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

Percentage Accurate: 99.4% → 99.8%

Time: 13.3s

Alternatives: 8

Speedup: 1.0×

• 43.9% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt((exp(pow(t, 2.0)) * (2.0 * z))) * ((0.5 * x) - 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((exp((t ** 2.0d0)) * (2.0d0 * z))) * ((0.5d0 * x) - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 79.5%

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

   5. exp-sqrt 79.5%

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

   6. sqrt-unprod 79.5%

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

   7. associate-*l* 79.5%

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

   8. pow2 79.5%

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

3. Applied egg-rr 79.5%

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

   1. distribute-lft-out 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. *-commutative 99.8%

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

   6. sub-neg 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+17} \lor \neg \left(t \cdot t \leq 10^{+194}\right):\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \left(t_1 \cdot \left(t \cdot \left(t \cdot 0.5\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(0.5 \cdot x\right) \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (or (<= (* t t) 5e+17) (not (<= (* t t) 1e+194)))
     (* (- (* 0.5 x) y) (* t_1 (+ (* t (* t 0.5)) 1.0)))
     (* (exp (/ (* t t) 2.0)) (* (* 0.5 x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if (((t * t) <= 5e+17) || !((t * t) <= 1e+194)) {
		tmp = ((0.5 * x) - y) * (t_1 * ((t * (t * 0.5)) + 1.0));
	} else {
		tmp = exp(((t * t) / 2.0)) * ((0.5 * x) * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if (((t * t) <= 5d+17) .or. (.not. ((t * t) <= 1d+194))) then
        tmp = ((0.5d0 * x) - y) * (t_1 * ((t * (t * 0.5d0)) + 1.0d0))
    else
        tmp = exp(((t * t) / 2.0d0)) * ((0.5d0 * x) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if (((t * t) <= 5e+17) || !((t * t) <= 1e+194)) {
		tmp = ((0.5 * x) - y) * (t_1 * ((t * (t * 0.5)) + 1.0));
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * ((0.5 * x) * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if ((t * t) <= 5e+17) or not ((t * t) <= 1e+194):
		tmp = ((0.5 * x) - y) * (t_1 * ((t * (t * 0.5)) + 1.0))
	else:
		tmp = math.exp(((t * t) / 2.0)) * ((0.5 * x) * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if ((Float64(t * t) <= 5e+17) || !(Float64(t * t) <= 1e+194))
		tmp = Float64(Float64(Float64(0.5 * x) - y) * Float64(t_1 * Float64(Float64(t * Float64(t * 0.5)) + 1.0)));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(Float64(0.5 * x) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if (((t * t) <= 5e+17) || ~(((t * t) <= 1e+194)))
		tmp = ((0.5 * x) - y) * (t_1 * ((t * (t * 0.5)) + 1.0));
	else
		tmp = exp(((t * t) / 2.0)) * ((0.5 * x) * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[N[(t * t), $MachinePrecision], 5e+17], N[Not[LessEqual[N[(t * t), $MachinePrecision], 1e+194]], $MachinePrecision]], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[(t$95$1 * N[(N[(t * N[(t * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 5e17 or 9.99999999999999945e193 < (*.f64 t t)

   1. Initial program 98.9%

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

      1. sqr-neg 98.9%

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

      2. associate-/l* 98.9%

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

      3. distribute-frac-neg 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-*r/ 98.9%

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

      6. *-rgt-identity 98.9%

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

      7. associate-*r/ 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. associate-*r/ 99.8%

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

      10. exp-neg 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. associate-/l* 99.8%

