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

Percentage Accurate: 99.4% → 99.5%

Time: 19.2s

Alternatives: 15

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (pow (exp t_m) (* 0.5 t_m)) (* (- (* x 0.5) y) (sqrt (* z 2.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Power[N[Exp[t$95$m], $MachinePrecision], N[(0.5 * t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. exp-sqrt 99.8%

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

   2. pow-exp 99.9%

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

   3. pow1/2 99.9%

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

   4. pow-pow 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (* z 2.0) (pow (exp t_m) t_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[N[Exp[t$95$m], $MachinePrecision], t$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-udef 74.9%

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

   3. sqrt-unprod 74.9%

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

   4. associate-*l* 74.9%

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

   5. pow-exp 74.9%

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

   6. pow2 74.9%

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

6. Applied egg-rr 74.9%

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

   1. expm1-def 98.9%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. unpow2 99.8%

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

   2. pow-exp 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;t\_1 \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot t\_m\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t_m 6.5e-31)
     (* t_1 t_2)
     (if (<= t_m 1.05e+74)
       (* t_1 (cbrt (pow (* z 2.0) 1.5)))
       (* t_1 (* t_2 t_m))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t_m <= 6.5e-31) {
		tmp = t_1 * t_2;
	} else if (t_m <= 1.05e+74) {
		tmp = t_1 * cbrt(pow((z * 2.0), 1.5));
	} else {
		tmp = t_1 * (t_2 * t_m);
	}
	return tmp;
}

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t_m <= 6.5e-31) {
		tmp = t_1 * t_2;
	} else if (t_m <= 1.05e+74) {
		tmp = t_1 * Math.cbrt(Math.pow((z * 2.0), 1.5));
	} else {
		tmp = t_1 * (t_2 * t_m);
	}
	return tmp;
}

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t_m <= 6.5e-31)
		tmp = Float64(t_1 * t_2);
	elseif (t_m <= 1.05e+74)
		tmp = Float64(t_1 * cbrt((Float64(z * 2.0) ^ 1.5)));
	else
		tmp = Float64(t_1 * Float64(t_2 * t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := 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$95$m, 6.5e-31], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+74], N[(t$95$1 * N[Power[N[Power[N[(z * 2.0), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.49999999999999967e-31

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.3%

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

      3. exp-sqrt 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 69.5%

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

   if 6.49999999999999967e-31 < t < 1.0499999999999999e74

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around 0 25.1%

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

      1. add-cbrt-cube 42.8%

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

      2. add-sqr-sqrt 42.8%

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

      3. pow1 42.8%

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

      4. pow1/2 42.8%

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

      5. pow-prod-up 42.8%

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

      6. metadata-eval 42.8%

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

   7. Applied egg-rr 42.8%

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

   if 1.0499999999999999e74 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 85.5%

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

   10. Taylor expanded in t around inf 69.2%

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

       1. associate-*l* 55.2%

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

       2. *-commutative 55.2%

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

   12. Simplified 55.2%

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

   13. Taylor expanded in t around 0 69.2%

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

       1. *-commutative 69.2%

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

       2. *-commutative 69.2%

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

       3. sub-neg 69.2%

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

       4. *-commutative 69.2%

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

       5. sub-neg 69.2%

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

       6. associate-*r* 55.2%

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

   15. Simplified 56.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t_m 1.0)
     (* t_1 (sqrt (* z 2.0)))
     (* t_1 (sqrt (* (* z 2.0) (pow t_m 2.0)))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = t_1 * sqrt(((z * 2.0) * pow(t_m, 2.0)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t_m <= 1.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else
        tmp = t_1 * sqrt(((z * 2.0d0) * (t_m ** 2.0d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = t_1 * Math.sqrt(((z * 2.0) * Math.pow(t_m, 2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t_m <= 1.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	else:
		tmp = t_1 * math.sqrt(((z * 2.0) * math.pow(t_m, 2.0)))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t_m <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(z * 2.0) * (t_m ^ 2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t_m <= 1.0)
		tmp = t_1 * sqrt((z * 2.0));
	else
		tmp = t_1 * sqrt(((z * 2.0) * (t_m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$m, 1.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.3%

