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

Percentage Accurate: 99.2% → 99.8%

Time: 18.5s

Alternatives: 14

Speedup: 1.0×

• 44.4% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   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. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   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. Step-by-step derivation

   1. associate-*r* 99.4%

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

   2. *-commutative 99.4%

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

   3. expm1-log1p-u 58.8%

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

   4. expm1-udef 44.5%

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

5. Applied egg-rr 44.5%

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

   1. expm1-def 58.8%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l* 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 6 \cdot 10^{+18}:\\ \;\;\;\;t_1 \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left({t}^{2} \cdot \left(0.5 \cdot t_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 6e+18)
     (* t_1 (sqrt (* (* z 2.0) (fma t t 1.0))))
     (* (sqrt (* z 2.0)) (* (pow t 2.0) (* 0.5 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 6e+18) {
		tmp = t_1 * sqrt(((z * 2.0) * fma(t, t, 1.0)));
	} else {
		tmp = sqrt((z * 2.0)) * (pow(t, 2.0) * (0.5 * t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 6e+18)
		tmp = Float64(t_1 * sqrt(Float64(Float64(z * 2.0) * fma(t, t, 1.0))));
	else
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64((t ^ 2.0) * Float64(0.5 * t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 6e+18], N[(t$95$1 * N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[t, 2.0], $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 6e18

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      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. Step-by-step derivation

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. expm1-log1p-u 60.5%

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

      4. expm1-udef 42.2%

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

   5. Applied egg-rr 42.2%

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

      1. expm1-def 60.5%

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

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 87.8%

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

      1. distribute-lft-out 87.8%

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

      2. *-commutative 87.8%

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

   10. Simplified 87.8%

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

       1. expm1-log1p-u 86.2%

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

       2. expm1-udef 52.2%

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

       3. +-commutative 52.2%

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

       4. fma-def 52.2%

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

   12. Applied egg-rr 52.2%

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

       1. expm1-def 86.2%

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

       2. expm1-log1p 87.8%

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

       3. *-commutative 87.8%

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

       4. fma-udef 87.8%

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

       5. *-commutative 87.8%

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

       6. distribute-lft1-in 87.8%

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

       7. associate-*l* 87.8%

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

       8. unpow2 87.8%

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

       9. fma-def 87.8%

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

   14. Simplified 87.8%

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

   if 6e18 < t

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 86.5%

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

      1. *-commutative 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in t around inf 86.5%

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

      1. fma-neg 86.5%

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

      2. *-commutative 86.5%

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

      3. associate-*r* 86.5%

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

      4. *-commutative 86.5%

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

      5. fma-neg 86.5%

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

   9. Simplified 86.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left({t}^{2} \cdot \left(0.5 \cdot \left(x \cdot 0.5 - y\right)\right)\right)\\ \end{array} \]

Alternative 4: 63.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot t_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {t_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t_1 \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 6.5e+18)
     (* (sqrt (* z 2.0)) t_1)
     (if (<= t 3.5e+151)
       (sqrt (* (* z 2.0) (pow t_1 2.0)))
       (* t (* t_1 (* (sqrt z) (sqrt 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 6.5e+18) {
		tmp = sqrt((z * 2.0)) * t_1;
	} else if (t <= 3.5e+151) {
		tmp = sqrt(((z * 2.0) * pow(t_1, 2.0)));
	} else {
		tmp = t * (t_1 * (sqrt(z) * sqrt(2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 6.5d+18) then
        tmp = sqrt((z * 2.0d0)) * t_1
    else if (t <= 3.5d+151) then
        tmp = sqrt(((z * 2.0d0) * (t_1 ** 2.0d0)))
    else
        tmp = t * (t_1 * (sqrt(z) * sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 6.5e+18) {
		tmp = Math.sqrt((z * 2.0)) * t_1;
	} else if (t <= 3.5e+151) {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(t_1, 2.0)));
	} else {
		tmp = t * (t_1 * (Math.sqrt(z) * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 6.5e+18:
		tmp = math.sqrt((z * 2.0)) * t_1
	elif t <= 3.5e+151:
		tmp = math.sqrt(((z * 2.0) * math.pow(t_1, 2.0)))
	else:
		tmp = t * (t_1 * (math.sqrt(z) * math.sqrt(2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 6.5e+18)
		tmp = Float64(sqrt(Float64(z * 2.0)) * t_1);
	elseif (t <= 3.5e+151)
		tmp = sqrt(Float64(Float64(z * 2.0) * (t_1 ^ 2.0)));
	else
		tmp = Float64(t * Float64(t_1 * Float64(sqrt(z) * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 6.5e+18)
		tmp = sqrt((z * 2.0)) * t_1;
	elseif (t <= 3.5e+151)
		tmp = sqrt(((z * 2.0) * (t_1 ^ 2.0)));
	else
		tmp = t * (t_1 * (sqrt(z) * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 6.5e+18], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 3.5e+151], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t * N[(t$95$1 * N[(N[Sqrt[z], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.5e18

