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

Percentage Accurate: 99.5% → 99.7%

Time: 16.0s

Alternatives: 14

Speedup: 1.0×

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

8. Simplified 99.8%

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

Alternative 2: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 10^{-19}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+266}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {\left(z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= (* t t) 1e-19)
     (* t_1 t_2)
     (if (<= (* t t) 2e+266)
       (* (exp (/ (* t t) 2.0)) (* 0.5 (* x t_2)))
       (* t_1 (pow (* z (* 2.0 (fma t t 1.0))) 0.5))))))

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 * t) <= 1e-19) {
		tmp = t_1 * t_2;
	} else if ((t * t) <= 2e+266) {
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * t_2));
	} else {
		tmp = t_1 * pow((z * (2.0 * fma(t, t, 1.0))), 0.5);
	}
	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 (Float64(t * t) <= 1e-19)
		tmp = Float64(t_1 * t_2);
	elseif (Float64(t * t) <= 2e+266)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(0.5 * Float64(x * t_2)));
	else
		tmp = Float64(t_1 * (Float64(z * Float64(2.0 * fma(t, t, 1.0))) ^ 0.5));
	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[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1e-19], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 2e+266], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[(z * N[(2.0 * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 9.9999999999999998e-20

   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. remove-double-neg 99.7%

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

      3. remove-double-neg 99.7%

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

      4. exp-sqrt 99.7%

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

      5. 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%

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

      2. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 99.3%

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

       1. unpow1/2 99.3%

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

       2. metadata-eval 99.3%

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

       3. pow-sqr 99.2%

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

       4. unpow1/2 99.2%

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

       5. metadata-eval 99.2%

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

       6. pow-sqr 99.0%

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

       7. unswap-sqr 99.0%

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

       8. exp-to-pow 94.5%

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

       9. exp-to-pow 94.5%

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

       10. exp-sum 94.5%

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

       11. distribute-rgt-in 94.5%

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

       12. +-commutative 94.5%

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

       13. *-commutative 94.5%

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

       14. exp-prod 94.6%

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

       15. exp-sum 94.5%

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

       16. rem-exp-log 94.5%

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

       17. rem-exp-log 99.4%

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

       18. *-commutative 99.4%

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

       19. exp-to-pow 94.5%

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

       20. exp-to-pow 94.5%

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

       21. exp-sum 94.5%

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

       22. distribute-rgt-in 94.5%

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

   11. Simplified 99.7%

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

   if 9.9999999999999998e-20 < (*.f64 t t) < 2.0000000000000001e266

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf 74.2%

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

   4. Taylor expanded in z around -inf 0.0%

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

   6. Simplified 74.2%

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

   if 2.0000000000000001e266 < (*.f64 t t)

   1. Initial program 100.0%

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

      1. associate-*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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 97.6%

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

       1. +-commutative 97.6%

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

       2. unpow2 97.6%

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

       3. fma-define 97.6%

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

   11. Simplified 97.6%

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

       1. pow1/2 97.6%

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

       2. associate-*l* 98.8%

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

   13. Applied egg-rr 98.8%

       \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{{\left(z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-19}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+266}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot {\left(z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 10^{-19}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+266}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= (* t t) 1e-19)
     (* t_1 t_2)
     (if (<= (* t t) 2e+266)
       (* (exp (/ (* t t) 2.0)) (* 0.5 (* x t_2)))
       (* t_1 (sqrt (* (* z 2.0) (fma t t 1.0))))))))

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 * t) <= 1e-19) {
		tmp = t_1 * t_2;
	} else if ((t * t) <= 2e+266) {
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * t_2));
	} else {
		tmp = t_1 * sqrt(((z * 2.0) * fma(t, t, 1.0)));
	}
	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 (Float64(t * t) <= 1e-19)
		tmp = Float64(t_1 * t_2);
	elseif (Float64(t * t) <= 2e+266)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(0.5 * Float64(x * t_2)));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(z * 2.0) * fma(t, t, 1.0))));
	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[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1e-19], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 2e+266], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 9.9999999999999998e-20

