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

Percentage Accurate: 99.4% → 98.8%

Time: 14.7s

Alternatives: 11

Speedup: 1.0×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (* (- (* (* x 0.5) t_1) (* t_1 y)) (exp (/ (* t t) 2.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	return (((x * 0.5) * t_1) - (t_1 * y)) * 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
    real(8) :: t_1
    t_1 = sqrt((z * 2.0d0))
    code = (((x * 0.5d0) * t_1) - (t_1 * y)) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	return (((x * 0.5) * t_1) - (t_1 * y)) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	return (((x * 0.5) * t_1) - (t_1 * y)) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	return Float64(Float64(Float64(Float64(x * 0.5) * t_1) - Float64(t_1 * y)) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = (((x * 0.5) * t_1) - (t_1 * y)) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. *-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. sub-neg 99.8%

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

   3. distribute-rgt-in 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {t\_2}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (- (* x 0.5) y)))
   (if (<= t 4.2e+19)
     (* t_1 t_2)
     (if (<= t 8e+210)
       (sqrt (* (* z 2.0) (pow t_2 2.0)))
       (* t_1 (* x (- 0.5 (/ y x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = (x * 0.5) - y;
	double tmp;
	if (t <= 4.2e+19) {
		tmp = t_1 * t_2;
	} else if (t <= 8e+210) {
		tmp = sqrt(((z * 2.0) * pow(t_2, 2.0)));
	} else {
		tmp = t_1 * (x * (0.5 - (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = (x * 0.5d0) - y
    if (t <= 4.2d+19) then
        tmp = t_1 * t_2
    else if (t <= 8d+210) then
        tmp = sqrt(((z * 2.0d0) * (t_2 ** 2.0d0)))
    else
        tmp = t_1 * (x * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double t_2 = (x * 0.5) - y;
	double tmp;
	if (t <= 4.2e+19) {
		tmp = t_1 * t_2;
	} else if (t <= 8e+210) {
		tmp = Math.sqrt(((z * 2.0) * Math.pow(t_2, 2.0)));
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	t_2 = (x * 0.5) - y
	tmp = 0
	if t <= 4.2e+19:
		tmp = t_1 * t_2
	elif t <= 8e+210:
		tmp = math.sqrt(((z * 2.0) * math.pow(t_2, 2.0)))
	else:
		tmp = t_1 * (x * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 4.2e+19)
		tmp = Float64(t_1 * t_2);
	elseif (t <= 8e+210)
		tmp = sqrt(Float64(Float64(z * 2.0) * (t_2 ^ 2.0)));
	else
		tmp = Float64(t_1 * Float64(x * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	t_2 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 4.2e+19)
		tmp = t_1 * t_2;
	elseif (t <= 8e+210)
		tmp = sqrt(((z * 2.0) * (t_2 ^ 2.0)));
	else
		tmp = t_1 * (x * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 4.2e+19], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t, 8e+210], N[Sqrt[N[(N[(z * 2.0), $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.2e19

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 69.9%

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

      1. *-rgt-identity 69.9%

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

      2. *-commutative 69.9%

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

      3. sub-neg 69.9%

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

      4. sub-neg 69.9%

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

      5. add-sqr-sqrt 38.8%

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

      6. sqrt-unprod 55.0%

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

      7. sqr-neg 55.0%

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

      8. sqrt-unprod 19.4%

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

      9. add-sqr-sqrt 42.0%

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

      10. distribute-rgt-out-- 42.0%

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

      11. *-commutative 42.0%

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

      12. cancel-sign-sub-inv 42.0%

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

      13. associate-*l* 42.0%

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

      14. add-sqr-sqrt 19.4%

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

      15. sqrt-unprod 55.0%

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

      16. sqr-neg 55.0%

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

      17. sqrt-unprod 38.8%

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

      18. add-sqr-sqrt 70.0%

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

   5. Applied egg-rr 70.0%

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

      1. fma-define 70.0%

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

      2. distribute-lft-neg-out 70.0%

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

      3. add-sqr-sqrt 70.0%

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

      4. sqr-neg 70.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 42.0%

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

      7. fma-neg 42.0%

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

      8. *-commutative 42.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 70.0%

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

      11. sqr-neg 70.0%

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

      12. add-sqr-sqrt 70.0%

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

   7. Applied egg-rr 70.0%

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

      1. associate-*r* 70.0%

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

      2. *-commutative 70.0%

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

      3. distribute-rgt-out-- 69.9%

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

   9. Simplified 69.9%

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

   if 4.2e19 < t < 7.99999999999999942e210

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 15.1%

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

      1. add-sqr-sqrt 5.5%

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

      2. sqrt-unprod 34.4%

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

      3. *-commutative 34.4%

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

      4. *-commutative 34.4%

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

      5. swap-sqr 42.4%

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

      6. add-sqr-sqrt 42.4%

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

      7. pow2 42.4%

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

   5. Applied egg-rr 42.4%

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

   if 7.99999999999999942e210 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 16.3%

