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

Percentage Accurate: 99.5% → 99.8%

Time: 17.8s

Alternatives: 14

Speedup: 1.0×

• 45.5% 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: 99.5% 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 99.0%

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

   1. associate-*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. exp-sqrt 99.8%

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

   3. 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. 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 99.0%

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

   1. associate-*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. exp-sqrt 99.8%

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

   3. 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. Add Preprocessing

Alternative 3: 91.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 0.0001:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{elif}\;t \cdot t \leq 10^{+268}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 0.0001)
     (* t_1 (sqrt (* (* 2.0 z) (fma t t 1.0))))
     (if (<= (* t t) 1e+268)
       (* (exp (/ (* t t) 2.0)) (* 0.5 (* x (sqrt (* 2.0 z)))))
       (* t_1 (sqrt (* (* 2.0 z) (pow t 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 0.0001) {
		tmp = t_1 * sqrt(((2.0 * z) * fma(t, t, 1.0)));
	} else if ((t * t) <= 1e+268) {
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * sqrt((2.0 * z))));
	} else {
		tmp = t_1 * sqrt(((2.0 * z) * pow(t, 2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 0.0001)
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))));
	elseif (Float64(t * t) <= 1e+268)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(0.5 * Float64(x * sqrt(Float64(2.0 * z)))));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * (t ^ 2.0))));
	end
	return 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], 0.0001], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+268], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(x * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 1.00000000000000005e-4

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. exp-sqrt 99.6%

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

      3. exp-prod 99.6%

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

   3. Simplified 99.6%

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

      1. pow1 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. associate-*l* 99.6%

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

      4. pow-exp 99.6%

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

      5. pow2 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow1 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in t around 0 99.3%

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

       1. +-commutative 99.3%

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

       2. unpow2 99.3%

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

       3. fma-define 99.3%

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

   11. Simplified 99.3%

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

   if 1.00000000000000005e-4 < (*.f64 t t) < 9.9999999999999997e267

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf 75.9%

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

      1. pow1 75.9%

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

      2. associate-*l* 75.9%

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

      3. *-commutative 75.9%

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

      4. sqrt-unprod 75.9%

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

      5. *-commutative 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. unpow1 75.9%

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

   7. Simplified 75.9%

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

   if 9.9999999999999997e267 < (*.f64 t t)

   1. Initial program 98.7%

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

      1. associate-*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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 96.3%

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

       1. +-commutative 96.3%

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

       2. unpow2 96.3%

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

       3. fma-define 96.3%

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

   11. Simplified 96.3%

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

   12. Taylor expanded in t around inf 96.3%

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

       1. *-commutative 96.3%

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

       2. associate-*l* 96.3%

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

       3. *-commutative 96.3%

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

   14. Simplified 96.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.0001:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{elif}\;t \cdot t \leq 10^{+268}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ t_2 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 0.0001:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+268}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))) (t_2 (- (* x 0.5) y)))
   (if (<= (* t t) 0.0001)
     (* t_2 (* t_1 (hypot 1.0 t)))
     (if (<= (* t t) 1e+268)
       (* (exp (/ (* t t) 2.0)) (* 0.5 (* x t_1)))
       (* t_2 (sqrt (* (* 2.0 z) (pow t 2.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double t_2 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 0.0001) {
		tmp = t_2 * (t_1 * hypot(1.0, t));
	} else if ((t * t) <= 1e+268) {
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * t_1));
	} else {
		tmp = t_2 * sqrt(((2.0 * z) * pow(t, 2.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double t_2 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 0.0001) {
		tmp = t_2 * (t_1 * Math.hypot(1.0, t));
	} else if ((t * t) <= 1e+268) {
		tmp = Math.exp(((t * t) / 2.0)) * (0.5 * (x * t_1));
	} else {
		tmp = t_2 * Math.sqrt(((2.0 * z) * Math.pow(t, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	t_2 = (x * 0.5) - y
	tmp = 0
	if (t * t) <= 0.0001:
		tmp = t_2 * (t_1 * math.hypot(1.0, t))
	elif (t * t) <= 1e+268:
		tmp = math.exp(((t * t) / 2.0)) * (0.5 * (x * t_1))
	else:
		tmp = t_2 * math.sqrt(((2.0 * z) * math.pow(t, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	t_2 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 0.0001)
		tmp = Float64(t_2 * Float64(t_1 * hypot(1.0, t)));
	elseif (Float64(t * t) <= 1e+268)
		tmp = Float64(exp(Float64(Float64(t * t) / 2.0)) * Float64(0.5 * Float64(x * t_1)));
	else
		tmp = Float64(t_2 * sqrt(Float64(Float64(2.0 * z) * (t ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	t_2 = (x * 0.5) - y;
	tmp = 0.0;
	if ((t * t) <= 0.0001)
		tmp = t_2 * (t_1 * hypot(1.0, t));
	elseif ((t * t) <= 1e+268)
		tmp = exp(((t * t) / 2.0)) * (0.5 * (x * t_1));
	else
		tmp = t_2 * sqrt(((2.0 * z) * (t ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 0.0001], N[(t$95$2 * N[(t$95$1 * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 1e+268], N[(N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 1.00000000000000005e-4