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      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. Taylor expanded in t around 0 71.9%

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

   if 6.5e18 < t < 3.5000000000000003e151

   1. Initial program 95.8%

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

      1. *-commutative 95.8%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 16.9%

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

      1. *-commutative 16.9%

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

      2. associate-*l* 16.9%

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

   6. Simplified 16.9%

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

      1. add-sqr-sqrt 6.3%

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

      2. sqrt-unprod 25.6%

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

      3. swap-sqr 25.6%

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

      4. rem-square-sqrt 25.6%

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

      5. *-commutative 25.6%

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

      6. *-commutative 25.6%

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

      7. *-commutative 25.6%

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

      8. *-commutative 25.6%

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

      9. swap-sqr 25.5%

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

      10. add-sqr-sqrt 25.5%

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

      11. pow2 25.5%

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

      12. *-commutative 25.5%

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

      13. fma-neg 25.5%

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

   8. Applied egg-rr 25.5%

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

      1. associate-*r* 25.5%

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

      2. *-commutative 25.5%

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

      3. fma-neg 25.5%

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

   10. Simplified 25.5%

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

   if 3.5000000000000003e151 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. expm1-log1p-u 60.6%

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

      4. expm1-udef 60.6%

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

   5. Applied egg-rr 60.6%

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

      1. expm1-def 60.6%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in t around inf 62.4%

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

       1. associate-*l* 51.2%

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

       2. *-commutative 51.2%

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

       3. associate-*l* 51.2%

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

       4. *-commutative 51.2%

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

   13. Simplified 51.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\\ \end{array} \]

Alternative 5: 65.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {t_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(t_1 \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 4e+18)
     (* (sqrt (* z 2.0)) t_1)
     (if (<= t 4.2e+78)
       (sqrt (* (* z 2.0) (pow t_1 2.0)))
       (* (sqrt z) (* t_1 (* t (sqrt 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 4e+18) {
		tmp = sqrt((z * 2.0)) * t_1;
	} else if (t <= 4.2e+78) {
		tmp = sqrt(((z * 2.0) * pow(t_1, 2.0)));
	} else {
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 4d+18) then
        tmp = sqrt((z * 2.0d0)) * t_1
    else if (t <= 4.2d+78) then
        tmp = sqrt(((z * 2.0d0) * (t_1 ** 2.0d0)))
    else
        tmp = sqrt(z) * (t_1 * (t * sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 4e+18) {
		tmp = Math.sqrt((z * 2.0)) * t_1;
	} else if (t <= 4.2e+78) {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(t_1, 2.0)));
	} else {
		tmp = Math.sqrt(z) * (t_1 * (t * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 4e+18:
		tmp = math.sqrt((z * 2.0)) * t_1
	elif t <= 4.2e+78:
		tmp = math.sqrt(((z * 2.0) * math.pow(t_1, 2.0)))
	else:
		tmp = math.sqrt(z) * (t_1 * (t * math.sqrt(2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 4e+18)
		tmp = Float64(sqrt(Float64(z * 2.0)) * t_1);
	elseif (t <= 4.2e+78)
		tmp = sqrt(Float64(Float64(z * 2.0) * (t_1 ^ 2.0)));
	else
		tmp = Float64(sqrt(z) * Float64(t_1 * Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 4e+18)
		tmp = sqrt((z * 2.0)) * t_1;
	elseif (t <= 4.2e+78)
		tmp = sqrt(((z * 2.0) * (t_1 ^ 2.0)));
	else
		tmp = sqrt(z) * (t_1 * (t * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 4e+18], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 4.2e+78], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(t$95$1 * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4e18