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

   4. Taylor expanded in x around inf 28.1%

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

      1. mul-1-neg 28.1%

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

      2. unsub-neg 28.1%

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

   6. Simplified 28.1%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{\left(z \cdot 2\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 3 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(0.5 \cdot t\_1\right) - t\_1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(t\_1 \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 3e-13)
     (- (* x (* 0.5 t_1)) (* t_1 y))
     (* (exp (/ (* t t) 2.0)) (* t_1 (- y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 3e-13) {
		tmp = (x * (0.5 * t_1)) - (t_1 * y);
	} else {
		tmp = exp(((t * t) / 2.0)) * (t_1 * -y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t <= 3d-13) then
        tmp = (x * (0.5d0 * t_1)) - (t_1 * y)
    else
        tmp = exp(((t * t) / 2.0d0)) * (t_1 * -y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 3e-13) {
		tmp = (x * (0.5 * t_1)) - (t_1 * y);
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (t_1 * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 3e-13:
		tmp = (x * (0.5 * t_1)) - (t_1 * y)
	else:
		tmp = math.exp(((t * t) / 2.0)) * (t_1 * -y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 3e-13)
		tmp = Float64(Float64(x * Float64(0.5 * t_1)) - Float64(t_1 * y));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(t_1 * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 3e-13)
		tmp = (x * (0.5 * t_1)) - (t_1 * y);
	else
		tmp = exp(((t * t) / 2.0)) * (t_1 * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 3e-13], N[(N[(x * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.99999999999999984e-13

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 70.9%

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

      1. *-rgt-identity 70.9%

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

      2. *-commutative 70.9%

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

      3. sub-neg 70.9%

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

      4. sub-neg 70.9%

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

      5. add-sqr-sqrt 39.1%

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

      6. sqrt-unprod 55.7%

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

      7. sqr-neg 55.7%

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

      8. sqrt-unprod 19.8%

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

      9. add-sqr-sqrt 42.9%

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

      10. distribute-rgt-out-- 42.9%

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

      11. *-commutative 42.9%

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

      12. cancel-sign-sub-inv 42.9%

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

      13. associate-*l* 42.9%

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

      14. add-sqr-sqrt 19.8%

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

      15. sqrt-unprod 55.7%

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

      16. sqr-neg 55.7%

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

      17. sqrt-unprod 39.1%

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

      18. add-sqr-sqrt 70.9%

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

   5. Applied egg-rr 70.9%

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

   if 2.99999999999999984e-13 < 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. Add Preprocessing
   3. 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. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. associate-*r* 98.6%

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

      3. distribute-rgt-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 65.3%

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

      1. mul-1-neg 65.3%

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

      2. rem-square-sqrt 27.8%

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

      3. fabs-sqr 27.8%

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

      4. rem-square-sqrt 41.7%

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

      5. rem-sqrt-square 34.7%

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

      6. associate-*l* 34.7%

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

      7. *-commutative 34.7%

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

      8. associate-*l* 34.7%

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

      9. *-commutative 34.7%

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

      10. swap-sqr 31.9%

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

      11. swap-sqr 31.9%

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

      12. rem-square-sqrt 31.9%

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

      13. rem-square-sqrt 31.9%

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

      14. rem-square-sqrt 31.9%

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

      15. swap-sqr 34.7%

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

   10. Simplified 65.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \sqrt{z \cdot 2}\right) - \sqrt{z \cdot 2} \cdot y\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 3 \cdot 10^{-13}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(t\_1 \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 3e-13)
     (* t_1 (- (* x 0.5) y))
     (* (exp (/ (* t t) 2.0)) (* t_1 (- y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 3e-13) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = exp(((t * t) / 2.0)) * (t_1 * -y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t <= 3d-13) then
        tmp = t_1 * ((x * 0.5d0) - y)
    else
        tmp = exp(((t * t) / 2.0d0)) * (t_1 * -y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 3e-13) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (t_1 * -y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 3e-13:
		tmp = t_1 * ((x * 0.5) - y)
	else:
		tmp = math.exp(((t * t) / 2.0)) * (t_1 * -y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 3e-13)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(t_1 * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 3e-13)
		tmp = t_1 * ((x * 0.5) - y);
	else
		tmp = exp(((t * t) / 2.0)) * (t_1 * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 3e-13], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * (-y)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.99999999999999984e-13