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. exp-sqrt 99.6%

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

      3. exp-prod 99.6%

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

   3. Simplified 99.6%

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

      1. pow1 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. associate-*l* 99.6%

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

      4. pow-exp 99.6%

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

      5. pow2 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow1 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in t around 0 99.3%

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

       1. +-commutative 99.3%

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

       2. unpow2 99.3%

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

       3. fma-define 99.3%

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

   11. Simplified 99.3%

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

       1. sqrt-prod 99.2%

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

   13. Applied egg-rr 99.2%

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

       1. fma-undefine 99.2%

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

       2. unpow2 99.2%

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

       3. +-commutative 99.2%

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

       4. unpow2 99.2%

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

       5. hypot-1-def 99.3%

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

   15. Simplified 99.3%

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

   if 1.00000000000000005e-4 < (*.f64 t t) < 9.9999999999999997e267

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf 75.9%

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

      1. pow1 75.9%

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

      2. associate-*l* 75.9%

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

      3. *-commutative 75.9%

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

      4. sqrt-unprod 75.9%

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

      5. *-commutative 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. unpow1 75.9%

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

   7. Simplified 75.9%

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

   if 9.9999999999999997e267 < (*.f64 t t)

   1. Initial program 98.7%

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

      1. associate-*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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 96.3%

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

       1. +-commutative 96.3%

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

       2. unpow2 96.3%

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

       3. fma-define 96.3%

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

   11. Simplified 96.3%

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

   12. Taylor expanded in t around inf 96.3%

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

       1. *-commutative 96.3%

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

       2. associate-*l* 96.3%

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

       3. *-commutative 96.3%

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

   14. Simplified 96.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.0001:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+268}:\\ \;\;\;\;e^{\frac{t \cdot t}{2}} \cdot \left(0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 10^{+87}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1e+87)
     (* t_1 (* (sqrt (* 2.0 z)) (hypot 1.0 t)))
     (* t_1 (sqrt (* (* 2.0 z) (pow t 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1e+87) {
		tmp = t_1 * (sqrt((2.0 * z)) * hypot(1.0, t));
	} else {
		tmp = t_1 * sqrt(((2.0 * z) * pow(t, 2.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1e+87) {
		tmp = t_1 * (Math.sqrt((2.0 * z)) * Math.hypot(1.0, t));
	} else {
		tmp = t_1 * Math.sqrt(((2.0 * z) * Math.pow(t, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 1e+87:
		tmp = t_1 * (math.sqrt((2.0 * z)) * math.hypot(1.0, t))
	else:
		tmp = t_1 * math.sqrt(((2.0 * z) * math.pow(t, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1e+87)
		tmp = Float64(t_1 * Float64(sqrt(Float64(2.0 * z)) * hypot(1.0, t)));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * (t ^ 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 <= 1e+87)
		tmp = t_1 * (sqrt((2.0 * z)) * hypot(1.0, t));
	else
		tmp = t_1 * sqrt(((2.0 * z) * (t ^ 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, 1e+87], N[(t$95$1 * N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999996e86

   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. 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. exp-sqrt 99.8%