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      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. Taylor expanded in t around 0 71.9%

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

   if 4e18 < t < 4.2000000000000002e78

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 14.2%

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

      1. *-commutative 14.2%

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

      2. associate-*l* 14.2%

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

   6. Simplified 14.2%

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

      1. add-sqr-sqrt 12.2%

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

      2. sqrt-unprod 36.8%

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

      3. swap-sqr 36.8%

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

      4. rem-square-sqrt 36.8%

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

      5. *-commutative 36.8%

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

      6. *-commutative 36.8%

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

      7. *-commutative 36.8%

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

      8. *-commutative 36.8%

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

      9. swap-sqr 36.8%

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

      10. add-sqr-sqrt 36.8%

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

      11. pow2 36.8%

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

      12. *-commutative 36.8%

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

      13. fma-neg 36.8%

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

   8. Applied egg-rr 36.8%

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

      1. associate-*r* 36.8%

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

      2. *-commutative 36.8%

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

      3. fma-neg 36.8%

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

   10. Simplified 36.8%

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

   if 4.2000000000000002e78 < t

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

      3. expm1-log1p-u 52.2%

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

      4. expm1-udef 52.2%

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

   5. Applied egg-rr 52.2%

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

      1. expm1-def 52.2%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 83.5%

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

      1. distribute-lft-out 83.5%

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

      2. *-commutative 83.5%

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

   10. Simplified 83.5%

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

   11. Taylor expanded in t around inf 56.4%

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

       1. *-commutative 56.4%

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

       2. associate-*r* 56.4%

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

       3. *-commutative 56.4%

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

   13. Simplified 56.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \end{array} \]

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Final simplification 99.4%

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

Alternative 7: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   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. Step-by-step derivation

   1. associate-*r* 99.4%

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

   2. *-commutative 99.4%

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

   3. expm1-log1p-u 58.8%

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

   4. expm1-udef 44.5%

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

5. Applied egg-rr 44.5%

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

   1. expm1-def 58.8%

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

   2. expm1-log1p 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l* 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in t around 0 85.0%