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

      3. exp-sqrt 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 69.8%

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

   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. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.2%

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

   10. Taylor expanded in t around inf 76.2%

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

       1. *-commutative 76.2%

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

       2. associate-*l* 76.2%

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

       3. *-commutative 76.2%

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

   12. Simplified 76.2%

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

4. Final simplification 71.6%

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

Alternative 5: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{hypot}\left(1, t\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(z \cdot 2\right) \cdot {t\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t_m 1.4e+24)
     (* t_1 (* (sqrt (* z 2.0)) (hypot 1.0 t_m)))
     (* t_1 (sqrt (* (* z 2.0) (pow t_m 2.0)))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.4e+24) {
		tmp = t_1 * (sqrt((z * 2.0)) * hypot(1.0, t_m));
	} else {
		tmp = t_1 * sqrt(((z * 2.0) * pow(t_m, 2.0)));
	}
	return tmp;
}

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.4e+24) {
		tmp = t_1 * (Math.sqrt((z * 2.0)) * Math.hypot(1.0, t_m));
	} else {
		tmp = t_1 * Math.sqrt(((z * 2.0) * Math.pow(t_m, 2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t_m <= 1.4e+24:
		tmp = t_1 * (math.sqrt((z * 2.0)) * math.hypot(1.0, t_m))
	else:
		tmp = t_1 * math.sqrt(((z * 2.0) * math.pow(t_m, 2.0)))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t_m <= 1.4e+24)
		tmp = Float64(t_1 * Float64(sqrt(Float64(z * 2.0)) * hypot(1.0, t_m)));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(z * 2.0) * (t_m ^ 2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t_m <= 1.4e+24)
		tmp = t_1 * (sqrt((z * 2.0)) * hypot(1.0, t_m));
	else
		tmp = t_1 * sqrt(((z * 2.0) * (t_m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$m, 1.4e+24], N[(t$95$1 * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + t$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4000000000000001e24

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

      3. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 66.2%

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

      3. sqrt-unprod 66.2%

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

      4. associate-*l* 66.2%

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

      5. pow-exp 66.2%

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

      6. pow2 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 87.3%

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

       1. *-commutative 87.3%

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

       2. sqrt-prod 86.8%

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

       3. unpow2 86.8%

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

       4. hypot-1-def 82.9%

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

   11. Applied egg-rr 82.9%

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

   if 1.4000000000000001e24 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 81.6%

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

   10. Taylor expanded in t around inf 81.6%

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

       1. *-commutative 81.6%

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

       2. associate-*l* 81.6%

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

       3. *-commutative 81.6%

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

   12. Simplified 81.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot {t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t_m 1.0)
     (* t_1 (sqrt (* z 2.0)))
     (* (* t_m (* t_1 (sqrt 2.0))) (sqrt z)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.0) {
		tmp = t_1 * sqrt((z * 2.0));
	} else {
		tmp = (t_m * (t_1 * sqrt(2.0))) * sqrt(z);
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t_m <= 1.0d0) then
        tmp = t_1 * sqrt((z * 2.0d0))
    else
        tmp = (t_m * (t_1 * sqrt(2.0d0))) * sqrt(z)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t_m <= 1.0) {
		tmp = t_1 * Math.sqrt((z * 2.0));
	} else {
		tmp = (t_m * (t_1 * Math.sqrt(2.0))) * Math.sqrt(z);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t_m <= 1.0:
		tmp = t_1 * math.sqrt((z * 2.0))
	else:
		tmp = (t_m * (t_1 * math.sqrt(2.0))) * math.sqrt(z)
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t_m <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(Float64(t_m * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t_m <= 1.0)
		tmp = t_1 * sqrt((z * 2.0));
	else
		tmp = (t_m * (t_1 * sqrt(2.0))) * sqrt(z);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$m, 1.0], N[(t$95$1 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.3%

