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

Percentage Accurate: 99.5% → 99.8%

Time: 12.1s

Alternatives: 10

Speedup: 1.0×

• 44.3% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(Float64(2.0 * z) * (exp(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[(N[(2.0 * z), $MachinePrecision] * N[Power[N[Exp[t], $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 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. remove-double-neg 99.8%

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

8. Simplified 99.8%

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

   1. pow2 99.8%

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

   2. exp-prod 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 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. remove-double-neg 99.8%

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

   3. remove-double-neg 99.8%

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

   4. exp-sqrt 99.8%

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

   5. exp-prod 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. sqrt-unprod 99.8%

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

   3. associate-*l* 99.8%

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

   4. pow-exp 99.8%

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

   5. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r* 99.8%

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

   3. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 4 \cdot 10^{-7} \lor \neg \left(t \cdot t \leq 5 \cdot 10^{+286}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{2 \cdot z}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* t t) 4e-7) (not (<= (* t t) 5e+286)))
   (* (- (* x 0.5) y) (sqrt (* (* 2.0 z) (fma t t 1.0))))
   (* (exp (/ (* t t) 2.0)) (* y (- (sqrt (* 2.0 z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((t * t) <= 4e-7) || !((t * t) <= 5e+286)) {
		tmp = ((x * 0.5) - y) * sqrt(((2.0 * z) * fma(t, t, 1.0)));
	} else {
		tmp = exp(((t * t) / 2.0)) * (y * -sqrt((2.0 * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(t * t) <= 4e-7) || !(Float64(t * t) <= 5e+286))
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))));
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * Float64(-sqrt(Float64(2.0 * z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(t * t), $MachinePrecision], 4e-7], N[Not[LessEqual[N[(t * t), $MachinePrecision], 5e+286]], $MachinePrecision]], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * (-N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 3.9999999999999998e-7 or 5.0000000000000004e286 < (*.f64 t t)

   1. Initial program 99.3%

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

      1. associate-*l* 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. remove-double-neg 99.8%

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

      3. remove-double-neg 99.8%

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

      4. exp-sqrt 99.8%

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

      5. exp-prod 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. associate-*l* 99.8%

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

      4. pow-exp 99.8%

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

      5. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in t around 0 99.7%

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

       1. +-commutative 99.7%

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

       2. unpow2 99.7%

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

       3. fma-define 99.7%

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

   11. Simplified 99.7%

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

   if 3.9999999999999998e-7 < (*.f64 t t) < 5.0000000000000004e286

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.4%

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

      1. mul-1-neg 78.4%

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

      2. distribute-rgt-neg-in 78.4%

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

   5. Simplified 78.4%

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

      1. distribute-rgt-neg-out 78.4%

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

      2. neg-sub0 78.4%

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

      3. associate-*l* 78.4%

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

      4. sqrt-prod 78.4%

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

      5. *-commutative 78.4%

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

      6. *-commutative 78.4%

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

   7. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 78.4%

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

      2. distribute-rgt-neg-in 78.4%

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

   9. Simplified 78.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 4 \cdot 10^{-7} \lor \neg \left(t \cdot t \leq 5 \cdot 10^{+286}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{2 \cdot z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \cdot t \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= (* t t) 6e-7)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* y (- 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) <= 6e-7) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * (y * -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) <= 6d-7) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * (y * -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) <= 6e-7) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (y * -t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (t * t) <= 6e-7:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * (y * -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) <= 6e-7)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * Float64(-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) <= 6e-7)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * (y * -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[LessEqual[N[(t * t), $MachinePrecision], 6e-7], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * (-t$95$1)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 5.9999999999999997e-7

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 99.3%

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

   if 5.9999999999999997e-7 < (*.f64 t t)

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 73.1%

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

      1. mul-1-neg 73.1%

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

      2. distribute-rgt-neg-in 73.1%

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

   5. Simplified 73.1%

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

      1. distribute-rgt-neg-out 73.1%

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

      2. neg-sub0 73.1%

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

      3. associate-*l* 73.0%

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

      4. sqrt-prod 73.1%

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

      5. *-commutative 73.1%

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

      6. *-commutative 73.1%

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

   7. Applied egg-rr 73.1%

      \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
   8. Step-by-step derivation

