -2.05e-140) || !(y <= 3.1e-44)) {
		tmp = t_1 * -y;
	} else {
		tmp = (x * 0.5) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (y <= (-1.3d-50)) then
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    else if (y <= (-5.2d-135)) then
        tmp = 0.5d0 * (x * t_1)
    else if ((y <= (-2.05d-140)) .or. (.not. (y <= 3.1d-44))) then
        tmp = t_1 * -y
    else
        tmp = (x * 0.5d0) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (y <= -1.3e-50) {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	} else if (y <= -5.2e-135) {
		tmp = 0.5 * (x * t_1);
	} else if ((y <= -2.05e-140) || !(y <= 3.1e-44)) {
		tmp = t_1 * -y;
	} else {
		tmp = (x * 0.5) * t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if y <= -1.3e-50:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	elif y <= -5.2e-135:
		tmp = 0.5 * (x * t_1)
	elif (y <= -2.05e-140) or not (y <= 3.1e-44):
		tmp = t_1 * -y
	else:
		tmp = (x * 0.5) * t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (y <= -1.3e-50)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	elseif (y <= -5.2e-135)
		tmp = Float64(0.5 * Float64(x * t_1));
	elseif ((y <= -2.05e-140) || !(y <= 3.1e-44))
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = Float64(Float64(x * 0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (y <= -1.3e-50)
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	elseif (y <= -5.2e-135)
		tmp = 0.5 * (x * t_1);
	elseif ((y <= -2.05e-140) || ~((y <= 3.1e-44)))
		tmp = t_1 * -y;
	else
		tmp = (x * 0.5) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.3e-50], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, -5.2e-135], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -2.05e-140], N[Not[LessEqual[y, 3.1e-44]], $MachinePrecision]], N[(t$95$1 * (-y)), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.3000000000000001e-50

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.9%

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

      2. pow-exp 99.9%

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

      3. pow1/2 99.9%

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

      4. pow-pow 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 59.5%

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

      1. associate-*r* 59.5%

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

      2. sqrt-prod 59.6%

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

      3. *-commutative 59.6%

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

      4. *-commutative 59.6%

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

      5. add-sqr-sqrt 55.3%

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

      6. sqrt-unprod 64.6%

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

      7. *-commutative 64.6%

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

      8. sub-neg 64.6%

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

      9. *-commutative 64.6%

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

      10. sub-neg 64.6%

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

      11. *-commutative 64.6%

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

      12. sub-neg 64.6%

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

      13. *-commutative 64.6%

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

      14. sub-neg 64.6%

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

   7. Applied egg-rr 68.5%

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

   8. Taylor expanded in x around 0 59.4%

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

      1. +-commutative 59.4%

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

      2. mul-1-neg 59.4%

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

      3. unsub-neg 59.4%

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

      4. unpow2 59.4%

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

      5. distribute-rgt-out-- 64.7%

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

   10. Simplified 64.7%

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

   if -1.3000000000000001e-50 < y < -5.20000000000000008e-135

   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. Add Preprocessing

   3. Taylor expanded in x around inf 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-*l* 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in t around 0 35.6%

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

      1. *-commutative 35.6%

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

      2. associate-*l* 35.6%

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

      3. *-commutative 35.6%

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

   8. Simplified 35.6%

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

      1. pow1 35.6%

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

      2. *-commutative 35.6%

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

      3. *-commutative 35.6%

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

      4. associate-*l* 35.6%

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

      5. sqrt-prod 35.5%

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

      6. *-commutative 35.5%

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

   10. Applied egg-rr 35.5%

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

       1. unpow1 35.5%

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

   12. Simplified 35.5%

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

   if -5.20000000000000008e-135 < y < -2.0500000000000001e-140 or 3.09999999999999984e-44 < y