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

      3. exp-sqrt 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 69.8%

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

   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. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.2%

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

   10. Taylor expanded in t around inf 62.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (exp (/ (* t_m t_m) 2.0)) (* (- (* x 0.5) y) (sqrt (* z 2.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[Exp[N[(N[(t$95$m * t$95$m), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 8: 85.3% accurate, 1.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (* (- (* x 0.5) y) (sqrt (* (* z 2.0) (fma t_m t_m 1.0)))))

C

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

Julia

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t$95$m * t$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. exp-sqrt 99.8%

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

   3. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-udef 74.9%

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

   3. sqrt-unprod 74.9%

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

   4. associate-*l* 74.9%

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

   5. pow-exp 74.9%

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

   6. pow2 74.9%

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

6. Applied egg-rr 74.9%

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

   1. expm1-def 98.9%

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

   2. expm1-log1p 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in t around 0 85.8%

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

    1. +-commutative 85.8%

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

    2. unpow2 85.8%

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

    3. fma-def 85.8%

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

11. Simplified 85.8%

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

12. Final simplification 85.8%

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

Alternative 9: 41.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+107}:\\ \;\;\;\;\left(y \cdot t\_1\right) \cdot \left(-t\_m\right)\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{+24} \lor \neg \left(y \leq 8 \cdot 10^{+83}\right):\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= y -1.7e+107)
     (* (* y t_1) (- t_m))
     (if (or (<= y -2.4e+24) (not (<= y 8e+83)))
       (* y (- t_1))
       (* 0.5 (* x t_1))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (y <= -1.7e+107) {
		tmp = (y * t_1) * -t_m;
	} else if ((y <= -2.4e+24) || !(y <= 8e+83)) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (y <= (-1.7d+107)) then
        tmp = (y * t_1) * -t_m
    else if ((y <= (-2.4d+24)) .or. (.not. (y <= 8d+83))) then
        tmp = y * -t_1
    else
        tmp = 0.5d0 * (x * t_1)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (y <= -1.7e+107) {
		tmp = (y * t_1) * -t_m;
	} else if ((y <= -2.4e+24) || !(y <= 8e+83)) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * (x * t_1);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if y <= -1.7e+107:
		tmp = (y * t_1) * -t_m
	elif (y <= -2.4e+24) or not (y <= 8e+83):
		tmp = y * -t_1
	else:
		tmp = 0.5 * (x * t_1)
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (y <= -1.7e+107)
		tmp = Float64(Float64(y * t_1) * Float64(-t_m));
	elseif ((y <= -2.4e+24) || !(y <= 8e+83))
		tmp = Float64(y * Float64(-t_1));
	else
		tmp = Float64(0.5 * Float64(x * t_1));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (y <= -1.7e+107)
		tmp = (y * t_1) * -t_m;
	elseif ((y <= -2.4e+24) || ~((y <= 8e+83)))
		tmp = y * -t_1;
	else
		tmp = 0.5 * (x * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.7e+107], N[(N[(y * t$95$1), $MachinePrecision] * (-t$95$m)), $MachinePrecision], If[Or[LessEqual[y, -2.4e+24], N[Not[LessEqual[y, 8e+83]], $MachinePrecision]], N[(y * (-t$95$1)), $MachinePrecision], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999998e107