   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. remove-double-neg 99.7%

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

      3. remove-double-neg 99.7%

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

      4. exp-sqrt 99.7%

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

      5. 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%

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

      2. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 99.3%

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

       1. unpow1/2 99.3%

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

       2. metadata-eval 99.3%

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

       3. pow-sqr 99.2%

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

       4. unpow1/2 99.2%

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

       5. metadata-eval 99.2%

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

       6. pow-sqr 99.0%

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

       7. unswap-sqr 99.0%

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

       8. exp-to-pow 94.5%

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

       9. exp-to-pow 94.5%

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

       10. exp-sum 94.5%

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

       11. distribute-rgt-in 94.5%

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

       12. +-commutative 94.5%

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

       13. *-commutative 94.5%

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

       14. exp-prod 94.6%

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

       15. exp-sum 94.5%

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

       16. rem-exp-log 94.5%

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

       17. rem-exp-log 99.4%

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

       18. *-commutative 99.4%

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

       19. exp-to-pow 94.5%

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

       20. exp-to-pow 94.5%

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

       21. exp-sum 94.5%

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

       22. distribute-rgt-in 94.5%

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

   11. Simplified 99.7%

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

   if 9.9999999999999998e-20 < (*.f64 t t) < 2.0000000000000001e266

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf 74.2%

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

   4. Taylor expanded in z around -inf 0.0%

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

   6. Simplified 74.2%

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

   if 2.0000000000000001e266 < (*.f64 t t)

   1. Initial program 100.0%

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

      1. associate-*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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 97.6%

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

       1. +-commutative 97.6%

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

       2. unpow2 97.6%

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

       3. fma-define 97.6%

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

   11. Simplified 97.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-19}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \cdot t \leq 2 \cdot 10^{+266}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.0)
     (* t_1 (sqrt (* z 2.0)))
     (if (<= t 4e+154)
       (* (* t (* t_1 (sqrt 2.0))) (sqrt z))
       (* (sqrt (* (* z 2.0) (fma t t 1.0))) (* x 0.5))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else if (t <= 4e+154) {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	} else {
		tmp = sqrt(((z * 2.0) * fma(t, t, 1.0))) * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	elseif (t <= 4e+154)
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	else
		tmp = Float64(sqrt(Float64(Float64(z * 2.0) * fma(t, t, 1.0))) * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+154], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1

   1. Initial program 99.2%

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

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 71.6%

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

       1. unpow1/2 71.6%

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

       2. metadata-eval 71.6%

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

       3. pow-sqr 71.5%

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

       4. unpow1/2 71.5%

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

       5. metadata-eval 71.5%

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

       6. pow-sqr 71.4%

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

       7. unswap-sqr 71.4%

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

       8. exp-to-pow 68.4%

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

       9. exp-to-pow 68.4%

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

       10. exp-sum 68.5%

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

       11. distribute-rgt-in 68.5%

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

       12. +-commutative 68.5%

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

       13. *-commutative 68.5%

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

       14. exp-prod 68.5%

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

       15. exp-sum 68.4%

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

       16. rem-exp-log 68.4%

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

       17. rem-exp-log 71.6%

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

       18. *-commutative 71.6%

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

       19. exp-to-pow 68.4%

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

       20. exp-to-pow 68.4%

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

       21. exp-sum 68.4%

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

       22. distribute-rgt-in 68.4%

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

   11. Simplified 71.8%

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

   if 1 < t < 4.00000000000000015e154

   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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 39.0%

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

       1. +-commutative 39.0%

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

       2. unpow2 39.0%

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

       3. fma-define 39.0%

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

   11. Simplified 39.0%

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

   12. Taylor expanded in t around inf 41.5%

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

   if 4.00000000000000015e154 < 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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. unpow2 100.0%