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.7%

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

      2. pow-exp 99.8%

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

      3. pow1/2 99.8%

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

      4. pow-pow 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 62.6%

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

   6. Taylor expanded in x around 0 42.4%

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

      1. mul-1-neg 42.4%

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

      2. associate-*l* 42.4%

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

      3. *-commutative 42.4%

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

      4. distribute-rgt-neg-in 42.4%

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

      5. distribute-rgt-neg-in 42.4%

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

   8. Simplified 42.4%

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

   10. Applied egg-rr 42.5%

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

   if -2.0500000000000001e-140 < y < 3.09999999999999984e-44

   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. Add Preprocessing

   3. Taylor expanded in x around inf 88.5%

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

      1. *-commutative 88.5%

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

      2. associate-*l* 88.4%

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

   5. Simplified 88.4%

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

   6. Taylor expanded in t around 0 51.7%

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

      1. *-commutative 51.7%

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

      2. associate-*l* 51.5%

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

      3. *-commutative 51.5%

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

   8. Simplified 51.5%

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

      1. pow1 51.5%

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

      2. *-commutative 51.5%

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

      3. *-commutative 51.5%

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

      4. associate-*l* 51.6%

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

      5. sqrt-prod 51.7%

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

      6. *-commutative 51.7%

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

   10. Applied egg-rr 51.7%

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

       1. unpow1 51.7%

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

   12. Simplified 51.7%

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

   14. Applied egg-rr 51.7%

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

       1. unpow1 51.7%

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

       2. *-commutative 51.7%

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

       3. *-commutative 51.7%

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

       4. *-commutative 51.7%

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

   16. Simplified 51.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right) \cdot \sqrt{2 \cdot z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-140} \lor \neg \left(y \leq 3.1 \cdot 10^{-44}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := t\_1 \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{-50}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \mathbf{elif}\;y \leq -1.96 \cdot 10^{-140} \lor \neg \left(y \leq 1.05 \cdot 10^{-81}\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (* t_1 (- y))))
   (if (<= y -3.9e-50)
     t_2
     (if (<= y -4.5e-132)
       (* 0.5 (* x t_1))
       (if (or (<= y -1.96e-140) (not (<= y 1.05e-81)))
         t_2
         (* (* x 0.5) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = t_1 * -y;
	double tmp;
	if (y <= -3.9e-50) {
		tmp = t_2;
	} else if (y <= -4.5e-132) {
		tmp = 0.5 * (x * t_1);
	} else if ((y <= -1.96e-140) || !(y <= 1.05e-81)) {
		tmp = t_2;
	} else {
		tmp = (x * 0.5) * 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) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = t_1 * -y
    if (y <= (-3.9d-50)) then
        tmp = t_2
    else if (y <= (-4.5d-132)) then
        tmp = 0.5d0 * (x * t_1)
    else if ((y <= (-1.96d-140)) .or. (.not. (y <= 1.05d-81))) then
        tmp = t_2
    else
        tmp = (x * 0.5d0) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double t_2 = t_1 * -y;
	double tmp;
	if (y <= -3.9e-50) {
		tmp = t_2;
	} else if (y <= -4.5e-132) {
		tmp = 0.5 * (x * t_1);
	} else if ((y <= -1.96e-140) || !(y <= 1.05e-81)) {
		tmp = t_2;
	} else {
		tmp = (x * 0.5) * t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	t_2 = t_1 * -y
	tmp = 0
	if y <= -3.9e-50:
		tmp = t_2
	elif y <= -4.5e-132:
		tmp = 0.5 * (x * t_1)
	elif (y <= -1.96e-140) or not (y <= 1.05e-81):
		tmp = t_2
	else:
		tmp = (x * 0.5) * t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = Float64(t_1 * Float64(-y))
	tmp = 0.0
	if (y <= -3.9e-50)
		tmp = t_2;
	elseif (y <= -4.5e-132)
		tmp = Float64(0.5 * Float64(x * t_1));
	elseif ((y <= -1.96e-140) || !(y <= 1.05e-81))
		tmp = t_2;
	else
		tmp = Float64(Float64(x * 0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	t_2 = t_1 * -y;
	tmp = 0.0;
	if (y <= -3.9e-50)
		tmp = t_2;
	elseif (y <= -4.5e-132)
		tmp = 0.5 * (x * t_1);
	elseif ((y <= -1.96e-140) || ~((y <= 1.05e-81)))
		tmp = t_2;
	else
		tmp = (x * 0.5) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * (-y)), $MachinePrecision]}, If[LessEqual[y, -3.9e-50], t$95$2, If[LessEqual[y, -4.5e-132], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -1.96e-140], N[Not[LessEqual[y, 1.05e-81]], $MachinePrecision]], t$95$2, N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.90000000000000021e-50 or -4.4999999999999999e-132 < y < -1.96000000000000002e-140 or 1.05e-81 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.8%