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 77.7%

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

      3. sqrt-unprod 77.7%

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

      4. associate-*l* 77.7%

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

      5. pow-exp 77.7%

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

      6. pow2 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 89.6%

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

   10. Taylor expanded in t around inf 39.7%

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

       1. associate-*l* 31.8%

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

       2. *-commutative 31.8%

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

   12. Simplified 31.8%

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

   13. Taylor expanded in x around 0 41.9%

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

       1. mul-1-neg 41.9%

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

       2. associate-*l* 34.0%

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

       3. distribute-rgt-neg-in 34.0%

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

       4. associate-*l* 34.0%

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

       5. *-commutative 34.0%

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

       6. distribute-lft-neg-in 34.0%

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

       7. unpow1/2 34.0%

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

       8. exp-to-pow 34.0%

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

       9. unpow1/2 34.0%

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

       10. exp-to-pow 34.0%

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

       11. exp-sum 34.0%

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

       12. distribute-rgt-in 34.0%

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

       13. log-prod 34.0%

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

       14. log-pow 34.0%

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

       15. unpow1/2 34.0%

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

       16. rem-exp-log 34.0%

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

       17. distribute-lft-neg-out 34.0%

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

       18. distribute-rgt-neg-in 34.0%

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

   15. Simplified 34.0%

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

   if -1.6999999999999998e107 < y < -2.4000000000000001e24 or 8.00000000000000025e83 < y

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 61.8%

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

   6. Taylor expanded in x around 0 52.7%

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

      1. associate-*r* 52.7%

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

      2. mul-1-neg 52.7%

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

      3. distribute-lft-neg-out 52.7%

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

      4. *-commutative 52.7%

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

      5. associate-*l* 52.7%

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

   8. Simplified 52.7%

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

      1. *-commutative 52.7%

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

      2. distribute-lft-neg-out 52.7%

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

      3. distribute-lft-neg-out 52.7%

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

      4. add-sqr-sqrt 43.9%

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

      5. sqrt-unprod 52.4%

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

      6. sqr-neg 52.4%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 1.0%

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

      9. associate-*l* 1.0%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 52.3%

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

      12. sqr-neg 52.3%

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

      13. sqrt-unprod 43.7%

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

      14. add-sqr-sqrt 52.7%

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

      15. sqrt-prod 53.0%

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

      16. *-commutative 53.0%

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

   10. Applied egg-rr 53.0%

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

       1. *-commutative 53.0%

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

       2. distribute-rgt-neg-in 53.0%

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

   12. Simplified 53.0%

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

   if -2.4000000000000001e24 < y < 8.00000000000000025e83

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.2%

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

      3. exp-sqrt 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t around 0 54.3%

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

      1. associate-*r* 54.3%

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

      2. *-commutative 54.3%

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

      3. *-commutative 54.3%

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

      4. fma-neg 54.3%

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

      5. associate-*l* 54.3%

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

      6. fma-neg 54.3%

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

      7. *-commutative 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in x around inf 41.7%

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

      1. associate-*l* 41.6%

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

   10. Simplified 41.6%

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

   11. Taylor expanded in x around 0 41.7%

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

       1. associate-*l* 41.6%

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

       2. *-commutative 41.6%

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

       3. unpow1/2 41.6%

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

       4. exp-to-pow 39.6%

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

       5. unpow1/2 39.6%

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

       6. exp-to-pow 39.6%

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

       7. exp-sum 39.5%

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

       8. distribute-rgt-in 39.5%

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

       9. log-prod 39.7%

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

       10. log-pow 39.7%

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

       11. unpow1/2 39.7%

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

       12. rem-exp-log 41.8%

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

   13. Simplified 41.8%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+107}:\\ \;\;\;\;\left(y \cdot \sqrt{z \cdot 2}\right) \cdot \left(-t\right)\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{+24} \lor \neg \left(y \leq 8 \cdot 10^{+83}\right):\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+63} \lor \neg \left(x \leq 4.3 \cdot 10^{+110}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= x -1.05e+63) (not (<= x 4.3e+110)))
     (* 0.5 (* x t_1))
     (* y (- t_1)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((x <= -1.05e+63) || !(x <= 4.3e+110)) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((x <= (-1.05d+63)) .or. (.not. (x <= 4.3d+110))) then
        tmp = 0.5d0 * (x * t_1)
    else
        tmp = y * -t_1
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((x <= -1.05e+63) || !(x <= 4.3e+110)) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (x <= -1.05e+63) or not (x <= 4.3e+110):
		tmp = 0.5 * (x * t_1)
	else:
		tmp = y * -t_1
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((x <= -1.05e+63) || !(x <= 4.3e+110))
		tmp = Float64(0.5 * Float64(x * t_1));
	else
		tmp = Float64(y * Float64(-t_1));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((x <= -1.05e+63) || ~((x <= 4.3e+110)))
		tmp = 0.5 * (x * t_1);
	else
		tmp = y * -t_1;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -1.05e+63], N[Not[LessEqual[x, 4.3e+110]], $MachinePrecision]], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0500000000000001e63 or 4.30000000000000007e110 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 98.9%

