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

Percentage Accurate: 99.2% → 99.2%

Time: 12.3s

Alternatives: 8

Speedup: 1.0×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

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 / Float64(-2.0 / t)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(-(t / (-2.0 / t)));
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[(t / N[(-2.0 / t), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. frac-2neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. metadata-eval 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-/l* 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 200 \lor \neg \left(t \cdot t \leq 2 \cdot 10^{+230}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(1 + t \cdot \left(0.5 \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* t t) 200.0) (not (<= (* t t) 2e+230)))
   (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (+ 1.0 (* t (* 0.5 t))))
   (* (exp (/ (* t t) 2.0)) (* x (sqrt (* 0.5 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((t * t) <= 200.0) || !((t * t) <= 2e+230)) {
		tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (1.0 + (t * (0.5 * t)));
	} else {
		tmp = exp(((t * t) / 2.0)) * (x * sqrt((0.5 * z)));
	}
	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 * t) <= 200.0d0) .or. (.not. ((t * t) <= 2d+230))) then
        tmp = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (1.0d0 + (t * (0.5d0 * t)))
    else
        tmp = exp(((t * t) / 2.0d0)) * (x * sqrt((0.5d0 * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((t * t) <= 200.0) || !((t * t) <= 2e+230)) {
		tmp = (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * (1.0 + (t * (0.5 * t)));
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (x * Math.sqrt((0.5 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((t * t) <= 200.0) or not ((t * t) <= 2e+230):
		tmp = (((x * 0.5) - y) * math.sqrt((z * 2.0))) * (1.0 + (t * (0.5 * t)))
	else:
		tmp = math.exp(((t * t) / 2.0)) * (x * math.sqrt((0.5 * z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(t * t) <= 200.0) || !(Float64(t * t) <= 2e+230))
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * Float64(1.0 + Float64(t * Float64(0.5 * t))));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(x * sqrt(Float64(0.5 * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((t * t) <= 200.0) || ~(((t * t) <= 2e+230)))
		tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (1.0 + (t * (0.5 * t)));
	else
		tmp = exp(((t * t) / 2.0)) * (x * sqrt((0.5 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(t * t), $MachinePrecision], 200.0], N[Not[LessEqual[N[(t * t), $MachinePrecision], 2e+230]], $MachinePrecision]], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t * N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 200 or 2.0000000000000002e230 < (*.f64 t t)

   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. Taylor expanded in t around 0 98.0%

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

      1. unpow2 59.3%

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

      2. associate-*r* 59.3%

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

      3. *-commutative 59.3%

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

   4. Simplified 98.0%

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

   if 200 < (*.f64 t t) < 2.0000000000000002e230

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 44.0%

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

      2. sqrt-unprod 42.0%

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

      3. *-commutative 42.0%

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

      4. *-commutative 42.0%

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

      5. swap-sqr 42.0%

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

      6. add-sqr-sqrt 42.0%

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

      7. pow2 42.0%

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

   3. Applied egg-rr 42.0%

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

   4. Taylor expanded in x around inf 34.0%

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

      1. associate-*r* 34.0%

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

      2. unpow2 34.0%

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

   6. Simplified 34.0%

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

      1. sqrt-prod 21.1%

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

      2. *-commutative 21.1%

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

   8. Applied egg-rr 36.0%

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

      1. *-commutative 21.1%

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

      2. rem-sqrt-square 9.8%

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

      3. unpow1 9.8%

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

      4. sqr-pow 9.4%

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

      5. fabs-sqr 9.4%

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

      6. sqr-pow 14.9%

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

      7. unpow1 14.9%

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

   10. Simplified 84.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 200 \lor \neg \left(t \cdot t \leq 2 \cdot 10^{+230}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(1 + t \cdot \left(0.5 \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Final simplification 99.8%

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

Alternative 4: 85.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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))) * (1.0d0 + (t * (0.5d0 * t)))
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))) * (1.0 + (t * (0.5 * t)));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (1.0 + (t * (0.5 * t)));
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[(1.0 + N[(t * N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 86.2%

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

   1. unpow2 53.0%

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

   2. associate-*r* 53.0%

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

   3. *-commutative 53.0%

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

4. Simplified 86.2%

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

5. Final simplification 86.2%

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

Alternative 5: 65.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+17}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t \cdot \left(0.5 \cdot t\right)\right) \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.3e+17)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (* (+ 1.0 (* t (* 0.5 t))) (* x (sqrt (* 0.5 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.3e+17) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = (1.0 + (t * (0.5 * t))) * (x * sqrt((0.5 * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 3.3d+17) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = (1.0d0 + (t * (0.5d0 * t))) * (x * sqrt((0.5d0 * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.3e+17) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = (1.0 + (t * (0.5 * t))) * (x * Math.sqrt((0.5 * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 3.3e+17:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = (1.0 + (t * (0.5 * t))) * (x * math.sqrt((0.5 * z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.3e+17)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(t * Float64(0.5 * t))) * Float64(x * sqrt(Float64(0.5 * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 3.3e+17)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = (1.0 + (t * (0.5 * t))) * (x * sqrt((0.5 * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3.3e+17], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(t * N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.3e17

   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. Taylor expanded in t around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*r* 71.1%