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 70.9%

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

      1. *-rgt-identity 70.9%

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

      2. *-commutative 70.9%

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

      3. sub-neg 70.9%

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

      4. sub-neg 70.9%

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

      5. add-sqr-sqrt 39.1%

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

      6. sqrt-unprod 55.7%

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

      7. sqr-neg 55.7%

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

      8. sqrt-unprod 19.8%

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

      9. add-sqr-sqrt 42.9%

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

      10. distribute-rgt-out-- 42.9%

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

      11. *-commutative 42.9%

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

      12. cancel-sign-sub-inv 42.9%

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

      13. associate-*l* 42.9%

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

      14. add-sqr-sqrt 19.8%

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

      15. sqrt-unprod 55.7%

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

      16. sqr-neg 55.7%

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

      17. sqrt-unprod 39.1%

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

      18. add-sqr-sqrt 70.9%

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

   5. Applied egg-rr 70.9%

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

      1. fma-define 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. add-sqr-sqrt 70.9%

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

      4. sqr-neg 70.9%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 42.9%

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

      7. fma-neg 42.9%

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

      8. *-commutative 42.9%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 70.9%

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

      11. sqr-neg 70.9%

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

      12. add-sqr-sqrt 70.9%

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

   7. Applied egg-rr 70.9%

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

      1. associate-*r* 70.9%

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

      2. *-commutative 70.9%

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

      3. distribute-rgt-out-- 70.9%

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

   9. Simplified 70.9%

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

   if 2.99999999999999984e-13 < 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. Add Preprocessing
   3. 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. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. associate-*r* 98.6%

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

      3. distribute-rgt-out 100.0%

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

      4. mul-1-neg 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 65.3%

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

      1. mul-1-neg 65.3%

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

      2. rem-square-sqrt 27.8%

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

      3. fabs-sqr 27.8%

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

      4. rem-square-sqrt 41.7%

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

      5. rem-sqrt-square 34.7%

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

      6. associate-*l* 34.7%

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

      7. *-commutative 34.7%

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

      8. associate-*l* 34.7%

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

      9. *-commutative 34.7%

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

      10. swap-sqr 31.9%

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

      11. swap-sqr 31.9%

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

      12. rem-square-sqrt 31.9%

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

      13. rem-square-sqrt 31.9%

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

      14. rem-square-sqrt 31.9%

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

      15. swap-sqr 34.7%

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

   10. Simplified 65.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\sqrt{z \cdot 2} \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 42.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;x \leq -2.3 \lor \neg \left(x \leq 4.8 \cdot 10^{-31}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= x -2.3) (not (<= x 4.8e-31))) (* (* x 0.5) t_1) (* t_1 (- y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((x <= -2.3) || !(x <= 4.8e-31)) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if ((x <= (-2.3d0)) .or. (.not. (x <= 4.8d-31))) then
        tmp = (x * 0.5d0) * t_1
    else
        tmp = t_1 * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if ((x <= -2.3) || !(x <= 4.8e-31)) {
		tmp = (x * 0.5) * t_1;
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if (x <= -2.3) or not (x <= 4.8e-31):
		tmp = (x * 0.5) * t_1
	else:
		tmp = t_1 * -y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((x <= -2.3) || !(x <= 4.8e-31))
		tmp = Float64(Float64(x * 0.5) * t_1);
	else
		tmp = Float64(t_1 * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if ((x <= -2.3) || ~((x <= 4.8e-31)))
		tmp = (x * 0.5) * t_1;
	else
		tmp = t_1 * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2.3], N[Not[LessEqual[x, 4.8e-31]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999998 or 4.8e-31 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 60.6%

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

      1. add-cbrt-cube 50.1%

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

      2. add-sqr-sqrt 50.2%

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

      3. pow1 50.2%

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

      4. pow1/2 50.2%

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

      5. pow-prod-up 50.2%

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

      6. metadata-eval 50.2%

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{\color{blue}{1.5}}}\right) \cdot 1 \]

   5. Applied egg-rr 50.2%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}}\right) \cdot 1 \]

   6. Taylor expanded in x around inf 42.0%

      \[\leadsto \left(\color{blue}{\left(0.5 \cdot x\right)} \cdot \sqrt[3]{{\left(z \cdot 2\right)}^{1.5}}\right) \cdot 1 \]
   7. Step-by-step derivation

      1. *-rgt-identity 42.0%

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

      2. *-commutative 42.0%

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

      3. pow1/3 40.4%

         \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{{\left({\left(z \cdot 2\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      4. pow-pow 52.4%