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

      2. pow-exp 99.8%

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

      3. pow1/2 99.8%

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

      4. pow-pow 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around 0 59.0%

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

   6. Taylor expanded in x around 0 41.8%

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

      1. mul-1-neg 41.8%

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

      2. associate-*l* 41.8%

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

      3. *-commutative 41.8%

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

      4. distribute-rgt-neg-in 41.8%

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

      5. distribute-rgt-neg-in 41.8%

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

   8. Simplified 41.8%

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

   10. Applied egg-rr 41.9%

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

   if -3.90000000000000021e-50 < y < -4.4999999999999999e-132

   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. Add Preprocessing

   3. Taylor expanded in x around inf 85.9%

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

      1. *-commutative 85.9%

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

      2. associate-*l* 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in t around 0 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 37.9%

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

      3. *-commutative 37.9%

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

   8. Simplified 37.9%

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

      1. pow1 37.9%

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

      2. *-commutative 37.9%

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

      3. *-commutative 37.9%

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

      4. associate-*l* 37.9%

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

      5. sqrt-prod 37.8%

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

      6. *-commutative 37.8%

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

   10. Applied egg-rr 37.8%

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

       1. unpow1 37.8%

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

   12. Simplified 37.8%

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

   if -1.96000000000000002e-140 < y < 1.05e-81

   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. Add Preprocessing

   3. Taylor expanded in x around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*l* 89.4%

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

   5. Simplified 89.4%

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

   6. Taylor expanded in t around 0 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-*l* 55.4%

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

      3. *-commutative 55.4%

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

   8. Simplified 55.4%

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

      1. pow1 55.4%

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

      2. *-commutative 55.4%

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

      3. *-commutative 55.4%

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

      4. associate-*l* 55.4%

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

      5. sqrt-prod 55.6%

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

      6. *-commutative 55.6%

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

   10. Applied egg-rr 55.6%

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

       1. unpow1 55.6%

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

   12. Simplified 55.6%

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

   14. Applied egg-rr 55.6%

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

       1. unpow1 55.6%

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

       2. *-commutative 55.6%

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

       3. *-commutative 55.6%

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

       4. *-commutative 55.6%

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

   16. Simplified 55.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{elif}\;y \leq -1.96 \cdot 10^{-140} \lor \neg \left(y \leq 1.05 \cdot 10^{-81}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(t \cdot t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 0.095)
     (* t_1 t_2)
     (if (<= t 3.5e+150)
       (sqrt (* (* z 2.0) (* t_1 t_1)))
       (* 0.5 (* x (* t t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * t_2;
	} else if (t <= 3.5e+150) {
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	} else {
		tmp = 0.5 * (x * (t * t_2));
	}
	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) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((z * 2.0d0))
    if (t <= 0.095d0) then
        tmp = t_1 * t_2
    else if (t <= 3.5d+150) then
        tmp = sqrt(((z * 2.0d0) * (t_1 * t_1)))
    else
        tmp = 0.5d0 * (x * (t * t_2))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 0.095) {
		tmp = t_1 * t_2;
	} else if (t <= 3.5e+150) {
		tmp = Math.sqrt(((z * 2.0) * (t_1 * t_1)));
	} else {
		tmp = 0.5 * (x * (t * t_2));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 0.095:
		tmp = t_1 * t_2
	elif t <= 3.5e+150:
		tmp = math.sqrt(((z * 2.0) * (t_1 * t_1)))
	else:
		tmp = 0.5 * (x * (t * t_2))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 0.095)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 3.5e+150)
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(t_1 * t_1)));
	else
		tmp = Float64(0.5 * Float64(x * Float64(t * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 0.095)
		tmp = t_1 * t_2;
	elseif (t <= 3.5e+150)
		tmp = sqrt(((z * 2.0) * (t_1 * t_1)));
	else
		tmp = 0.5 * (x * (t * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.095], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 3.5e+150], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(x * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 0.095000000000000001