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

   1. distribute-lft-out 85.0%

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

   2. *-commutative 85.0%

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

10. Simplified 85.0%

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

    1. expm1-log1p-u 83.7%

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

    2. expm1-udef 57.3%

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

    3. +-commutative 57.3%

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

    4. fma-def 57.3%

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

12. Applied egg-rr 57.3%

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

    1. expm1-def 83.7%

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

    2. expm1-log1p 85.0%

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

    3. *-commutative 85.0%

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

    4. fma-udef 85.0%

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

    5. *-commutative 85.0%

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

    6. distribute-lft1-in 85.0%

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

    7. associate-*l* 85.0%

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

    8. unpow2 85.0%

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

    9. fma-def 85.0%

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

14. Simplified 85.0%

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

15. Final simplification 85.0%

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

Alternative 8: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+142} \lor \neg \left(t \leq 1.7 \cdot 10^{+223}\right):\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.5e+18)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (if (or (<= t 2.15e+142) (not (<= t 1.7e+223)))
     (sqrt (* z (* 0.5 (pow x 2.0))))
     (sqrt (* (* z 2.0) (* y (- y x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+18) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else if ((t <= 2.15e+142) || !(t <= 1.7e+223)) {
		tmp = sqrt((z * (0.5 * pow(x, 2.0))));
	} else {
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 3.5d+18) then
        tmp = sqrt((z * 2.0d0)) * ((x * 0.5d0) - y)
    else if ((t <= 2.15d+142) .or. (.not. (t <= 1.7d+223))) then
        tmp = sqrt((z * (0.5d0 * (x ** 2.0d0))))
    else
        tmp = sqrt(((z * 2.0d0) * (y * (y - x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.5e+18) {
		tmp = Math.sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else if ((t <= 2.15e+142) || !(t <= 1.7e+223)) {
		tmp = Math.sqrt((z * (0.5 * Math.pow(x, 2.0))));
	} else {
		tmp = Math.sqrt(((z * 2.0) * (y * (y - x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 3.5e+18:
		tmp = math.sqrt((z * 2.0)) * ((x * 0.5) - y)
	elif (t <= 2.15e+142) or not (t <= 1.7e+223):
		tmp = math.sqrt((z * (0.5 * math.pow(x, 2.0))))
	else:
		tmp = math.sqrt(((z * 2.0) * (y * (y - x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.5e+18)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	elseif ((t <= 2.15e+142) || !(t <= 1.7e+223))
		tmp = sqrt(Float64(z * Float64(0.5 * (x ^ 2.0))));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * Float64(y * Float64(y - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 3.5e+18)
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	elseif ((t <= 2.15e+142) || ~((t <= 1.7e+223)))
		tmp = sqrt((z * (0.5 * (x ^ 2.0))));
	else
		tmp = sqrt(((z * 2.0) * (y * (y - x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3.5e+18], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 2.15e+142], N[Not[LessEqual[t, 1.7e+223]], $MachinePrecision]], N[Sqrt[N[(z * N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.5e18

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      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. Taylor expanded in t around 0 71.9%

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

   if 3.5e18 < t < 2.15000000000000006e142 or 1.6999999999999999e223 < t

   1. Initial program 97.4%

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

      1. *-commutative 97.4%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 11.8%

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

      1. *-commutative 11.8%

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

      2. associate-*l* 11.8%

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

   6. Simplified 11.8%

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

      1. add-sqr-sqrt 7.3%

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

      2. sqrt-unprod 24.1%

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

      3. swap-sqr 24.1%

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

      4. rem-square-sqrt 24.1%

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

      5. *-commutative 24.1%

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

      6. *-commutative 24.1%

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

      7. *-commutative 24.1%

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

      8. *-commutative 24.1%

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

      9. swap-sqr 24.0%

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

      10. add-sqr-sqrt 24.0%

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

      11. pow2 24.0%

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

      12. *-commutative 24.0%

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

      13. fma-neg 24.0%

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

   8. Applied egg-rr 24.0%

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

      1. associate-*r* 24.0%

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

      2. *-commutative 24.0%

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

      3. fma-neg 24.0%

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

   10. Simplified 24.0%

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

   11. Taylor expanded in x around inf 21.6%

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

       1. associate-*r* 21.6%

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

       2. *-commutative 21.6%

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

   13. Simplified 21.6%

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

   if 2.15000000000000006e142 < t < 1.6999999999999999e223

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 20.1%

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

      1. *-commutative 20.1%

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

      2. associate-*l* 20.1%

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

   6. Simplified 20.1%

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

      1. add-sqr-sqrt 7.6%

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

      2. sqrt-unprod 28.6%

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

      3. swap-sqr 28.6%

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

      4. rem-square-sqrt 28.6%

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

      5. *-commutative 28.6%

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

      6. *-commutative 28.6%

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

      7. *-commutative 28.6%

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

      8. *-commutative 28.6%

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

      9. swap-sqr 28.6%

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

      10. add-sqr-sqrt 28.6%

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

      11. pow2 28.6%

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

      12. *-commutative 28.6%

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

      13. fma-neg 28.6%

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

   8. Applied egg-rr 28.6%

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

      1. associate-*r* 28.6%

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

      2. *-commutative 28.6%

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

      3. fma-neg 28.6%

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

   10. Simplified 28.6%

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

   11. Taylor expanded in x around 0 22.9%

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

       1. +-commutative 22.9%

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

       2. unpow2 22.9%

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

       3. associate-*r* 22.9%

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

       4. distribute-rgt-out 22.9%

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

       5. mul-1-neg 22.9%

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

   13. Simplified 22.9%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+142} \lor \neg \left(t \leq 1.7 \cdot 10^{+223}\right):\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \end{array} \]