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

       3. fma-define 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 78.6%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 2.6 \cdot 10^{+30}:\\ \;\;\;\;t\_1 \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot {t\_1}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 2.6e+30)
     (* t_1 (sqrt (* z 2.0)))
     (if (<= t 9.8e+69)
       (sqrt (* z (* 2.0 (pow t_1 2.0))))
       (* (sqrt (* (* z 2.0) (fma t t 1.0))) (* x 0.5))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 2.6e+30) {
		tmp = t_1 * sqrt((z * 2.0));
	} else if (t <= 9.8e+69) {
		tmp = sqrt((z * (2.0 * pow(t_1, 2.0))));
	} else {
		tmp = sqrt(((z * 2.0) * fma(t, t, 1.0))) * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 2.6e+30)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	elseif (t <= 9.8e+69)
		tmp = sqrt(Float64(z * Float64(2.0 * (t_1 ^ 2.0))));
	else
		tmp = Float64(sqrt(Float64(Float64(z * 2.0) * fma(t, t, 1.0))) * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 2.6e+30], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.8e+69], N[Sqrt[N[(z * N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.59999999999999988e30

   1. Initial program 99.2%

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

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 70.6%

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

       1. unpow1/2 70.6%

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

       2. metadata-eval 70.6%

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

       3. pow-sqr 70.5%

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

       4. unpow1/2 70.5%

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

       5. metadata-eval 70.5%

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

       6. pow-sqr 70.4%

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

       7. unswap-sqr 70.4%

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

       8. exp-to-pow 67.5%

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

       9. exp-to-pow 67.5%

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

       10. exp-sum 67.6%

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

       11. distribute-rgt-in 67.6%

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

       12. +-commutative 67.6%

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

       13. *-commutative 67.6%

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

       14. exp-prod 67.6%

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

       15. exp-sum 67.5%

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

       16. rem-exp-log 67.5%

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

       17. rem-exp-log 70.7%

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

       18. *-commutative 70.7%

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

       19. exp-to-pow 67.5%

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

       20. exp-to-pow 67.5%

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

       21. exp-sum 67.5%

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

       22. distribute-rgt-in 67.5%

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

   11. Simplified 70.8%

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

   if 2.59999999999999988e30 < t < 9.7999999999999999e69

   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 t around 0 13.0%

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

      1. *-rgt-identity 13.0%

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

      2. add-sqr-sqrt 11.6%

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

      3. sqrt-unprod 35.4%

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

      4. *-commutative 35.4%

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

      5. *-commutative 35.4%

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

      6. swap-sqr 51.4%

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

      7. add-sqr-sqrt 51.4%

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

      8. pow2 51.4%

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

      9. fmm-def 51.4%

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

   5. Applied egg-rr 51.4%

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

      1. associate-*l* 51.4%

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

      2. fmm-undef 51.4%

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

      3. *-commutative 51.4%

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

   7. Simplified 51.4%

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

   if 9.7999999999999999e69 < 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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 83.0%

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

       1. +-commutative 83.0%

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

       2. unpow2 83.0%

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

       3. fma-define 83.0%

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

   11. Simplified 83.0%

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

   12. Taylor expanded in x around inf 64.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{+30}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+69}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot {t\_1}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 2.3e+30)
     (* t_1 t_2)
     (if (<= t 1.4e+127)
       (sqrt (* z (* 2.0 (pow t_1 2.0))))
       (* t_1 (* 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 <= 2.3e+30) {
		tmp = t_1 * t_2;
	} else if (t <= 1.4e+127) {
		tmp = sqrt((z * (2.0 * pow(t_1, 2.0))));
	} else {
		tmp = t_1 * (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 <= 2.3d+30) then
        tmp = t_1 * t_2
    else if (t <= 1.4d+127) then
        tmp = sqrt((z * (2.0d0 * (t_1 ** 2.0d0))))
    else
        tmp = t_1 * (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 <= 2.3e+30) {
		tmp = t_1 * t_2;
	} else if (t <= 1.4e+127) {
		tmp = Math.sqrt((z * (2.0 * Math.pow(t_1, 2.0))));
	} else {
		tmp = t_1 * (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 <= 2.3e+30:
		tmp = t_1 * t_2
	elif t <= 1.4e+127:
		tmp = math.sqrt((z * (2.0 * math.pow(t_1, 2.0))))
	else:
		tmp = t_1 * (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 <= 2.3e+30)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 1.4e+127)
		tmp = sqrt(Float64(z * Float64(2.0 * (t_1 ^ 2.0))));
	else
		tmp = Float64(t_1 * 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 <= 2.3e+30)
		tmp = t_1 * t_2;
	elseif (t <= 1.4e+127)
		tmp = sqrt((z * (2.0 * (t_1 ^ 2.0))));
	else
		tmp = t_1 * (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, 2.3e+30], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 1.4e+127], N[Sqrt[N[(z * N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3e30