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

      3. 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 84.0%

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

       1. +-commutative 84.0%

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

       2. unpow2 84.0%

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

       3. fma-define 84.0%

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

   11. Simplified 84.0%

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

       1. sqrt-prod 82.6%

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

   13. Applied egg-rr 82.6%

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

       1. fma-undefine 82.6%

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

       2. unpow2 82.6%

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

       3. +-commutative 82.6%

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

       4. unpow2 82.6%

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

       5. hypot-1-def 77.0%

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

   15. Simplified 77.0%

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

   if 9.9999999999999996e86 < t

   1. Initial program 96.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 100.0%

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

      5. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around 0 85.2%

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

       1. +-commutative 85.2%

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

       2. unpow2 85.2%

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

       3. fma-define 85.2%

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

   11. Simplified 85.2%

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

   12. Taylor expanded in t around inf 85.2%

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

       1. *-commutative 85.2%

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

       2. associate-*l* 85.2%

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

       3. *-commutative 85.2%

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

   14. Simplified 85.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{+87}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.0)
     (* t_1 (sqrt (* 2.0 z)))
     (* t_1 (sqrt (* (* 2.0 z) (pow t 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = t_1 * sqrt(((2.0 * z) * pow(t, 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 <= 1.0d0) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = t_1 * sqrt(((2.0d0 * z) * (t ** 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 <= 1.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = t_1 * Math.sqrt(((2.0 * z) * Math.pow(t, 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 1.0:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = t_1 * math.sqrt(((2.0 * z) * math.pow(t, 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(t_1 * sqrt(Float64(Float64(2.0 * z) * (t ^ 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 <= 1.0)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = t_1 * sqrt(((2.0 * z) * (t ^ 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, 1.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   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. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      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 68.7%

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

   if 1 < t

   1. Initial program 97.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 99.9%

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

      5. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in t around 0 68.9%

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

       1. +-commutative 68.9%

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

       2. unpow2 68.9%

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

       3. fma-define 68.9%

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

   11. Simplified 68.9%

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

   12. Taylor expanded in t around inf 68.9%

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

       1. *-commutative 68.9%

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

       2. associate-*l* 68.9%

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

       3. *-commutative 68.9%

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

   14. Simplified 68.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot {t}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 1.0)
     (* t_1 (sqrt (* 2.0 z)))
     (* (* t (* t_1 (sqrt 2.0))) (sqrt z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * sqrt((2.0 * z));
	} else {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(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) :: t_1
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    if (t <= 1.0d0) then
        tmp = t_1 * sqrt((2.0d0 * z))
    else
        tmp = (t * (t_1 * sqrt(2.0d0))) * sqrt(z)
    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 <= 1.0) {
		tmp = t_1 * Math.sqrt((2.0 * z));
	} else {
		tmp = (t * (t_1 * Math.sqrt(2.0))) * Math.sqrt(z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	tmp = 0
	if t <= 1.0:
		tmp = t_1 * math.sqrt((2.0 * z))
	else:
		tmp = (t * (t_1 * math.sqrt(2.0))) * math.sqrt(z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 1.0)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	tmp = 0.0;
	if (t <= 1.0)
		tmp = t_1 * sqrt((2.0 * z));
	else
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	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, 1.0], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   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. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      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 68.7%

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

   if 1 < t

   1. Initial program 97.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 99.9%

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

      5. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in t around 0 68.9%

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

       1. +-commutative 68.9%

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

       2. unpow2 68.9%

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

       3. fma-define 68.9%

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

   11. Simplified 68.9%

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

   12. Taylor expanded in t around inf 51.3%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 99.0%

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

3. Final simplification 99.0%

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

Alternative 9: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(t \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* 2.0 z))))
   (if (<= t 1.0) (* t_1 t_2) (* t_1 (* t t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double t_2 = sqrt((2.0 * z));
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_1 * (t * t_2);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * 0.5d0) - y
    t_2 = sqrt((2.0d0 * z))
    if (t <= 1.0d0) then
        tmp = t_1 * t_2
    else
        tmp = t_1 * (t * t_2)
    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 t_2 = Math.sqrt((2.0 * z));
	double tmp;
	if (t <= 1.0) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_1 * (t * t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * 0.5) - y
	t_2 = math.sqrt((2.0 * z))
	tmp = 0
	if t <= 1.0:
		tmp = t_1 * t_2
	else:
		tmp = t_1 * (t * t_2)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	t_2 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 1.0)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(t_1 * Float64(t * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * 0.5) - y;
	t_2 = sqrt((2.0 * z));
	tmp = 0.0;
	if (t <= 1.0)
		tmp = t_1 * t_2;
	else
		tmp = t_1 * (t * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1.0], N[(t$95$1 * t$95$2), $MachinePrecision], N[(t$95$1 * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   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. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      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 68.7%