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

      13. sqr-neg 99.8%

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

      14. exp-sqrt 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 95.5%

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

      1. +-commutative 95.5%

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

   6. Simplified 95.5%

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

      1. *-commutative 95.5%

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

      2. metadata-eval 95.5%

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

      3. div-inv 95.5%

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

      4. pow2 95.5%

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

      5. associate-/l* 95.5%

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

   8. Applied egg-rr 95.5%

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

      1. associate-/r/ 95.5%

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

      2. div-inv 95.5%

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

      3. metadata-eval 95.5%

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

   10. Applied egg-rr 95.5%

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

   if 5e17 < (*.f64 t t) < 9.99999999999999945e193

   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. add-cbrt-cube 87.2%

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

      2. pow3 87.2%

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

      3. fma-neg 87.2%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\mathsf{fma}\left(x, 0.5, -y\right)} \cdot \sqrt{z \cdot 2}\right)}^{3}} \cdot e^{\frac{t \cdot t}{2}} \]

   3. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x, 0.5, -y\right) \cdot \sqrt{z \cdot 2}\right)}^{3}}} \cdot e^{\frac{t \cdot t}{2}} \]
   4. Step-by-step derivation

      1. rem-cbrt-cube 97.4%

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

      2. fma-neg 97.4%

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

      3. *-commutative 97.4%

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

      4. add-sqr-sqrt 97.4%

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

      5. associate-*l* 97.4%

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

      6. pow1/2 97.4%

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

      7. sqrt-pow1 97.4%

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

      8. *-commutative 97.4%

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

      9. metadata-eval 97.4%

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

      10. pow1/2 97.4%

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

      11. sqrt-pow1 97.4%

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

      12. *-commutative 97.4%

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

      13. metadata-eval 97.4%

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

      14. *-commutative 97.4%

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

      15. fma-neg 97.4%

          \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \color{blue}{\mathsf{fma}\left(0.5, x, -y\right)}\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      16. add-sqr-sqrt 59.0%

          \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \mathsf{fma}\left(0.5, x, \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      17. sqrt-unprod 74.4%

          \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \mathsf{fma}\left(0.5, x, \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      18. sqr-neg 74.4%

          \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \mathsf{fma}\left(0.5, x, \sqrt{\color{blue}{y \cdot y}}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      19. sqrt-unprod 17.9%

          \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \mathsf{fma}\left(0.5, x, \color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

      20. add-sqr-sqrt 51.3%

          \[\leadsto \left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \mathsf{fma}\left(0.5, x, \color{blue}{y}\right)\right)\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Applied egg-rr 51.3%

      \[\leadsto \color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(2 \cdot z\right)}^{0.25} \cdot \mathsf{fma}\left(0.5, x, y\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

   6. Taylor expanded in x around inf 82.1%

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

      1. *-commutative 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*r* 82.1%

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

      4. *-commutative 82.1%

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

   8. Simplified 82.1%

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

      1. expm1-log1p-u 59.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(z \cdot 2\right)}^{0.25} \cdot \left(0.5 \cdot x\right)\right)\right)\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      2. expm1-udef 30.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(2 \cdot z\right)}^{0.25} \cdot \left({\left(z \cdot 2\right)}^{0.25} \cdot \left(0.5 \cdot x\right)\right)\right)} - 1\right)} \cdot e^{\frac{t \cdot t}{2}} \]

      3. associate-*r* 30.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left({\left(2 \cdot z\right)}^{0.25} \cdot {\left(z \cdot 2\right)}^{0.25}\right) \cdot \left(0.5 \cdot x\right)}\right)} - 1\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative 30.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\left({\color{blue}{\left(z \cdot 2\right)}}^{0.25} \cdot {\left(z \cdot 2\right)}^{0.25}\right) \cdot \left(0.5 \cdot x\right)\right)} - 1\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. pow-prod-up 30.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(z \cdot 2\right)}^{\left(0.25 + 0.25\right)}} \cdot \left(0.5 \cdot x\right)\right)} - 1\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. metadata-eval 30.8%

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

      7. pow1/2 30.8%

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

   10. Applied egg-rr 30.8%

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

       1. expm1-def 59.0%

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

       2. expm1-log1p 82.1%

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

       3. *-commutative 82.1%

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

   12. Simplified 82.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 5 \cdot 10^{+17} \lor \neg \left(t \cdot t \leq 10^{+194}\right):\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \left(t \cdot \left(t \cdot 0.5\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(\left(0.5 \cdot x\right) \cdot \sqrt{2 \cdot z}\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