   1. Initial program 99.2%

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

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 70.6%

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

       1. unpow1/2 70.6%

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

       2. metadata-eval 70.6%

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

       3. pow-sqr 70.5%

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

       4. unpow1/2 70.5%

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

       5. metadata-eval 70.5%

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

       6. pow-sqr 70.4%

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

       7. unswap-sqr 70.4%

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

       8. exp-to-pow 67.5%

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

       9. exp-to-pow 67.5%

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

       10. exp-sum 67.6%

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

       11. distribute-rgt-in 67.6%

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

       12. +-commutative 67.6%

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

       13. *-commutative 67.6%

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

       14. exp-prod 67.6%

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

       15. exp-sum 67.5%

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

       16. rem-exp-log 67.5%

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

       17. rem-exp-log 70.7%

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

       18. *-commutative 70.7%

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

       19. exp-to-pow 67.5%

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

       20. exp-to-pow 67.5%

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

       21. exp-sum 67.5%

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

       22. distribute-rgt-in 67.5%

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

   11. Simplified 70.8%

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

   if 2.3e30 < t < 1.4000000000000001e127

   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 t around 0 15.1%

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

      1. *-rgt-identity 15.1%

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

      2. add-sqr-sqrt 13.6%

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

      3. sqrt-unprod 31.2%

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

      4. *-commutative 31.2%

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

      5. *-commutative 31.2%

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

      6. swap-sqr 38.3%

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

      7. add-sqr-sqrt 38.3%

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

      8. pow2 38.3%

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

      9. fmm-def 38.3%

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

   5. Applied egg-rr 38.3%

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

      1. associate-*l* 38.3%

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

      2. fmm-undef 38.3%

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

      3. *-commutative 38.3%

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

   7. Simplified 38.3%

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

   if 1.4000000000000001e127 < 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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 98.0%

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

       1. +-commutative 98.0%

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

       2. unpow2 98.0%

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

       3. fma-define 98.0%

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

   11. Simplified 98.0%

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

   12. Taylor expanded in t around inf 55.6%

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

       1. associate-*l* 55.6%

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

   14. Simplified 55.6%

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

       1. *-commutative 55.6%

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

       2. sqrt-prod 55.6%

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

   16. Applied egg-rr 55.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot {\left(x \cdot 0.5 - y\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(t \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 10^{-19}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 1e-19)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* 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 ((t * t) <= 1e-19) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * (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 ((t * t) <= 1d-19) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * (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 ((t * t) <= 1e-19) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (0.5 * (x * t_1));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (t * t) <= 1e-19:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * (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 (Float64(t * t) <= 1e-19)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * 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 ((t * t) <= 1e-19)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * (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[LessEqual[N[(t * t), $MachinePrecision], 1e-19], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 9.9999999999999998e-20

   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. remove-double-neg 99.7%

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

      3. remove-double-neg 99.7%

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

      4. exp-sqrt 99.7%

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

      5. 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%

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

      2. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 99.3%

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

       1. unpow1/2 99.3%

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

       2. metadata-eval 99.3%

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

       3. pow-sqr 99.2%

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

       4. unpow1/2 99.2%

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

       5. metadata-eval 99.2%

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

       6. pow-sqr 99.0%

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

       7. unswap-sqr 99.0%

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

       8. exp-to-pow 94.5%

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

       9. exp-to-pow 94.5%

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

       10. exp-sum 94.5%

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

       11. distribute-rgt-in 94.5%

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

       12. +-commutative 94.5%

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

       13. *-commutative 94.5%

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

       14. exp-prod 94.6%

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

       15. exp-sum 94.5%

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

       16. rem-exp-log 94.5%

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

       17. rem-exp-log 99.4%

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

       18. *-commutative 99.4%

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

       19. exp-to-pow 94.5%

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

       20. exp-to-pow 94.5%

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

       21. exp-sum 94.5%

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

       22. distribute-rgt-in 94.5%

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

   11. Simplified 99.7%

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

   if 9.9999999999999998e-20 < (*.f64 t t)