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

   if 1 < t

   1. Initial program 97.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 99.9%

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

      5. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in t around 0 68.9%

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

       1. +-commutative 68.9%

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

       2. unpow2 68.9%

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

       3. fma-define 68.9%

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

   11. Simplified 68.9%

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

   12. Taylor expanded in t around inf 47.2%

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

       1. associate-*l* 47.2%

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

   14. Simplified 47.2%

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

       1. sqrt-prod 47.2%

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

       2. pow1/2 47.2%

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

   16. Applied egg-rr 47.2%

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

       1. unpow1/2 47.2%

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

   18. Simplified 47.2%

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

Alternative 10: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- (* x 0.5) y) (sqrt (* 2.0 z)))))
   (if (<= t 1.0) t_1 (* t t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * 0.5) - y) * sqrt((2.0 * z));
	double tmp;
	if (t <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t * 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 = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    if (t <= 1.0d0) then
        tmp = t_1
    else
        tmp = t * t_1
    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) * Math.sqrt((2.0 * z));
	double tmp;
	if (t <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t * t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	tmp = 0
	if t <= 1.0:
		tmp = t_1
	else:
		tmp = t * t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)))
	tmp = 0.0
	if (t <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(t * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x * 0.5) - y) * sqrt((2.0 * z));
	tmp = 0.0;
	if (t <= 1.0)
		tmp = t_1;
	else
		tmp = t * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.0], t$95$1, N[(t * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   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. 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. exp-prod 99.7%

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

   3. Simplified 99.7%

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

      1. pow1 99.7%

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

      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 68.7%

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

   if 1 < t

   1. Initial program 97.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   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. exp-prod 100.0%

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

   3. Simplified 100.0%

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

      1. pow1 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-exp 99.9%

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

      5. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. unpow1 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in t around 0 68.9%

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

       1. +-commutative 68.9%

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

       2. unpow2 68.9%

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

       3. fma-define 68.9%

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

   11. Simplified 68.9%

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

   12. Taylor expanded in t around inf 47.2%

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

       1. associate-*l* 47.2%

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

   14. Simplified 47.2%

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

       1. *-commutative 47.2%

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

       2. sub-neg 47.2%

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

       3. *-commutative 47.2%

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

       4. distribute-lft-in 43.0%

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

       5. sqrt-prod 43.0%

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

       6. *-commutative 43.0%

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

       7. *-commutative 43.0%

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

       8. sqrt-prod 43.0%

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

       9. *-commutative 43.0%

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

   16. Applied egg-rr 43.0%

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

       1. *-commutative 43.0%

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

       2. *-commutative 43.0%

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

       3. associate-*r* 42.9%

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

       4. associate-*r* 42.9%

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

       5. associate-*r* 43.0%

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

       6. distribute-rgt-out 47.2%

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

       7. associate-*r* 47.2%

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

       8. distribute-rgt-in 47.2%

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

       9. *-commutative 47.2%

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

       10. sub-neg 47.2%

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

   18. Simplified 47.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.7 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.7e+114)
   (sqrt (* (* 2.0 z) (* y (- y x))))
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.7e+114) {
		tmp = sqrt(((2.0 * z) * (y * (y - x))));
	} else {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * 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 (y <= (-9.7d+114)) then
        tmp = sqrt(((2.0d0 * z) * (y * (y - x))))
    else
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.7e+114) {
		tmp = Math.sqrt(((2.0 * z) * (y * (y - x))));
	} else {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9.7e+114:
		tmp = math.sqrt(((2.0 * z) * (y * (y - x))))
	else:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.7e+114)
		tmp = sqrt(Float64(Float64(2.0 * z) * Float64(y * Float64(y - x))));
	else
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9.7e+114)
		tmp = sqrt(((2.0 * z) * (y * (y - x))));
	else
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.7e+114], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.7000000000000002e114