Alternative 9: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+129} \lor \neg \left(t \leq 1.8 \cdot 10^{+224}\right):\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 7.2e+18)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (if (or (<= t 1.6e+129) (not (<= t 1.8e+224)))
     (sqrt (* z (* 0.5 (pow x 2.0))))
     (sqrt (* (* z 2.0) (pow y 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 7.2e+18) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else if ((t <= 1.6e+129) || !(t <= 1.8e+224)) {
		tmp = sqrt((z * (0.5 * pow(x, 2.0))));
	} else {
		tmp = sqrt(((z * 2.0) * pow(y, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 7.2d+18) then
        tmp = sqrt((z * 2.0d0)) * ((x * 0.5d0) - y)
    else if ((t <= 1.6d+129) .or. (.not. (t <= 1.8d+224))) then
        tmp = sqrt((z * (0.5d0 * (x ** 2.0d0))))
    else
        tmp = sqrt(((z * 2.0d0) * (y ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 7.2e+18) {
		tmp = Math.sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else if ((t <= 1.6e+129) || !(t <= 1.8e+224)) {
		tmp = Math.sqrt((z * (0.5 * Math.pow(x, 2.0))));
	} else {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(y, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 7.2e+18:
		tmp = math.sqrt((z * 2.0)) * ((x * 0.5) - y)
	elif (t <= 1.6e+129) or not (t <= 1.8e+224):
		tmp = math.sqrt((z * (0.5 * math.pow(x, 2.0))))
	else:
		tmp = math.sqrt(((z * 2.0) * math.pow(y, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 7.2e+18)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	elseif ((t <= 1.6e+129) || !(t <= 1.8e+224))
		tmp = sqrt(Float64(z * Float64(0.5 * (x ^ 2.0))));
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * (y ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 7.2e+18)
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	elseif ((t <= 1.6e+129) || ~((t <= 1.8e+224)))
		tmp = sqrt((z * (0.5 * (x ^ 2.0))));
	else
		tmp = sqrt(((z * 2.0) * (y ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 7.2e+18], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 1.6e+129], N[Not[LessEqual[t, 1.8e+224]], $MachinePrecision]], N[Sqrt[N[(z * N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e18

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      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. Taylor expanded in t around 0 71.9%

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

   if 7.2e18 < t < 1.6000000000000001e129 or 1.8e224 < t

   1. Initial program 97.3%

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

      1. *-commutative 97.3%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 12.3%