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 75.9%

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

   4. Taylor expanded in z around -inf 0.0%

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

   6. Simplified 75.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-19}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+118}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.25e+118)
   (* (- (* x 0.5) y) (* (sqrt (* z 2.0)) (hypot 1.0 t)))
   (* (sqrt (* (* z 2.0) (fma t t 1.0))) (* x 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.25e+118) {
		tmp = ((x * 0.5) - y) * (sqrt((z * 2.0)) * hypot(1.0, t));
	} else {
		tmp = sqrt(((z * 2.0) * fma(t, t, 1.0))) * (x * 0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.25e+118)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * Float64(sqrt(Float64(z * 2.0)) * hypot(1.0, t)));
	else
		tmp = Float64(sqrt(Float64(Float64(z * 2.0) * fma(t, t, 1.0))) * Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.25e+118], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.24999999999999993e118

   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. 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. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 80.5%

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

       1. +-commutative 80.5%

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

       2. unpow2 80.5%

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

       3. fma-define 80.5%

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

   11. Simplified 80.5%

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

       1. sqrt-prod 78.3%

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

   13. Applied egg-rr 78.3%

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

       1. *-commutative 78.3%

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

       2. fma-undefine 78.3%

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

       3. unpow2 78.3%

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

       4. +-commutative 78.3%

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

       5. unpow2 78.3%

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

       6. hypot-1-def 74.2%

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

   15. Simplified 74.2%

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

   if 1.24999999999999993e118 < 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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 96.1%

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

       1. +-commutative 96.1%

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

       2. unpow2 96.1%

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

       3. fma-define 96.1%

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

   11. Simplified 96.1%

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

   12. Taylor expanded in x around inf 75.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{+118}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% 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]

Derivation

1. Initial program 99.5%

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

Alternative 10: 43.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+59} \lor \neg \left(x \leq 1.1 \cdot 10^{+24}\right):\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= x -3e+59) (not (<= x 1.1e+24)))
     (* t_1 (* x 0.5))
     (* y (- t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((x <= -3e+59) || !(x <= 1.1e+24)) {
		tmp = t_1 * (x * 0.5);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((x <= (-3d+59)) .or. (.not. (x <= 1.1d+24))) then
        tmp = t_1 * (x * 0.5d0)
    else
        tmp = y * -t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((x <= -3e+59) || !(x <= 1.1e+24)) {
		tmp = t_1 * (x * 0.5);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (x <= -3e+59) or not (x <= 1.1e+24):
		tmp = t_1 * (x * 0.5)
	else:
		tmp = y * -t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((x <= -3e+59) || !(x <= 1.1e+24))
		tmp = Float64(t_1 * Float64(x * 0.5));
	else
		tmp = Float64(y * Float64(-t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((x <= -3e+59) || ~((x <= 1.1e+24)))
		tmp = t_1 * (x * 0.5);
	else
		tmp = y * -t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -3e+59], N[Not[LessEqual[x, 1.1e+24]], $MachinePrecision]], N[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3e59 or 1.10000000000000001e24 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0 58.3%

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

   4. Taylor expanded in x around inf 45.6%

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

      1. *-commutative 45.6%

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

   6. Simplified 46.5%

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

   if -3e59 < x < 1.10000000000000001e24

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0 50.2%

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

   4. Taylor expanded in x around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. associate-*l* 43.1%

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

      3. *-commutative 43.1%

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

      4. unpow1/2 43.1%

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

      5. metadata-eval 43.1%

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

      6. pow-sqr 43.0%

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

      7. unpow1/2 43.0%

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

      8. metadata-eval 43.0%

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

      9. pow-sqr 43.0%

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

      10. unswap-sqr 43.0%

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

      11. exp-to-pow 41.3%

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

      12. exp-to-pow 41.3%

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

      13. exp-sum 41.2%

          \[\leadsto -y \cdot \left(\color{blue}{e^{\log z \cdot 0.25 + \log 2 \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