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0 48.8%

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

      1. associate-*r* 48.7%

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

      2. *-commutative 48.7%

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

      3. *-commutative 48.7%

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

      4. add-sqr-sqrt 48.3%

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

      5. sqrt-unprod 61.0%

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

      6. *-commutative 61.0%

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

      7. *-commutative 61.0%

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

      8. swap-sqr 63.7%

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

      9. sqrt-unprod 63.7%

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

      10. sqrt-unprod 63.7%

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

      11. add-sqr-sqrt 63.7%

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

      12. *-commutative 63.7%

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

      13. pow2 63.7%

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

      14. fma-neg 63.7%

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

   5. Applied egg-rr 63.7%

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

      1. fma-neg 63.7%

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

      2. *-commutative 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in x around 0 55.3%

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

      1. +-commutative 55.3%

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

      2. mul-1-neg 55.3%

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

      3. unsub-neg 55.3%

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

      4. unpow2 55.3%

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

      5. distribute-rgt-out-- 63.7%

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

   10. Simplified 63.7%

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

   if -9.7000000000000002e114 < y

   1. Initial program 98.9%

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

      1. 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. exp-sqrt 99.8%

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

      3. 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 54.6%

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

Alternative 12: 37.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.8e-19)
   (sqrt (* (* 2.0 z) (* y (- y x))))
   (* (sqrt (* 2.0 z)) (* x 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.8e-19) {
		tmp = sqrt(((2.0 * z) * (y * (y - x))));
	} else {
		tmp = sqrt((2.0 * z)) * (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.8d-19)) then
        tmp = sqrt(((2.0d0 * z) * (y * (y - x))))
    else
        tmp = sqrt((2.0d0 * z)) * (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.8e-19) {
		tmp = Math.sqrt(((2.0 * z) * (y * (y - x))));
	} else {
		tmp = Math.sqrt((2.0 * z)) * (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.8e-19:
		tmp = math.sqrt(((2.0 * z) * (y * (y - x))))
	else:
		tmp = math.sqrt((2.0 * z)) * (x * 0.5)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.8e-19)
		tmp = sqrt(Float64(Float64(2.0 * z) * Float64(y * Float64(y - x))));
	else
		tmp = Float64(sqrt(Float64(2.0 * z)) * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.8e-19)
		tmp = sqrt(((2.0 * z) * (y * (y - x))));
	else
		tmp = sqrt((2.0 * z)) * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.8e-19], N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(y * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000003e-19

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0 50.5%

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

      1. associate-*r* 50.4%

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

      2. *-commutative 50.4%

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

      3. *-commutative 50.4%

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

      4. add-sqr-sqrt 42.6%

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

      5. sqrt-unprod 49.7%

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

      6. *-commutative 49.7%

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

      7. *-commutative 49.7%

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

      8. swap-sqr 52.6%

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

      9. sqrt-unprod 52.7%

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

      10. sqrt-unprod 52.7%

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

      11. add-sqr-sqrt 52.7%

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

      12. *-commutative 52.7%

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

      13. pow2 52.7%

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

      14. fma-neg 52.7%

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

   5. Applied egg-rr 52.7%

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

      1. fma-neg 52.7%

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

      2. *-commutative 52.7%

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

   7. Simplified 52.7%

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

   8. Taylor expanded in x around 0 46.6%

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

      1. +-commutative 46.6%

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

      2. mul-1-neg 46.6%

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

      3. unsub-neg 46.6%

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

      4. unpow2 46.6%

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

      5. distribute-rgt-out-- 51.3%

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

   10. Simplified 51.3%

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

   if -2.80000000000000003e-19 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in t around 0 54.3%