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

      3. exp-sqrt 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t around 0 54.4%

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

      1. associate-*r* 54.3%

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

      2. *-commutative 54.3%

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

      3. *-commutative 54.3%

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

      4. fma-neg 54.3%

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

      5. associate-*l* 54.4%

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

      6. fma-neg 54.4%

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

      7. *-commutative 54.4%

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

   7. Simplified 54.4%

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

   8. Taylor expanded in x around inf 51.1%

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

      1. associate-*l* 51.0%

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

   10. Simplified 51.0%

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

   11. Taylor expanded in x around 0 51.1%

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

       1. associate-*l* 51.0%

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

       2. *-commutative 51.0%

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

       3. unpow1/2 51.0%

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

       4. exp-to-pow 48.8%

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

       5. unpow1/2 48.8%

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

       6. exp-to-pow 48.8%

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

       7. exp-sum 48.8%

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

       8. distribute-rgt-in 48.8%

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

       9. log-prod 49.0%

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

       10. log-pow 49.0%

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

       11. unpow1/2 49.0%

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

       12. rem-exp-log 51.2%

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

   13. Simplified 51.2%

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

   if -1.0500000000000001e63 < x < 4.30000000000000007e110

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 54.2%

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

   6. Taylor expanded in x around 0 40.8%

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

      1. associate-*r* 40.8%

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

      2. mul-1-neg 40.8%

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

      3. distribute-lft-neg-out 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-*l* 40.8%