      14. distribute-rgt-in 41.2%

          \[\leadsto -y \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log z + \log 2\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

      15. +-commutative 41.2%

          \[\leadsto -y \cdot \left(e^{0.25 \cdot \color{blue}{\left(\log 2 + \log z\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

      16. *-commutative 41.2%

          \[\leadsto -y \cdot \left(e^{\color{blue}{\left(\log 2 + \log z\right) \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

      17. exp-prod 41.2%

          \[\leadsto -y \cdot \left(\color{blue}{{\left(e^{\log 2 + \log z}\right)}^{0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

      18. exp-sum 41.3%

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

      19. rem-exp-log 41.3%

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

      20. rem-exp-log 43.1%

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

      21. *-commutative 43.1%

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

   6. Simplified 43.2%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+59} \lor \neg \left(x \leq 1.1 \cdot 10^{+24}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t 1.0) (* t_1 t_2) (* t_1 (* 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 <= 1.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_1 * (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 <= 1.0d0) then
        tmp = t_1 * t_2
    else
        tmp = t_1 * (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 <= 1.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_1 * (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 <= 1.0:
		tmp = t_1 * t_2
	else:
		tmp = t_1 * (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 <= 1.0)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(t_1 * 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 <= 1.0)
		tmp = t_1 * t_2;
	else
		tmp = t_1 * (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, 1.0], N[(t$95$1 * t$95$2), $MachinePrecision], N[(t$95$1 * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.2%

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

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 71.6%

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

       1. unpow1/2 71.6%

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

       2. metadata-eval 71.6%

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

       3. pow-sqr 71.5%

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

       4. unpow1/2 71.5%

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

       5. metadata-eval 71.5%

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

       6. pow-sqr 71.4%

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

       7. unswap-sqr 71.4%

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

       8. exp-to-pow 68.4%

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

       9. exp-to-pow 68.4%

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

       10. exp-sum 68.5%

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

       11. distribute-rgt-in 68.5%

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

       12. +-commutative 68.5%

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

       13. *-commutative 68.5%

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

       14. exp-prod 68.5%

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

       15. exp-sum 68.4%

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

       16. rem-exp-log 68.4%

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

       17. rem-exp-log 71.6%

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

       18. *-commutative 71.6%

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

       19. exp-to-pow 68.4%

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

       20. exp-to-pow 68.4%

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

       21. exp-sum 68.4%

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

       22. distribute-rgt-in 68.4%

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

   11. Simplified 71.8%

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

   if 1 < 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. remove-double-neg 100.0%

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

      3. remove-double-neg 100.0%

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

      4. exp-sqrt 100.0%

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

      5. 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%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 71.9%

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

       1. +-commutative 71.9%

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

       2. unpow2 71.9%

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

       3. fma-define 71.9%

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

   11. Simplified 71.9%

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

   12. Taylor expanded in t around inf 43.9%

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

       1. associate-*l* 43.9%

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

   14. Simplified 43.9%

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

       1. *-commutative 43.9%

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

       2. sqrt-prod 43.9%

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

   16. Applied egg-rr 43.9%

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

Alternative 12: 56.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in t around 0 53.7%

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

    1. unpow1/2 53.7%

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

    2. metadata-eval 53.7%

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

    3. pow-sqr 53.7%

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

    4. unpow1/2 53.7%

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

    5. metadata-eval 53.7%

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

    6. pow-sqr 53.6%

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

    7. unswap-sqr 53.6%

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

    8. exp-to-pow 51.5%

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

    9. exp-to-pow 51.5%

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

    10. exp-sum 51.5%

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

    11. distribute-rgt-in 51.5%

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

    12. +-commutative 51.5%

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

    13. *-commutative 51.5%

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

    14. exp-prod 51.5%

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

    15. exp-sum 51.5%

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

    16. rem-exp-log 51.5%

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

    17. rem-exp-log 53.7%

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

    18. *-commutative 53.7%

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

    19. exp-to-pow 51.5%

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

    20. exp-to-pow 51.5%

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

    21. exp-sum 51.5%

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

    22. distribute-rgt-in 51.5%

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

11. Simplified 53.9%

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

Alternative 13: 30.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in t around 0 53.9%