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

      1. *-commutative 12.3%

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

      2. associate-*l* 12.3%

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

   6. Simplified 12.3%

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

      1. add-sqr-sqrt 7.7%

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

      2. sqrt-unprod 25.3%

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

      3. swap-sqr 25.3%

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

      4. rem-square-sqrt 25.3%

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

      5. *-commutative 25.3%

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

      6. *-commutative 25.3%

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

      7. *-commutative 25.3%

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

      8. *-commutative 25.3%

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

      9. swap-sqr 25.3%

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

      10. add-sqr-sqrt 25.3%

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

      11. pow2 25.3%

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

      12. *-commutative 25.3%

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

      13. fma-neg 25.3%

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

   8. Applied egg-rr 25.3%

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

      1. associate-*r* 25.3%

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

      2. *-commutative 25.3%

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

      3. fma-neg 25.3%

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

   10. Simplified 25.3%

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

   11. Taylor expanded in x around inf 22.7%

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

       1. associate-*r* 22.7%

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

       2. *-commutative 22.7%

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

   13. Simplified 22.7%

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

   if 1.6000000000000001e129 < t < 1.8e224

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 18.4%

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

   5. Taylor expanded in x around 0 8.0%

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

      1. mul-1-neg 8.0%

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

      2. associate-*l* 8.0%

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

      3. *-commutative 8.0%

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

      4. distribute-rgt-neg-in 8.0%

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

      5. *-commutative 8.0%

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

   7. Simplified 8.0%

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

      1. add-sqr-sqrt 1.7%

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

      2. sqrt-unprod 20.8%

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

      3. *-commutative 20.8%

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

      4. *-commutative 20.8%

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

      5. swap-sqr 20.8%

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

      6. *-commutative 20.8%

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

      7. sqrt-prod 20.8%

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

      8. *-commutative 20.8%

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

      9. sqrt-prod 20.8%

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

      10. sqr-neg 20.8%

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

      11. add-sqr-sqrt 20.8%

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

      12. *-commutative 20.8%

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

      13. pow2 20.8%

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

   9. Applied egg-rr 20.8%

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

       1. *-commutative 20.8%

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

       2. *-commutative 20.8%

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

   11. Simplified 20.8%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+129} \lor \neg \left(t \leq 1.8 \cdot 10^{+224}\right):\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {y}^{2}}\\ \end{array} \]

Alternative 10: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 2.95 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {t_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 2.95e+18)
     (* (sqrt (* z 2.0)) t_1)
     (sqrt (* (* z 2.0) (pow t_1 2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 2.95e+18) {
		tmp = sqrt((z * 2.0)) * t_1;
	} else {
		tmp = sqrt(((z * 2.0) * pow(t_1, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 2.95d+18) then
        tmp = sqrt((z * 2.0d0)) * t_1
    else
        tmp = sqrt(((z * 2.0d0) * (t_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 2.95e+18) {
		tmp = Math.sqrt((z * 2.0)) * t_1;
	} else {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(t_1, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 2.95e+18:
		tmp = math.sqrt((z * 2.0)) * t_1
	else:
		tmp = math.sqrt(((z * 2.0) * math.pow(t_1, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 2.95e+18)
		tmp = Float64(sqrt(Float64(z * 2.0)) * t_1);
	else
		tmp = sqrt(Float64(Float64(z * 2.0) * (t_1 ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 2.95e+18)
		tmp = sqrt((z * 2.0)) * t_1;
	else
		tmp = sqrt(((z * 2.0) * (t_1 ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 2.95e+18], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.95e18

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      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. Taylor expanded in t around 0 71.9%

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

   if 2.95e18 < t

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l* 100.0%

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

      3. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 14.4%

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

      1. *-commutative 14.4%

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

      2. associate-*l* 14.4%

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

   6. Simplified 14.4%

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

      1. add-sqr-sqrt 7.4%

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

      2. sqrt-unprod 25.5%

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

      3. swap-sqr 25.5%

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

      4. rem-square-sqrt 25.5%

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

      5. *-commutative 25.5%

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

      6. *-commutative 25.5%

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

      7. *-commutative 25.5%

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

      8. *-commutative 25.5%

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

      9. swap-sqr 25.5%

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

      10. add-sqr-sqrt 25.5%

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

      11. pow2 25.5%

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

      12. *-commutative 25.5%

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

      13. fma-neg 25.5%

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

   8. Applied egg-rr 25.5%

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

      1. associate-*r* 25.5%

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

      2. *-commutative 25.5%

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

      3. fma-neg 25.5%

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

   10. Simplified 25.5%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.95 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \end{array} \]

Alternative 11: 42.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+89} \lor \neg \left(y \leq 7.2 \cdot 10^{-38}\right):\\ \;\;\;\;t_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= y -2.1e+89) (not (<= y 7.2e-38)))
     (* t_1 (- y))
     (* t_1 (* x 0.5)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((y <= -2.1e+89) || !(y <= 7.2e-38)) {
		tmp = t_1 * -y;
	} else {
		tmp = t_1 * (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((y <= (-2.1d+89)) .or. (.not. (y <= 7.2d-38))) then
        tmp = t_1 * -y
    else
        tmp = t_1 * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((y <= -2.1e+89) || !(y <= 7.2e-38)) {
		tmp = t_1 * -y;
	} else {
		tmp = t_1 * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (y <= -2.1e+89) or not (y <= 7.2e-38):
		tmp = t_1 * -y
	else:
		tmp = t_1 * (x * 0.5)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((y <= -2.1e+89) || !(y <= 7.2e-38))
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = Float64(t_1 * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((y <= -2.1e+89) || ~((y <= 7.2e-38)))
		tmp = t_1 * -y;
	else
		tmp = t_1 * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y, -2.1e+89], N[Not[LessEqual[y, 7.2e-38]], $MachinePrecision]], N[(t$95$1 * (-y)), $MachinePrecision], N[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999986e89 or 7.2000000000000001e-38 < 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. Taylor expanded in t around 0 58.9%