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

   8. Simplified 40.8%

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

      1. *-commutative 40.8%

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

      2. distribute-lft-neg-out 40.8%

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

      3. distribute-lft-neg-out 40.8%

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

      4. add-sqr-sqrt 22.5%

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

      5. sqrt-unprod 24.6%

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

      6. sqr-neg 24.6%

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

      7. sqrt-unprod 1.5%

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

      8. add-sqr-sqrt 2.7%

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

      9. associate-*l* 2.7%

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

      10. add-sqr-sqrt 1.5%

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

      11. sqrt-unprod 24.6%

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

      12. sqr-neg 24.6%

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

      13. sqrt-unprod 22.5%

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

      14. add-sqr-sqrt 40.8%

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

      15. sqrt-prod 40.9%

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

      16. *-commutative 40.9%

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

   10. Applied egg-rr 40.9%

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

       1. *-commutative 40.9%

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

       2. distribute-rgt-neg-in 40.9%

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

   12. Simplified 40.9%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+63} \lor \neg \left(x \leq 4.3 \cdot 10^{+110}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := x \cdot t\_1\\ \mathbf{if}\;x \leq -4 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \left(t\_m \cdot t\_2\right)\\ \mathbf{elif}\;x \leq 1.44 \cdot 10^{+110}:\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_2\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (* x t_1)))
   (if (<= x -4e+63)
     (* 0.5 (* t_m t_2))
     (if (<= x 1.44e+110) (* y (- t_1)) (* 0.5 t_2)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = x * t_1;
	double tmp;
	if (x <= -4e+63) {
		tmp = 0.5 * (t_m * t_2);
	} else if (x <= 1.44e+110) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * t_2;
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = x * t_1
    if (x <= (-4d+63)) then
        tmp = 0.5d0 * (t_m * t_2)
    else if (x <= 1.44d+110) then
        tmp = y * -t_1
    else
        tmp = 0.5d0 * t_2
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = Math.sqrt((z * 2.0));
	double t_2 = x * t_1;
	double tmp;
	if (x <= -4e+63) {
		tmp = 0.5 * (t_m * t_2);
	} else if (x <= 1.44e+110) {
		tmp = y * -t_1;
	} else {
		tmp = 0.5 * t_2;
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = math.sqrt((z * 2.0))
	t_2 = x * t_1
	tmp = 0
	if x <= -4e+63:
		tmp = 0.5 * (t_m * t_2)
	elif x <= 1.44e+110:
		tmp = y * -t_1
	else:
		tmp = 0.5 * t_2
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = Float64(x * t_1)
	tmp = 0.0
	if (x <= -4e+63)
		tmp = Float64(0.5 * Float64(t_m * t_2));
	elseif (x <= 1.44e+110)
		tmp = Float64(y * Float64(-t_1));
	else
		tmp = Float64(0.5 * t_2);
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = sqrt((z * 2.0));
	t_2 = x * t_1;
	tmp = 0.0;
	if (x <= -4e+63)
		tmp = 0.5 * (t_m * t_2);
	elseif (x <= 1.44e+110)
		tmp = y * -t_1;
	else
		tmp = 0.5 * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$1), $MachinePrecision]}, If[LessEqual[x, -4e+63], N[(0.5 * N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.44e+110], N[(y * (-t$95$1)), $MachinePrecision], N[(0.5 * t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000023e63

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. exp-sqrt 99.8%

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

      3. exp-prod 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 84.9%

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

      3. sqrt-unprod 84.9%

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

      4. associate-*l* 84.9%

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

      5. pow-exp 84.8%

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

      6. pow2 84.8%

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

   6. Applied egg-rr 84.8%

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

      1. expm1-def 99.2%

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

      2. expm1-log1p 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 88.2%

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

   10. Taylor expanded in t around inf 36.6%

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

       1. associate-*l* 34.9%

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

       2. *-commutative 34.9%

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

   12. Simplified 34.9%

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

   13. Taylor expanded in x around inf 28.3%

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

       1. associate-*l* 32.3%

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

       2. associate-*l* 32.3%

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

       3. *-commutative 32.3%

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

       4. unpow1/2 32.3%

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

       5. exp-to-pow 32.3%

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

       6. unpow1/2 32.3%

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

       7. exp-to-pow 32.3%

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

       8. exp-sum 32.3%

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

       9. distribute-rgt-in 32.3%

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

       10. log-prod 32.3%

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

       11. log-pow 32.3%

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

       12. unpow1/2 32.3%

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

       13. rem-exp-log 32.3%

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

   15. Simplified 32.3%

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

   if -4.00000000000000023e63 < x < 1.44e110

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 54.2%

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

   6. Taylor expanded in x around 0 40.8%

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

      1. associate-*r* 40.8%

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

      2. mul-1-neg 40.8%

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

      3. distribute-lft-neg-out 40.8%

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

      4. *-commutative 40.8%

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

      5. associate-*l* 40.8%

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

   8. Simplified 40.8%

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

      1. *-commutative 40.8%

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

      2. distribute-lft-neg-out 40.8%

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

      3. distribute-lft-neg-out 40.8%

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

      4. add-sqr-sqrt 22.5%

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

      5. sqrt-unprod 24.6%

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

      6. sqr-neg 24.6%

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

      7. sqrt-unprod 1.5%

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

      8. add-sqr-sqrt 2.7%

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

      9. associate-*l* 2.7%

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

      10. add-sqr-sqrt 1.5%

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

      11. sqrt-unprod 24.6%

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

      12. sqr-neg 24.6%

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

      13. sqrt-unprod 22.5%

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

      14. add-sqr-sqrt 40.8%

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

      15. sqrt-prod 40.9%

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

      16. *-commutative 40.9%

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

   10. Applied egg-rr 40.9%

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

       1. *-commutative 40.9%

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

       2. distribute-rgt-neg-in 40.9%

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

   12. Simplified 40.9%

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

   if 1.44e110 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 60.1%

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

      1. associate-*r* 60.1%

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

      2. *-commutative 60.1%

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

      3. *-commutative 60.1%

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

      4. fma-neg 60.1%

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

      5. associate-*l* 60.2%

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

      6. fma-neg 60.2%

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

      7. *-commutative 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in x around inf 58.0%