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

4. Taylor expanded in x around 0 29.9%

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

   1. mul-1-neg 29.9%

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

   2. associate-*l* 29.9%

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

   3. *-commutative 29.9%

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

   4. unpow1/2 29.9%

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

   5. metadata-eval 29.9%

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

   6. pow-sqr 29.9%

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

   7. unpow1/2 29.9%

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

   8. metadata-eval 29.9%

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

   9. pow-sqr 29.9%

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

   10. unswap-sqr 29.9%

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

   11. exp-to-pow 28.7%

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

   12. exp-to-pow 28.7%

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

   13. exp-sum 28.7%

       \[\leadsto -y \cdot \left(\color{blue}{e^{\log z \cdot 0.25 + \log 2 \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   14. distribute-rgt-in 28.7%

       \[\leadsto -y \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log z + \log 2\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   15. +-commutative 28.7%

       \[\leadsto -y \cdot \left(e^{0.25 \cdot \color{blue}{\left(\log 2 + \log z\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   16. *-commutative 28.7%

       \[\leadsto -y \cdot \left(e^{\color{blue}{\left(\log 2 + \log z\right) \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   17. exp-prod 28.7%

       \[\leadsto -y \cdot \left(\color{blue}{{\left(e^{\log 2 + \log z}\right)}^{0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   18. exp-sum 28.7%

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

   19. rem-exp-log 28.7%

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

   20. rem-exp-log 29.9%

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

   21. *-commutative 29.9%

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

6. Simplified 30.0%

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

7. Final simplification 30.0%

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

Alternative 14: 2.5% 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 99.5%

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

3. Taylor expanded in t around 0 53.9%

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

4. Taylor expanded in x around 0 29.9%

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

   1. mul-1-neg 29.9%

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

   2. associate-*l* 29.9%

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

   3. *-commutative 29.9%

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

   4. unpow1/2 29.9%

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

   5. metadata-eval 29.9%

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

   6. pow-sqr 29.9%

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

   7. unpow1/2 29.9%

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

   8. metadata-eval 29.9%

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

   9. pow-sqr 29.9%

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

   10. unswap-sqr 29.9%

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

   11. exp-to-pow 28.7%

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

   12. exp-to-pow 28.7%

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

   13. exp-sum 28.7%

       \[\leadsto -y \cdot \left(\color{blue}{e^{\log z \cdot 0.25 + \log 2 \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   14. distribute-rgt-in 28.7%

       \[\leadsto -y \cdot \left(e^{\color{blue}{0.25 \cdot \left(\log z + \log 2\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   15. +-commutative 28.7%

       \[\leadsto -y \cdot \left(e^{0.25 \cdot \color{blue}{\left(\log 2 + \log z\right)}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   16. *-commutative 28.7%

       \[\leadsto -y \cdot \left(e^{\color{blue}{\left(\log 2 + \log z\right) \cdot 0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   17. exp-prod 28.7%

       \[\leadsto -y \cdot \left(\color{blue}{{\left(e^{\log 2 + \log z}\right)}^{0.25}} \cdot \left({z}^{0.25} \cdot {2}^{0.25}\right)\right) \]

   18. exp-sum 28.7%

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

   19. rem-exp-log 28.7%

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

   20. rem-exp-log 29.9%

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

   21. *-commutative 29.9%

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

6. Simplified 30.0%

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

   1. neg-sub0 30.0%

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

   2. sub-neg 30.0%

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

   3. add-sqr-sqrt 16.6%

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

   4. sqrt-unprod 16.2%

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

   5. sqr-neg 16.2%

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

   6. sqrt-unprod 1.3%

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

   7. add-sqr-sqrt 2.1%

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

8. Applied egg-rr 2.1%

   \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(0 + y\right)} \]
9. Step-by-step derivation

   1. +-lft-identity 2.1%

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

10. Simplified 2.1%

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

11. Final simplification 2.1%

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

Developer Target 1: 99.5% 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 2024157 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))