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.7%

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

      2. pow-exp 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. sqrt-prod 72.0%

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

      3. sub-neg 72.0%

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

      4. distribute-rgt-in 70.4%

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

      5. *-commutative 70.4%

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

      6. *-commutative 70.4%

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

   7. Applied egg-rr 70.4%

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

      1. distribute-rgt-out 72.0%

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

      2. sub-neg 72.0%

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

      3. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   if 0.095000000000000001 < t < 3.49999999999999984e150

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 96.9%

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

      2. pow-exp 96.9%

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

      3. pow1/2 96.9%

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

      4. pow-pow 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0 18.0%

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

      1. associate-*r* 18.0%

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

      2. sqrt-prod 18.0%

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

      3. *-commutative 18.0%

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

      4. *-commutative 18.0%

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

      5. add-sqr-sqrt 9.9%

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

      6. sqrt-unprod 31.3%

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

      7. *-commutative 31.3%

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

      8. sub-neg 31.3%

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

      9. *-commutative 31.3%

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

      10. sub-neg 31.3%

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

      11. *-commutative 31.3%

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

      12. sub-neg 31.3%

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

      13. *-commutative 31.3%

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

      14. sub-neg 31.3%

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

   7. Applied egg-rr 43.6%

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

   9. Applied egg-rr 43.6%

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

   if 3.49999999999999984e150 < 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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 97.1%

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

       1. +-commutative 97.1%

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

       2. unpow2 97.1%

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

       3. fma-define 97.1%

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

   11. Simplified 97.1%

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

   12. Taylor expanded in t around inf 97.1%

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

   13. Taylor expanded in x around inf 52.8%

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

       1. associate-*l* 52.8%

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

       2. *-commutative 52.8%

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

   15. Simplified 52.8%

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

       1. pow1 52.8%

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

       2. *-commutative 52.8%

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

       3. *-commutative 52.8%

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

       4. associate-*r* 52.8%

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

       5. *-commutative 52.8%

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

       6. associate-*l* 52.8%

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

       7. sqrt-unprod 52.8%

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

       8. *-commutative 52.8%

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

   17. Applied egg-rr 52.8%

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

       1. unpow1 52.8%

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

       2. associate-*r* 52.8%

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

       3. *-commutative 52.8%

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

       4. associate-*l* 55.9%

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

       5. *-commutative 55.9%

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

   19. Simplified 55.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.095:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(x \cdot 0.5 - y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(t \cdot \sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := \left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{if}\;t \leq 8 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{+122}:\\ \;\;\;\;\left(0.5 \cdot t\_1\right) \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+141}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(t \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (* (- (* x 0.5) y) t_1)))
   (if (<= t 8e+45)
     t_2
     (if (<= t 3.25e+122)
       (* (* 0.5 t_1) (* x t))
       (if (<= t 5.4e+141) t_2 (* 0.5 (* x (* t t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = ((x * 0.5) - y) * t_1;
	double tmp;
	if (t <= 8e+45) {
		tmp = t_2;
	} else if (t <= 3.25e+122) {
		tmp = (0.5 * t_1) * (x * t);
	} else if (t <= 5.4e+141) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (x * (t * 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) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = ((x * 0.5d0) - y) * t_1
    if (t <= 8d+45) then
        tmp = t_2
    else if (t <= 3.25d+122) then
        tmp = (0.5d0 * t_1) * (x * t)
    else if (t <= 5.4d+141) then
        tmp = t_2
    else
        tmp = 0.5d0 * (x * (t * t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double t_2 = ((x * 0.5) - y) * t_1;
	double tmp;
	if (t <= 8e+45) {
		tmp = t_2;
	} else if (t <= 3.25e+122) {
		tmp = (0.5 * t_1) * (x * t);
	} else if (t <= 5.4e+141) {
		tmp = t_2;
	} else {
		tmp = 0.5 * (x * (t * t_1));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	t_2 = ((x * 0.5) - y) * t_1
	tmp = 0
	if t <= 8e+45:
		tmp = t_2
	elif t <= 3.25e+122:
		tmp = (0.5 * t_1) * (x * t)
	elif t <= 5.4e+141:
		tmp = t_2
	else:
		tmp = 0.5 * (x * (t * t_1))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = Float64(Float64(Float64(x * 0.5) - y) * t_1)
	tmp = 0.0
	if (t <= 8e+45)
		tmp = t_2;
	elseif (t <= 3.25e+122)
		tmp = Float64(Float64(0.5 * t_1) * Float64(x * t));
	elseif (t <= 5.4e+141)
		tmp = t_2;
	else
		tmp = Float64(0.5 * Float64(x * Float64(t * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	t_2 = ((x * 0.5) - y) * t_1;
	tmp = 0.0;
	if (t <= 8e+45)
		tmp = t_2;
	elseif (t <= 3.25e+122)
		tmp = (0.5 * t_1) * (x * t);
	elseif (t <= 5.4e+141)
		tmp = t_2;
	else
		tmp = 0.5 * (x * (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t, 8e+45], t$95$2, If[LessEqual[t, 3.25e+122], N[(N[(0.5 * t$95$1), $MachinePrecision] * N[(x * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.4e+141], t$95$2, N[(0.5 * N[(x * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.9999999999999994e45 or 3.24999999999999982e122 < t < 5.4000000000000002e141