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

      4. sqrt-prod 71.3%

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

      5. sub-neg 71.3%

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

      6. distribute-lft-in 70.2%

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

   4. Applied egg-rr 70.2%

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

      1. distribute-lft-out 71.3%

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

      2. *-commutative 71.3%

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

      3. *-commutative 71.3%

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

      4. sub-neg 71.3%

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

   6. Simplified 71.3%

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

   if 3.3e17 < t

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 48.7%

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

      2. sqrt-unprod 48.7%

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

      3. *-commutative 48.7%

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

      4. *-commutative 48.7%

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

      5. swap-sqr 48.7%

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

      6. add-sqr-sqrt 48.7%

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

      7. pow2 48.7%

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

   3. Applied egg-rr 48.7%

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

   4. Taylor expanded in x around inf 39.5%

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

      1. associate-*r* 39.5%

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

      2. unpow2 39.5%

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

   6. Simplified 39.5%

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

      1. sqrt-prod 22.2%

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

      2. *-commutative 22.2%

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

   8. Applied egg-rr 43.4%

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

      1. *-commutative 22.2%

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

      2. rem-sqrt-square 8.6%

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

      3. unpow1 8.6%

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

      4. sqr-pow 7.9%

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

      5. fabs-sqr 7.9%

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

      6. sqr-pow 14.6%

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

      7. unpow1 14.6%

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

   10. Simplified 76.3%

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

   11. Taylor expanded in t around 0 55.2%

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

       1. unpow2 55.2%

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

       2. associate-*r* 55.2%

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

       3. *-commutative 55.2%

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

   13. Simplified 55.2%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+17}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t \cdot \left(0.5 \cdot t\right)\right) \cdot \left(x \cdot \sqrt{0.5 \cdot z}\right)\\ \end{array} \]

Alternative 6: 56.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot z\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x 0.5) 2e+160)
   (* (- (* x 0.5) y) (sqrt (* z 2.0)))
   (sqrt (* (* 0.5 z) (* x x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * 0.5) <= 2e+160) {
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	} else {
		tmp = sqrt(((0.5 * z) * (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) :: tmp
    if ((x * 0.5d0) <= 2d+160) then
        tmp = ((x * 0.5d0) - y) * sqrt((z * 2.0d0))
    else
        tmp = sqrt(((0.5d0 * z) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * 0.5) <= 2e+160) {
		tmp = ((x * 0.5) - y) * Math.sqrt((z * 2.0));
	} else {
		tmp = Math.sqrt(((0.5 * z) * (x * x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * 0.5) <= 2e+160:
		tmp = ((x * 0.5) - y) * math.sqrt((z * 2.0))
	else:
		tmp = math.sqrt(((0.5 * z) * (x * x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * 0.5) <= 2e+160)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0)));
	else
		tmp = sqrt(Float64(Float64(0.5 * z) * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * 0.5) <= 2e+160)
		tmp = ((x * 0.5) - y) * sqrt((z * 2.0));
	else
		tmp = sqrt(((0.5 * z) * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], 2e+160], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(0.5 * z), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < 2.00000000000000001e160

   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. Taylor expanded in t around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. *-commutative 55.0%

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

      3. associate-*r* 55.1%

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

      4. sqrt-prod 55.2%

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

      5. sub-neg 55.2%

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

      6. distribute-lft-in 54.7%

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

   4. Applied egg-rr 54.7%

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

      1. distribute-lft-out 55.2%

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

      2. *-commutative 55.2%

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

      3. *-commutative 55.2%

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

      4. sub-neg 55.2%

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

   6. Simplified 55.2%

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

   if 2.00000000000000001e160 < (*.f64 x 1/2)

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 80.2%

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

      3. *-commutative 80.2%

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

      4. *-commutative 80.2%

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

      5. swap-sqr 77.7%

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

      6. add-sqr-sqrt 77.7%

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

      7. pow2 77.7%

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

   3. Applied egg-rr 77.7%

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

   4. Taylor expanded in x around inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. unpow2 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in t around 0 77.7%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot z\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 7: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0 56.6%

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

   1. *-commutative 56.6%

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

   2. *-commutative 56.6%

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

   3. associate-*r* 56.6%

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

   4. sqrt-prod 56.7%

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

   5. sub-neg 56.7%

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

   6. distribute-lft-in 55.9%

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

4. Applied egg-rr 55.9%

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

   1. distribute-lft-out 56.7%

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

   2. *-commutative 56.7%

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

   3. *-commutative 56.7%

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

   4. sub-neg 56.7%

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

6. Simplified 56.7%

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

7. Final simplification 56.7%

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

Alternative 8: 29.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \sqrt{0.5 \cdot z} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. add-sqr-sqrt 50.6%

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

   2. sqrt-unprod 41.1%

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

   3. *-commutative 41.1%

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

   4. *-commutative 41.1%

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

   5. swap-sqr 38.1%

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

   6. add-sqr-sqrt 38.1%

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

   7. pow2 38.1%

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

3. Applied egg-rr 38.1%

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

4. Taylor expanded in x around inf 27.1%

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

   1. associate-*r* 27.1%

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

   2. unpow2 27.1%

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

6. Simplified 27.1%

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

7. Taylor expanded in t around 0 19.1%

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

   1. sqrt-prod 21.6%

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

   2. *-commutative 21.6%

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

9. Applied egg-rr 21.6%

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

    1. *-commutative 21.6%

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

    2. rem-sqrt-square 19.8%

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

    3. unpow1 19.8%

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

    4. sqr-pow 18.5%

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

    5. fabs-sqr 18.5%

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

    6. sqr-pow 33.1%

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

    7. unpow1 33.1%

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

11. Simplified 33.1%

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

12. Final simplification 33.1%

    \[\leadsto x \cdot \sqrt{0.5 \cdot z} \]

Developer target: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023200 
(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))))