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

Percentage Accurate: 99.5% → 99.8%

Time: 9.8s

Alternatives: 5

Speedup: 1.0×

• 45.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l* 99.9%

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

   2. exp-sqrt 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 78.4%

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

   3. sqrt-unprod 78.4%

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

   4. associate-*l* 78.4%

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

   5. exp-prod 78.4%

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

5. Applied egg-rr 78.4%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-*l* 99.9%

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

   5. *-commutative 99.9%

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

   6. exp-prod 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 10^{-16}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \cdot t \leq 10^{+229}:\\ \;\;\;\;y \cdot \left(-\sqrt{e^{t \cdot t} \cdot \left(2 \cdot z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 1e-16)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= (* t t) 1e+229)
       (* y (- (sqrt (* (exp (* t t)) (* 2.0 z)))))
       (* t_1 (sqrt (* 2.0 (+ z (* z (* t t))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 1e-16) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if ((t * t) <= 1e+229) {
		tmp = y * -sqrt((exp((t * t)) * (2.0 * z)));
	} else {
		tmp = t_1 * sqrt((2.0 * (z + (z * (t * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if ((t * t) <= 1d-16) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else if ((t * t) <= 1d+229) then
        tmp = y * -sqrt((exp((t * t)) * (2.0d0 * z)))
    else
        tmp = t_1 * sqrt((2.0d0 * (z + (z * (t * t)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 1e-16) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else if ((t * t) <= 1e+229) {
		tmp = y * -Math.sqrt((Math.exp((t * t)) * (2.0 * z)));
	} else {
		tmp = t_1 * Math.sqrt((2.0 * (z + (z * (t * t)))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if (t * t) <= 1e-16:
		tmp = t_1 * math.sqrt((2.0 * z))
	elif (t * t) <= 1e+229:
		tmp = y * -math.sqrt((math.exp((t * t)) * (2.0 * z)))
	else:
		tmp = t_1 * math.sqrt((2.0 * (z + (z * (t * t)))))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 1e-16)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (Float64(t * t) <= 1e+229)
		tmp = Float64(y * Float64(-sqrt(Float64(exp(Float64(t * t)) * Float64(2.0 * z)))));
	else
		tmp = Float64(t_1 * sqrt(Float64(2.0 * Float64(z + Float64(z * Float64(t * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if ((t * t) <= 1e-16)
		tmp = t_1 * sqrt((2.0 * z));
	elseif ((t * t) <= 1e+229)
		tmp = y * -sqrt((exp((t * t)) * (2.0 * z)));
	else
		tmp = t_1 * sqrt((2.0 * (z + (z * (t * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1e-16], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+229], N[(y * (-N[Sqrt[N[(N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision] * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(2.0 * N[(z + N[(z * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 9.9999999999999998e-17

   1. Initial program 99.7%

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

      1. associate-*l* 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. exp-sqrt 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 96.9%

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

      2. expm1-udef 54.0%

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

      3. sqrt-unprod 54.0%

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

      4. associate-*l* 54.0%

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

      5. exp-prod 54.0%

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

   5. Applied egg-rr 54.0%

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

      1. expm1-def 96.9%

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

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

      6. exp-prod 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in t around 0 99.7%

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

   if 9.9999999999999998e-17 < (*.f64 t t) < 9.9999999999999999e228

   1. Initial program 100.0%

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

      1. associate-*r* 100.0%

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

      2. exp-sqrt 100.0%

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

      3. *-commutative 100.0%

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

      4. flip-- 90.5%

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

      5. associate-*r/ 90.5%

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

   3. Applied egg-rr 90.5%

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

      1. associate-/l* 90.5%

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

      2. associate-*l* 90.5%

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

   5. Simplified 90.5%

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

   6. Taylor expanded in x around 0 75.4%

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

      1. associate-/r/ 75.4%

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

      2. div-inv 75.4%

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

      3. associate-*r* 75.4%

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

      4. *-commutative 75.4%

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

      5. pow-exp 75.4%

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

      6. metadata-eval 75.4%

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

   8. Applied egg-rr 75.4%

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

   if 9.9999999999999999e228 < (*.f64 t t)