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

   5. Taylor expanded in x around 0 46.1%

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

      1. mul-1-neg 46.1%

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

      2. associate-*l* 46.1%

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

      3. *-commutative 46.1%

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

      4. distribute-rgt-neg-in 46.1%

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

      5. *-commutative 46.1%

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

   7. Simplified 46.1%

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

      1. distribute-rgt-neg-out 46.1%

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

      2. neg-sub0 46.1%

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

      3. sqrt-unprod 46.3%

         \[\leadsto 0 - y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

   9. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{0 - y \cdot \sqrt{2 \cdot z}} \]
   10. Step-by-step derivation

       1. neg-sub0 46.3%

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

       2. distribute-lft-neg-in 46.3%

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

       3. *-commutative 46.3%

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

   11. Simplified 46.3%

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

   if -2.09999999999999986e89 < y < 7.2000000000000001e-38

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      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. Taylor expanded in t around 0 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l* 59.1%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in x around inf 49.4%

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

      1. associate-*r* 49.4%

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

      2. *-commutative 49.4%

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

   9. Simplified 49.4%

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

       1. expm1-log1p-u 30.3%

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

       2. expm1-udef 18.1%

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

       3. *-commutative 18.1%

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

       4. *-commutative 18.1%

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

       5. associate-*l* 18.1%

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

       6. sqrt-prod 18.1%

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

       7. *-commutative 18.1%

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

   11. Applied egg-rr 18.1%

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

       1. expm1-def 30.4%

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

       2. expm1-log1p 49.6%

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

       3. *-commutative 49.6%

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

   13. Simplified 49.6%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+89} \lor \neg \left(y \leq 7.2 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 56.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   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. Taylor expanded in t around 0 59.1%

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

5. Final simplification 59.1%

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

Alternative 13: 29.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   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. Taylor expanded in t around 0 59.1%

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

5. Taylor expanded in x around 0 27.2%

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

   1. mul-1-neg 27.2%

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

   2. associate-*l* 27.2%

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

   3. *-commutative 27.2%

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

   4. distribute-rgt-neg-in 27.2%

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

   5. *-commutative 27.2%

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

7. Simplified 27.2%

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

   1. distribute-rgt-neg-out 27.2%

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

   2. neg-sub0 27.2%

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

   3. sqrt-unprod 27.2%

      \[\leadsto 0 - y \cdot \color{blue}{\sqrt{2 \cdot z}} \]

9. Applied egg-rr 27.2%

   \[\leadsto \color{blue}{0 - y \cdot \sqrt{2 \cdot z}} \]
10. Step-by-step derivation

    1. neg-sub0 27.2%

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

    2. distribute-lft-neg-in 27.2%

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

    3. *-commutative 27.2%

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

11. Simplified 27.2%

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

12. Final simplification 27.2%

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

Alternative 14: 2.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{z \cdot 2} \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   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. Taylor expanded in t around 0 59.1%

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

5. Taylor expanded in x around 0 27.2%

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

   1. mul-1-neg 27.2%

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

   2. associate-*l* 27.2%

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

   3. *-commutative 27.2%

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

   4. distribute-rgt-neg-in 27.2%

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

   5. *-commutative 27.2%

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

7. Simplified 27.2%

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

   1. expm1-log1p-u 17.2%

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

   2. expm1-udef 10.2%

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

9. Applied egg-rr 2.2%

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

    1. expm1-def 2.1%

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

    2. expm1-log1p 2.3%

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

    3. *-commutative 2.3%

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

11. Simplified 2.3%

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

12. Final simplification 2.3%

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

Developer target: 99.2% 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 2023321 
(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))))