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

      1. associate-*l* 58.0%

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

   10. Simplified 58.0%

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

   11. Taylor expanded in x around 0 58.0%

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

       1. associate-*l* 58.0%

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

       2. *-commutative 58.0%

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

       3. unpow1/2 58.0%

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

       4. exp-to-pow 55.2%

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

       5. unpow1/2 55.2%

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

       6. exp-to-pow 55.2%

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

       7. exp-sum 55.3%

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

       8. distribute-rgt-in 55.3%

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

       9. log-prod 55.5%

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

       10. log-pow 55.5%

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

       11. unpow1/2 55.5%

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

       12. rem-exp-log 58.1%

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

   13. Simplified 58.1%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \left(t \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\right)\\ \mathbf{elif}\;x \leq 1.44 \cdot 10^{+110}:\\ \;\;\;\;y \cdot \left(-\sqrt{z \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.4% accurate, 1.9× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* z 2.0))))
   (if (<= t_m 1.0) (* t_1 t_2) (* t_1 (* t_2 t_m)))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t_m <= 1.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_1 * (t_2 * t_m);
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    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_m <= 1.0d0) then
        tmp = t_1 * t_2
    else
        tmp = t_1 * (t_2 * t_m)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = (x * 0.5) - y;
	double t_2 = Math.sqrt((z * 2.0));
	double tmp;
	if (t_m <= 1.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_1 * (t_2 * t_m);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((z * 2.0))
	tmp = 0
	if t_m <= 1.0:
		tmp = t_1 * t_2
	else:
		tmp = t_1 * (t_2 * t_m)
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t_m <= 1.0)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(t_1 * Float64(t_2 * t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t_m <= 1.0)
		tmp = t_1 * t_2;
	else
		tmp = t_1 * (t_2 * t_m);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := 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$95$m, 1.0], N[(t$95$1 * t$95$2), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.3%

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

      3. exp-sqrt 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 69.8%

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

   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. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 76.2%

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

   10. Taylor expanded in t around inf 62.3%

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

       1. associate-*l* 47.0%

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

       2. *-commutative 47.0%

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

   12. Simplified 47.0%

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

   13. Taylor expanded in t around 0 62.3%

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

       1. *-commutative 62.3%

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

       2. *-commutative 62.3%

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

       3. sub-neg 62.3%

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

       4. *-commutative 62.3%

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

       5. sub-neg 62.3%

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

       6. associate-*r* 47.0%

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

   15. Simplified 48.4%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_m \cdot \left(x \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t_m 1.15e+89) (* (- (* x 0.5) y) t_1) (* 0.5 (* t_m (* x t_1))))))

C

t_m = fabs(t);
double code(double x, double y, double z, double t_m) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t_m <= 1.15e+89) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = 0.5 * (t_m * (x * t_1));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(x, y, z, t_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t_m <= 1.15d+89) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = 0.5d0 * (t_m * (x * t_1))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double x, double y, double z, double t_m) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t_m <= 1.15e+89) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = 0.5 * (t_m * (x * t_1));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(x, y, z, t_m):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t_m <= 1.15e+89:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = 0.5 * (t_m * (x * t_1))
	return tmp

Julia

t_m = abs(t)
function code(x, y, z, t_m)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t_m <= 1.15e+89)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(0.5 * Float64(t_m * Float64(x * t_1)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(x, y, z, t_m)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t_m <= 1.15e+89)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = 0.5 * (t_m * (x * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$m, 1.15e+89], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(0.5 * N[(t$95$m * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1499999999999999e89