2. Final simplification 98.6%

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

Alternative 4: 86.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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 = ((0.5d0 * x) - y) * (sqrt((2.0d0 * z)) * ((t * (t * 0.5d0)) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. sqr-neg 98.6%

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

   2. associate-/l* 98.6%

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

   3. distribute-frac-neg 98.6%

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

   4. exp-neg 98.6%

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

   5. associate-*r/ 98.6%

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

   6. *-rgt-identity 98.6%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. associate-*r/ 99.8%

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

   10. exp-neg 99.8%

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

   11. distribute-frac-neg 99.8%

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

   12. associate-/l* 99.8%

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

   13. sqr-neg 99.8%

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

   14. exp-sqrt 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around 0 85.4%

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

   1. +-commutative 85.4%

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

6. Simplified 85.4%

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

   1. *-commutative 85.4%

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

   2. metadata-eval 85.4%

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

   3. div-inv 85.4%

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

   4. pow2 85.4%

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

   5. associate-/l* 85.4%

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

8. Applied egg-rr 85.4%

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

   1. associate-/r/ 85.4%

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

   2. div-inv 85.4%

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

   3. metadata-eval 85.4%

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

10. Applied egg-rr 85.4%

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

11. Final simplification 85.4%

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

Alternative 5: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{+15}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(2 \cdot \left(y \cdot \left(x + y\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.65e+15)
   (* (- (* 0.5 x) y) (sqrt (* 2.0 z)))
   (pow (* z (* 2.0 (* y (+ x y)))) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.65e+15) {
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	} else {
		tmp = pow((z * (2.0 * (y * (x + y)))), 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) :: tmp
    if (t <= 1.65d+15) then
        tmp = ((0.5d0 * x) - y) * sqrt((2.0d0 * z))
    else
        tmp = (z * (2.0d0 * (y * (x + y)))) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.65e+15) {
		tmp = ((0.5 * x) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.pow((z * (2.0 * (y * (x + y)))), 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.65e+15:
		tmp = ((0.5 * x) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.pow((z * (2.0 * (y * (x + y)))), 0.5)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.65e+15)
		tmp = Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(z * Float64(2.0 * Float64(y * Float64(x + y)))) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.65e+15)
		tmp = ((0.5 * x) - y) * sqrt((2.0 * z));
	else
		tmp = (z * (2.0 * (y * (x + y)))) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.65e+15], N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(z * N[(2.0 * N[(y * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.65e15