         \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{{\left(z \cdot 2\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      5. metadata-eval 52.4%

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

      6. pow1/2 52.4%

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

      7. pow1 52.4%

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

      8. *-commutative 52.4%

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

      9. associate-*r* 52.4%

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

   8. Applied egg-rr 52.4%

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

      1. unpow1 52.4%

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

      2. associate-*r* 52.4%

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

      3. *-commutative 52.4%

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

   10. Simplified 52.4%

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

   if -2.2999999999999998 < x < 4.8e-31

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 49.3%

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

   4. Taylor expanded in x around 0 38.0%

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

      1. mul-1-neg 38.0%

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

   6. Simplified 38.0%

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

      1. *-rgt-identity 38.0%

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

      2. distribute-lft-neg-out 38.0%

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

   8. Applied egg-rr 38.0%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \lor \neg \left(x \leq 4.8 \cdot 10^{-31}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 0.025:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 0.025) (* t_1 (- (* x 0.5) y)) (* t_1 (* x (- 0.5 (/ y x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 0.025) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * (x * (0.5 - (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) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t <= 0.025d0) then
        tmp = t_1 * ((x * 0.5d0) - y)
    else
        tmp = t_1 * (x * (0.5d0 - (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 0.025) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * (x * (0.5 - (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 0.025:
		tmp = t_1 * ((x * 0.5) - y)
	else:
		tmp = t_1 * (x * (0.5 - (y / x)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 0.025)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(t_1 * Float64(x * Float64(0.5 - Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 0.025)
		tmp = t_1 * ((x * 0.5) - y);
	else
		tmp = t_1 * (x * (0.5 - (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.025], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x * N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 0.025000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 71.0%

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

      1. *-rgt-identity 71.0%

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

      2. *-commutative 71.0%

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

      3. sub-neg 71.0%

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

      4. sub-neg 71.0%

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

      5. add-sqr-sqrt 39.4%

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

      6. sqrt-unprod 55.9%

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

      7. sqr-neg 55.9%

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

      8. sqrt-unprod 19.7%

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

      9. add-sqr-sqrt 42.6%

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

      10. distribute-rgt-out-- 42.6%

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

      11. *-commutative 42.6%

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

      12. cancel-sign-sub-inv 42.6%

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

      13. associate-*l* 42.6%

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

      14. add-sqr-sqrt 19.7%

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

      15. sqrt-unprod 55.9%

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

      16. sqr-neg 55.9%

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

      17. sqrt-unprod 39.4%

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

      18. add-sqr-sqrt 71.0%

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

   5. Applied egg-rr 71.0%

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

      1. fma-define 71.0%

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

      2. distribute-lft-neg-out 71.0%

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

      3. add-sqr-sqrt 71.0%

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

      4. sqr-neg 71.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 42.6%

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

      7. fma-neg 42.6%

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

      8. *-commutative 42.6%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 71.0%

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

      11. sqr-neg 71.0%

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

      12. add-sqr-sqrt 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

      3. distribute-rgt-out-- 71.0%

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

   9. Simplified 71.0%

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

   if 0.025000000000000001 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 15.2%

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

   4. Taylor expanded in x around inf 27.2%

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

      1. mul-1-neg 27.2%

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

      2. unsub-neg 27.2%

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

   6. Simplified 27.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.025:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot \left(0.5 - \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 1.45 \cdot 10^{+113}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\frac{y}{x} \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= t 1.45e+113) (* t_1 (- (* x 0.5) y)) (* t_1 (* (/ y x) (- x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if (t <= 1.45e+113) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * ((y / x) * -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    if (t <= 1.45d+113) then
        tmp = t_1 * ((x * 0.5d0) - y)
    else
        tmp = t_1 * ((y / x) * -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double tmp;
	if (t <= 1.45e+113) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = t_1 * ((y / x) * -x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	tmp = 0
	if t <= 1.45e+113:
		tmp = t_1 * ((x * 0.5) - y)
	else:
		tmp = t_1 * ((y / x) * -x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 1.45e+113)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(t_1 * Float64(Float64(y / x) * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	tmp = 0.0;
	if (t <= 1.45e+113)
		tmp = t_1 * ((x * 0.5) - y);
	else
		tmp = t_1 * ((y / x) * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.45e+113], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(y / x), $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.44999999999999992e113