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.4%

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

      3. exp-sqrt 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around 0 65.8%

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

   if 1.1499999999999999e89 < t

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. pow-exp 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 87.6%

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

   10. Taylor expanded in t around inf 68.1%

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

       1. associate-*l* 54.6%

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

       2. *-commutative 54.6%

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

   12. Simplified 54.6%

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

   13. Taylor expanded in x around inf 38.5%

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

       1. associate-*l* 33.3%

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

       2. associate-*l* 33.3%

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

       3. *-commutative 33.3%

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

       4. unpow1/2 33.3%

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

       5. exp-to-pow 33.3%

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

       6. unpow1/2 33.3%

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

       7. exp-to-pow 33.3%

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

       8. exp-sum 33.3%

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

       9. distribute-rgt-in 33.3%

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

       10. log-prod 33.3%

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

       11. log-pow 33.3%

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

       12. unpow1/2 33.3%

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

       13. rem-exp-log 33.3%

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

   15. Simplified 33.3%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t \cdot \left(x \cdot \sqrt{z \cdot 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.8% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m) :precision binary64 (* y (- (sqrt (* z 2.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(y * (-N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l* 99.5%

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

   3. exp-sqrt 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in t around 0 54.3%

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

6. Taylor expanded in x around 0 27.8%

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

   1. associate-*r* 27.8%

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

   2. mul-1-neg 27.8%

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

   3. distribute-lft-neg-out 27.8%

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

   4. *-commutative 27.8%

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

   5. associate-*l* 27.8%

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

8. Simplified 27.8%

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

   1. *-commutative 27.8%

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

   2. distribute-lft-neg-out 27.8%

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

   3. distribute-lft-neg-out 27.8%

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

   4. add-sqr-sqrt 14.4%

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

   5. sqrt-unprod 17.1%

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

   6. sqr-neg 17.1%

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

   7. sqrt-unprod 1.3%

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

   8. add-sqr-sqrt 2.4%

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

   9. associate-*l* 2.4%

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

   10. add-sqr-sqrt 1.3%

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

   11. sqrt-unprod 17.1%

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

   12. sqr-neg 17.1%

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

   13. sqrt-unprod 14.4%

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

   14. add-sqr-sqrt 27.8%

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

   15. sqrt-prod 27.9%

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

   16. *-commutative 27.9%

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

10. Applied egg-rr 27.9%

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

    1. *-commutative 27.9%

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

    2. distribute-rgt-neg-in 27.9%

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

12. Simplified 27.9%

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

13. Final simplification 27.9%

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

Alternative 15: 2.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ y \cdot \sqrt{z \cdot 2} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (x y z t_m) :precision binary64 (* y (sqrt (* z 2.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z_, t$95$m_] := N[(y * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l* 99.5%

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

   3. exp-sqrt 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in t around 0 54.3%

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

6. Taylor expanded in x around 0 27.8%

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

   1. associate-*r* 27.8%

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

   2. mul-1-neg 27.8%

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

   3. distribute-lft-neg-out 27.8%

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

   4. *-commutative 27.8%

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

   5. associate-*l* 27.8%

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

8. Simplified 27.8%

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

   1. expm1-log1p-u 16.6%

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

   2. expm1-udef 11.0%

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

   3. *-commutative 11.0%

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

   4. associate-*l* 11.0%

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

   5. add-sqr-sqrt 9.5%

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

   6. sqrt-unprod 15.1%

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

   7. sqr-neg 15.1%

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

   8. sqrt-unprod 1.2%

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

   9. add-sqr-sqrt 2.3%

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

   10. sqrt-prod 2.3%

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

   11. *-commutative 2.3%

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

10. Applied egg-rr 2.3%

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

    1. expm1-def 2.2%

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

    2. expm1-log1p 2.8%

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

12. Simplified 2.8%

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

13. Final simplification 2.8%

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

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024031 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

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