      1. neg-sub0 73.1%

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

      2. distribute-rgt-neg-in 73.1%

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

   9. Simplified 73.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \left(-\sqrt{2 \cdot z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= t 7.8e+31)
     (* (- (* x 0.5) y) t_1)
     (* (exp (/ (* t t) 2.0)) (* y t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if (t <= 7.8e+31) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = exp(((t * t) / 2.0)) * (y * 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 <= 7.8d+31) then
        tmp = ((x * 0.5d0) - y) * t_1
    else
        tmp = exp(((t * t) / 2.0d0)) * (y * 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 <= 7.8e+31) {
		tmp = ((x * 0.5) - y) * t_1;
	} else {
		tmp = Math.exp(((t * t) / 2.0)) * (y * t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if t <= 7.8e+31:
		tmp = ((x * 0.5) - y) * t_1
	else:
		tmp = math.exp(((t * t) / 2.0)) * (y * t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 7.8e+31)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * t_1);
	else
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(y * 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 <= 7.8e+31)
		tmp = ((x * 0.5) - y) * t_1;
	else
		tmp = exp(((t * t) / 2.0)) * (y * 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[LessEqual[t, 7.8e+31], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.79999999999999999e31

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0 67.4%

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

   if 7.79999999999999999e31 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. distribute-rgt-neg-in 76.9%

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

   5. Simplified 76.9%

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

      1. pow1 76.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 13.5%

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

      4. sqr-neg 13.5%

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

      5. add-sqr-sqrt 13.5%

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

      6. associate-*l* 13.5%

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

      7. sqrt-prod 13.5%

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

   7. Applied egg-rr 13.5%

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

      1. unpow1 13.5%

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

   9. Simplified 13.5%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(y \cdot \sqrt{2 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {t\_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.7e+26)
     (* t_1 (sqrt (* 2.0 z)))
     (sqrt (* (* 2.0 z) (pow t_1 2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.7e+26) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = sqrt(((2.0 * z) * pow(t_1, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 1.7d+26) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = sqrt(((2.0d0 * z) * (t_1 ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.7e+26) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(((2.0 * z) * Math.pow(t_1, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 1.7e+26:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(((2.0 * z) * math.pow(t_1, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.7e+26)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = sqrt(Float64(Float64(2.0 * z) * (t_1 ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 1.7e+26)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = sqrt(((2.0 * z) * (t_1 ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 1.7e+26], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.7000000000000001e26

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0 68.1%

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

   if 1.7000000000000001e26 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 9.8%

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

      1. add-sqr-sqrt 8.6%

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

      2. sqrt-unprod 29.2%

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

      3. *-commutative 29.2%

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

      4. *-commutative 29.2%

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

      5. swap-sqr 32.7%

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

      6. add-sqr-sqrt 32.7%

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

      7. pow2 32.7%

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

   5. Applied egg-rr 32.7%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot {\left(x \cdot 0.5 - y\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 8: 57.0% 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 99.4%

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

3. Taylor expanded in t around 0 55.8%

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

4. Final simplification 55.8%

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

Alternative 9: 30.4% accurate, 2.0× speedup

Math

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

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 * Float64(-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 99.4%

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

3. Taylor expanded in x around 0 60.7%

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

   1. mul-1-neg 60.7%

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

   2. distribute-rgt-neg-in 60.7%

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

5. Simplified 60.7%

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

   1. distribute-rgt-neg-out 60.7%

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

   2. neg-sub0 60.7%

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

   3. associate-*l* 60.7%

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

   4. sqrt-prod 60.8%

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

   5. *-commutative 60.8%

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

   6. *-commutative 60.8%

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

7. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\left(0 - y \cdot \sqrt{2 \cdot z}\right)} \cdot e^{\frac{t \cdot t}{2}} \]
8. Step-by-step derivation

   1. neg-sub0 60.8%

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

   2. distribute-rgt-neg-in 60.8%

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

9. Simplified 60.8%

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

10. Taylor expanded in t around 0 27.1%

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

11. Final simplification 27.1%

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

Alternative 10: 0.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0 60.7%

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

   1. mul-1-neg 60.7%

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

   2. distribute-rgt-neg-in 60.7%

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

5. Simplified 60.7%

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

6. Taylor expanded in t around 0 27.0%

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

   1. mul-1-neg 27.0%

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

   2. associate-*l* 27.0%

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

   3. *-commutative 27.0%

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

   4. distribute-rgt-neg-in 27.0%

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

   5. distribute-rgt-neg-in 27.0%

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

8. Simplified 27.0%

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

10. Applied egg-rr 0.0%

    \[\leadsto y \cdot \color{blue}{\left(0 + \sqrt{-2 \cdot z}\right)} \]
11. Step-by-step derivation

    1. +-lft-identity 0.0%

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

    2. *-commutative 0.0%

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

12. Simplified 0.0%

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

13. Final simplification 0.0%

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

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 2024100 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (* (* (- (* 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))))