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 65.7%

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

      1. *-rgt-identity 65.7%

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

      2. *-commutative 65.7%

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

      3. sub-neg 65.7%

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

      4. sub-neg 65.7%

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

      5. add-sqr-sqrt 36.4%

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

      6. sqrt-unprod 52.3%

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

      7. sqr-neg 52.3%

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

      8. sqrt-unprod 18.4%

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

      9. add-sqr-sqrt 39.8%

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

      10. distribute-rgt-out-- 39.8%

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

      11. *-commutative 39.8%

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

      12. cancel-sign-sub-inv 39.8%

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

      13. associate-*l* 39.8%

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

      14. add-sqr-sqrt 18.4%

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

      15. sqrt-unprod 52.3%

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

      16. sqr-neg 52.3%

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

      17. sqrt-unprod 36.4%

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

      18. add-sqr-sqrt 65.7%

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

   5. Applied egg-rr 65.7%

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

      1. fma-define 65.7%

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

      2. distribute-lft-neg-out 65.7%

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

      3. add-sqr-sqrt 65.7%

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

      4. sqr-neg 65.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 39.8%

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

      7. fma-neg 39.8%

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

      8. *-commutative 39.8%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 65.7%

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

      11. sqr-neg 65.7%

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

      12. add-sqr-sqrt 65.7%

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

   7. Applied egg-rr 65.7%

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

      1. associate-*r* 65.7%

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

      2. *-commutative 65.7%

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

      3. distribute-rgt-out-- 65.7%

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

   9. Simplified 65.7%

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

   if 1.44999999999999992e113 < 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. Add Preprocessing

   3. Taylor expanded in t around 0 15.6%

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

   4. Taylor expanded in x around inf 28.3%

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

      1. mul-1-neg 28.3%

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

      2. unsub-neg 28.3%

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

   6. Simplified 28.3%

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

   7. Taylor expanded in y around inf 21.3%

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

      1. neg-mul-1 21.3%

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

      2. distribute-neg-frac2 21.3%

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

   9. Simplified 21.3%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(\frac{y}{x} \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.4% 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 99.8%

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

3. Taylor expanded in t around 0 55.5%

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

   1. *-rgt-identity 55.5%

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

   2. *-commutative 55.5%

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

   3. sub-neg 55.5%

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

   4. sub-neg 55.5%

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

   5. add-sqr-sqrt 31.3%

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

   6. sqrt-unprod 46.1%

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

   7. sqr-neg 46.1%

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

   8. sqrt-unprod 15.3%

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

   9. add-sqr-sqrt 33.7%

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

   10. distribute-rgt-out-- 33.4%

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

   11. *-commutative 33.4%

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

   12. cancel-sign-sub-inv 33.4%

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

   13. associate-*l* 33.4%

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

   14. add-sqr-sqrt 15.3%

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

   15. sqrt-unprod 46.1%

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

   16. sqr-neg 46.1%

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

   17. sqrt-unprod 31.3%

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

   18. add-sqr-sqrt 55.5%

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

5. Applied egg-rr 55.5%

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

   1. fma-define 55.5%

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

   2. distribute-lft-neg-out 55.5%

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

   3. add-sqr-sqrt 55.5%

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

   4. sqr-neg 55.5%

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

   5. sqrt-unprod 0.0%

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

   6. add-sqr-sqrt 33.4%

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

   7. fma-neg 33.4%

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

   8. *-commutative 33.4%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 55.5%

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

   11. sqr-neg 55.5%

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

   12. add-sqr-sqrt 55.5%

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

7. Applied egg-rr 55.5%

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

   1. associate-*r* 55.5%

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

   2. *-commutative 55.5%

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

   3. distribute-rgt-out-- 55.5%

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

9. Simplified 55.5%

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

10. Final simplification 55.5%

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

Alternative 10: 29.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0 55.5%

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

4. Taylor expanded in x around 0 23.2%

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

   1. mul-1-neg 23.2%

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

6. Simplified 23.2%

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

   1. *-rgt-identity 23.2%

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

   2. distribute-lft-neg-out 23.2%

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

8. Applied egg-rr 23.2%

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

9. Final simplification 23.2%

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

Alternative 11: 2.5% 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 99.8%

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

3. Taylor expanded in t around 0 55.5%

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

4. Taylor expanded in x around 0 23.2%

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

   1. mul-1-neg 23.2%

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

6. Simplified 23.2%

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

   1. *-rgt-identity 23.2%

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

   2. add-sqr-sqrt 9.3%

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

   3. sqrt-unprod 15.2%

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

   4. sqr-neg 15.2%

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

   5. sqrt-unprod 1.4%

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

   6. add-sqr-sqrt 2.2%

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

   7. pow1 2.2%

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

8. Applied egg-rr 2.2%

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

   1. unpow1 2.2%

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

10. Simplified 2.2%

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

11. Final simplification 2.2%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024144 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

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

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