   1. Initial program 98.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. distribute-lft-in 86.7%

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

      5. exp-sqrt 86.7%

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

      6. sqrt-unprod 86.7%

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

      7. associate-*l* 86.7%

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

      8. pow2 86.7%

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

   3. Applied egg-rr 86.7%

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

      1. distribute-lft-out 99.7%

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

      2. *-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

      6. sub-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in t around 0 72.7%

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

      1. *-commutative 72.7%

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

   8. Simplified 72.7%

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

   if 1.65e15 < t

   1. Initial program 98.5%

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

   2. Taylor expanded in t around 0 17.6%

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

      1. add-sqr-sqrt 12.8%

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

      2. sqrt-unprod 31.7%

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

      3. *-commutative 31.7%

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

      4. *-commutative 31.7%

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

      5. swap-sqr 31.7%

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

      6. rem-square-sqrt 31.7%

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

      7. pow2 31.7%

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

      8. *-commutative 31.7%

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

      9. fma-neg 31.7%

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

      10. add-sqr-sqrt 25.2%

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

      11. sqrt-unprod 31.7%

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

      12. sqr-neg 31.7%

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

      13. sqrt-unprod 6.4%

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

      14. add-sqr-sqrt 31.7%

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

   4. Applied egg-rr 31.7%

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

   5. Taylor expanded in x around 0 11.6%

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

      1. +-commutative 11.6%

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

      2. unpow2 11.6%

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

      3. distribute-rgt-out 16.2%

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

   7. Simplified 16.2%

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

      1. sqrt-unprod 16.2%

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

      2. pow1/2 19.3%

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

      3. +-commutative 19.3%

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

   9. Applied egg-rr 19.3%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{+15}:\\ \;\;\;\;\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;{\left(z \cdot \left(2 \cdot \left(y \cdot \left(x + y\right)\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 6: 16.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt(((2.0 * z) * (y * (x + 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(((2.0d0 * z) * (y * (x + y))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in t around 0 58.6%

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

   1. add-sqr-sqrt 27.6%

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

   2. sqrt-unprod 28.6%

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

   3. *-commutative 28.6%

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

   4. *-commutative 28.6%

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

   5. swap-sqr 28.5%

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

   6. rem-square-sqrt 28.6%

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

   7. pow2 28.6%

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

   8. *-commutative 28.6%

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

   9. fma-neg 28.6%

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

   10. add-sqr-sqrt 22.1%

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

   11. sqrt-unprod 28.6%

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

   12. sqr-neg 28.6%

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

   13. sqrt-unprod 6.5%

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

   14. add-sqr-sqrt 28.1%

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

4. Applied egg-rr 28.1%

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

5. Taylor expanded in x around 0 15.2%

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

   1. +-commutative 15.2%

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

   2. unpow2 15.2%

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

   3. distribute-rgt-out 17.6%

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

7. Simplified 17.6%

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

   1. expm1-log1p-u 17.3%

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

   2. expm1-udef 15.3%

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

   3. sqrt-unprod 15.2%

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

   4. +-commutative 15.2%

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

9. Applied egg-rr 15.2%

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

    1. expm1-def 16.9%

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

    2. expm1-log1p 17.1%

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

    3. associate-*r* 17.1%

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

    4. +-commutative 17.1%

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

11. Simplified 17.1%

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

12. Final simplification 17.1%

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

Alternative 7: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 = ((0.5d0 * x) - y) * sqrt((2.0d0 * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. distribute-lft-in 79.5%

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

   5. exp-sqrt 79.5%

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

   6. sqrt-unprod 79.5%

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

   7. associate-*l* 79.5%

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

   8. pow2 79.5%

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

3. Applied egg-rr 79.5%

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

   1. distribute-lft-out 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. *-commutative 99.8%

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

   6. sub-neg 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around 0 58.7%

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

   1. *-commutative 58.7%

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

8. Simplified 58.7%

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

9. Final simplification 58.7%

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

Alternative 8: 2.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

2. Taylor expanded in t around 0 58.6%

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

   1. *-commutative 58.6%

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

   2. *-commutative 58.6%

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

   3. fma-neg 58.6%

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

   4. associate-*l* 58.5%

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

   5. fma-neg 58.5%

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

   6. *-commutative 58.5%

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

4. Simplified 58.5%

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

5. Taylor expanded in x around 0 30.3%

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

   1. mul-1-neg 30.3%

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

   2. distribute-lft-neg-out 30.3%

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

   3. *-commutative 30.3%

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

7. Simplified 30.3%

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

   1. expm1-log1p-u 20.9%

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

   2. expm1-udef 13.9%

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

   3. associate-*r* 13.9%

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

   4. sqrt-prod 13.9%

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

   5. *-commutative 13.9%

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

   6. add-sqr-sqrt 12.7%

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

   7. sqrt-unprod 15.7%

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

   8. sqr-neg 15.7%

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

   9. sqrt-unprod 1.7%

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

   10. add-sqr-sqrt 2.3%

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

9. Applied egg-rr 2.3%

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

    1. expm1-def 2.2%

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

    2. expm1-log1p 2.5%

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

    3. *-commutative 2.5%

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

11. Simplified 2.5%

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

12. Final simplification 2.5%

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

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(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))))