   1. Initial program 98.8%

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

      1. associate-*l* 100.0%

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

      2. exp-sqrt 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. associate-*l* 100.0%

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

      5. exp-prod 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. exp-prod 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around 0 93.1%

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

      1. *-commutative 93.1%

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

      2. unpow2 93.1%

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

   10. Simplified 93.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-16}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \cdot t \leq 10^{+229}:\\ \;\;\;\;y \cdot \left(-\sqrt{e^{t \cdot t} \cdot \left(2 \cdot z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]

Alternative 3: 83.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l* 99.9%

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

   2. exp-sqrt 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 78.4%

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

   3. sqrt-unprod 78.4%

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

   4. associate-*l* 78.4%

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

   5. exp-prod 78.4%

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

5. Applied egg-rr 78.4%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-*l* 99.9%

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

   5. *-commutative 99.9%

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

   6. exp-prod 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in t around 0 83.3%

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

   1. *-commutative 83.3%

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

   2. unpow2 83.3%

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

10. Simplified 83.3%

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

11. Final simplification 83.3%

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

Alternative 4: 56.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l* 99.9%

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

   2. exp-sqrt 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 78.4%

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

   3. sqrt-unprod 78.4%

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

   4. associate-*l* 78.4%

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

   5. exp-prod 78.4%

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

5. Applied egg-rr 78.4%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-*l* 99.9%

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

   5. *-commutative 99.9%

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

   6. exp-prod 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in t around 0 52.4%

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

9. Final simplification 52.4%

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

Alternative 5: 29.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*r* 99.9%

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

   2. exp-sqrt 99.9%

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

   3. *-commutative 99.9%

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

   4. flip-- 74.0%

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

   5. associate-*r/ 71.2%

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

3. Applied egg-rr 71.2%

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

   1. associate-/l* 74.0%

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

   2. associate-*l* 74.0%

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

5. Simplified 74.0%

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

6. Taylor expanded in x around 0 61.8%

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

7. Taylor expanded in t around 0 25.1%

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

   1. rem-log-exp 25.9%

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

   2. log-pow 25.9%

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

   3. unpow2 25.9%

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

   4. prod-exp 25.9%

      \[\leadsto \frac{\sqrt{\log \color{blue}{\left(e^{z + z}\right)}}}{\frac{-1}{y}} \]

   5. rem-log-exp 25.1%

      \[\leadsto \frac{\sqrt{\color{blue}{z + z}}}{\frac{-1}{y}} \]

9. Simplified 25.1%

   \[\leadsto \frac{\sqrt{\color{blue}{z + z}}}{\frac{-1}{y}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 15.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{z + z}}{\frac{-1}{y}}\right)\right)} \]

    2. expm1-udef 11.9%

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

    3. associate-/r/ 11.9%

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

11. Applied egg-rr 11.9%

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

    1. expm1-def 15.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{z + z}}{-1} \cdot y\right)\right)} \]

    2. expm1-log1p 25.1%

       \[\leadsto \color{blue}{\frac{\sqrt{z + z}}{-1} \cdot y} \]

    3. associate-/r/ 25.1%

       \[\leadsto \color{blue}{\frac{\sqrt{z + z}}{\frac{-1}{y}}} \]

    4. *-rgt-identity 25.1%

       \[\leadsto \frac{\color{blue}{\sqrt{z + z} \cdot 1}}{\frac{-1}{y}} \]

    5. associate-*r/ 25.1%

       \[\leadsto \color{blue}{\sqrt{z + z} \cdot \frac{1}{\frac{-1}{y}}} \]

    6. associate-/r/ 25.1%

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

    7. metadata-eval 25.1%

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

    8. neg-mul-1 25.1%

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

13. Simplified 25.1%

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

14. Final simplification 25.1%

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

Developer target: 99.5% 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 2023199 
(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))))