   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. Add Preprocessing
   3. Step-by-step derivation

      1. exp-sqrt 99.3%

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

      2. pow-exp 99.3%

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

      3. pow1/2 99.3%

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

      4. pow-pow 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in t around 0 68.6%

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

      1. associate-*r* 68.6%

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

      2. sqrt-prod 68.8%

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

      3. sub-neg 68.8%

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

      4. distribute-rgt-in 67.3%

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

      5. *-commutative 67.3%

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

      6. *-commutative 67.3%

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

   7. Applied egg-rr 67.3%

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

      1. distribute-rgt-out 68.8%

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

      2. sub-neg 68.8%

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

      3. *-commutative 68.8%

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

   9. Simplified 68.8%

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

   if 7.9999999999999994e45 < t < 3.24999999999999982e122

   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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 40.7%

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

       1. +-commutative 40.7%

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

       2. unpow2 40.7%

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

       3. fma-define 40.7%

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

   11. Simplified 40.7%

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

   12. Taylor expanded in t around inf 40.7%

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

   13. Taylor expanded in x around inf 33.4%

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

       1. associate-*l* 27.5%

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

       2. *-commutative 27.5%

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

   15. Simplified 27.5%

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

       1. pow1 27.5%

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

       2. *-commutative 27.5%

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

       3. *-commutative 27.5%

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

       4. associate-*r* 27.5%

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

       5. *-commutative 27.5%

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

       6. associate-*l* 33.4%

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

       7. sqrt-unprod 33.4%

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

       8. *-commutative 33.4%

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

   17. Applied egg-rr 33.4%

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

       1. unpow1 33.4%

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

       2. associate-*r* 33.4%

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

       3. *-commutative 33.4%

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

   19. Simplified 33.4%

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

   if 5.4000000000000002e141 < 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-*l* 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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 97.3%

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

       1. +-commutative 97.3%

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

       2. unpow2 97.3%

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

       3. fma-define 97.3%

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

   11. Simplified 97.3%

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

   12. Taylor expanded in t around inf 97.3%

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

   13. Taylor expanded in x around inf 49.6%

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

       1. associate-*l* 49.6%

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

       2. *-commutative 49.6%

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

   15. Simplified 49.6%

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

       1. pow1 49.6%

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

       2. *-commutative 49.6%

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

       3. *-commutative 49.6%

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

       4. associate-*r* 49.6%

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

       5. *-commutative 49.6%

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

       6. associate-*l* 49.6%

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

       7. sqrt-unprod 49.6%

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

       8. *-commutative 49.6%

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

   17. Applied egg-rr 49.6%

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

       1. unpow1 49.6%

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

       2. associate-*r* 49.6%

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

       3. *-commutative 49.6%

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

       4. associate-*l* 52.5%

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

       5. *-commutative 52.5%

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

   19. Simplified 52.5%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+45}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{+122}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{z \cdot 2}\right) \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+141}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(t \cdot \sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = sqrt((z * 2.0)) * -y;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. exp-sqrt 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{t \cdot t}}} \]

   2. pow-exp 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   3. pow1/2 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left({\left(e^{t}\right)}^{t}\right)}^{0.5}} \]

   4. pow-pow 99.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{t}\right)}^{\left(t \cdot 0.5\right)}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{t}\right)}^{\left(t \cdot 0.5\right)}} \]

5. Taylor expanded in t around 0 59.4%

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)} \]

6. Taylor expanded in x around 0 31.6%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 31.6%

      \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

   2. associate-*l* 31.6%

      \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   3. *-commutative 31.6%

      \[\leadsto -y \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \]

   4. distribute-rgt-neg-in 31.6%

      \[\leadsto \color{blue}{y \cdot \left(-\sqrt{z} \cdot \sqrt{2}\right)} \]

   5. distribute-rgt-neg-in 31.6%

      \[\leadsto y \cdot \color{blue}{\left(\sqrt{z} \cdot \left(-\sqrt{2}\right)\right)} \]

8. Simplified 31.6%

   \[\leadsto \color{blue}{y \cdot \left(\sqrt{z} \cdot \left(-\sqrt{2}\right)\right)} \]

10. Applied egg-rr 31.7%

    \[\leadsto y \cdot \color{blue}{\left(-\sqrt{z \cdot 2}\right)} \]

11. Final simplification 31.7%

    \[\leadsto \sqrt{z \cdot 2} \cdot \left(-y\right) \]
12. Add Preprocessing

Alternative 15: 2.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \sqrt{z \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y (sqrt (* z 2.0))))

C

double code(double x, double y, double z, double t) {
	return y * sqrt((z * 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * sqrt((z * 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * Math.sqrt((z * 2.0));
}

Python

def code(x, y, z, t):
	return y * math.sqrt((z * 2.0))

Julia

function code(x, y, z, t)
	return Float64(y * sqrt(Float64(z * 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y * sqrt((z * 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(y * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. exp-sqrt 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{t \cdot t}}} \]

   2. pow-exp 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   3. pow1/2 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left({\left(e^{t}\right)}^{t}\right)}^{0.5}} \]

   4. pow-pow 99.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{t}\right)}^{\left(t \cdot 0.5\right)}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{t}\right)}^{\left(t \cdot 0.5\right)}} \]

5. Taylor expanded in t around 0 59.4%

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(0.5 \cdot x - y\right)\right)} \]

6. Taylor expanded in x around 0 31.6%

   \[\leadsto \color{blue}{-1 \cdot \left(\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 31.6%

      \[\leadsto \color{blue}{-\left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}} \]

   2. associate-*l* 31.6%

      \[\leadsto -\color{blue}{y \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)} \]

   3. *-commutative 31.6%

      \[\leadsto -y \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)} \]

   4. distribute-rgt-neg-in 31.6%

      \[\leadsto \color{blue}{y \cdot \left(-\sqrt{z} \cdot \sqrt{2}\right)} \]

   5. distribute-rgt-neg-in 31.6%

      \[\leadsto y \cdot \color{blue}{\left(\sqrt{z} \cdot \left(-\sqrt{2}\right)\right)} \]

8. Simplified 31.6%

   \[\leadsto \color{blue}{y \cdot \left(\sqrt{z} \cdot \left(-\sqrt{2}\right)\right)} \]

10. Applied egg-rr 3.9%

    \[\leadsto y \cdot \color{blue}{\sqrt{z \cdot 2}} \]
11. Add Preprocessing

Developer target: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(exp(1.0), ((t * t) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (exp(1.0d0) ** ((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.pow(Math.exp(1.0), ((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.exp(1.0), ((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * (exp(1.0) ^ Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(1.0) ^ ((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[1.0], $MachinePrecision], N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024107 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))