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

   4. Taylor expanded in x around inf 34.4%

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

      1. associate-*r* 34.4%

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

      2. *-commutative 34.4%

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

      3. *-commutative 34.4%

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

      4. associate-*r* 34.4%

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

      5. *-commutative 34.4%

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

   6. Simplified 34.4%

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

      1. pow1 34.4%

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

      2. *-commutative 34.4%

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

      3. *-commutative 34.4%

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

      4. associate-*l* 34.4%

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

      5. sqrt-prod 35.0%

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

   8. Applied egg-rr 35.0%

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

      1. unpow1 35.0%

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

      2. *-commutative 35.0%

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

   10. Simplified 35.0%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \left(y \cdot \left(y - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.26e+48) (* x (sqrt (* 0.5 z))) (sqrt (* z (* 0.5 (* x x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.26e+48) {
		tmp = x * sqrt((0.5 * z));
	} else {
		tmp = sqrt((z * (0.5 * (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 (t <= 1.26d+48) then
        tmp = x * sqrt((0.5d0 * z))
    else
        tmp = sqrt((z * (0.5d0 * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.26e+48) {
		tmp = x * Math.sqrt((0.5 * z));
	} else {
		tmp = Math.sqrt((z * (0.5 * (x * x))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.26e+48:
		tmp = x * math.sqrt((0.5 * z))
	else:
		tmp = math.sqrt((z * (0.5 * (x * x))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.26e+48)
		tmp = Float64(x * sqrt(Float64(0.5 * z)));
	else
		tmp = sqrt(Float64(z * Float64(0.5 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.26e+48)
		tmp = x * sqrt((0.5 * z));
	else
		tmp = sqrt((z * (0.5 * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.26e+48], N[(x * N[Sqrt[N[(0.5 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(z * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.26000000000000001e48

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 65.2%

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

   4. Taylor expanded in x around inf 35.9%

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

      1. associate-*r* 35.9%

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

      2. *-commutative 35.9%

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

      3. *-commutative 35.9%

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

      4. associate-*r* 35.9%

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

      5. *-commutative 35.9%

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

   6. Simplified 35.9%

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

      1. add-sqr-sqrt 16.4%

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

      2. sqrt-unprod 17.2%

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

      3. *-commutative 17.2%

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

      4. *-commutative 17.2%

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

      5. swap-sqr 15.4%

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

      6. add-sqr-sqrt 15.5%

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

      7. *-commutative 15.5%

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

      8. *-commutative 15.5%

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

      9. swap-sqr 15.4%

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

      10. rem-square-sqrt 15.5%

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

      11. pow2 15.5%

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

   8. Applied egg-rr 15.5%

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

      1. *-commutative 15.5%

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

      2. unpow2 15.5%

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

      3. rem-square-sqrt 15.4%

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

      4. swap-sqr 15.5%

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

      5. associate-*r* 15.5%

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

      6. associate-*r* 15.5%

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

      7. *-commutative 15.5%

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

      8. *-commutative 15.5%

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

      9. swap-sqr 15.4%

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

      10. swap-sqr 15.4%

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

      11. metadata-eval 15.4%

          \[\leadsto \sqrt{z \cdot \left(\left(\color{blue}{0.25} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right) \cdot \left(x \cdot x\right)\right)} \]

      12. rem-square-sqrt 15.5%

          \[\leadsto \sqrt{z \cdot \left(\left(0.25 \cdot \color{blue}{2}\right) \cdot \left(x \cdot x\right)\right)} \]

      13. metadata-eval 15.5%

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

   10. Simplified 15.5%

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

       1. associate-*r* 15.5%

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

       2. sqrt-prod 14.0%

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

       3. *-commutative 14.0%

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

       4. sqrt-prod 16.3%

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

       5. add-sqr-sqrt 36.0%

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

   12. Applied egg-rr 36.0%

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

   if 1.26000000000000001e48 < t

   1. Initial program 96.7%

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

   3. Taylor expanded in t around 0 15.4%

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

   4. Taylor expanded in x around inf 8.0%

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

      1. associate-*r* 8.0%

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

      2. *-commutative 8.0%

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

      3. *-commutative 8.0%

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

      4. associate-*r* 8.0%

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

      5. *-commutative 8.0%

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

   6. Simplified 8.0%

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

      1. add-sqr-sqrt 3.2%

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

      2. sqrt-unprod 11.1%

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

      3. *-commutative 11.1%

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

      4. *-commutative 11.1%

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

      5. swap-sqr 11.1%

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

      6. add-sqr-sqrt 11.1%

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

      7. *-commutative 11.1%

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

      8. *-commutative 11.1%

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

      9. swap-sqr 11.1%

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

      10. rem-square-sqrt 11.1%

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

      11. pow2 11.1%

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

   8. Applied egg-rr 11.1%

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

      1. *-commutative 11.1%

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

      2. unpow2 11.1%

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

      3. rem-square-sqrt 11.1%

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

      4. swap-sqr 11.1%

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

      5. associate-*r* 11.1%

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

      6. associate-*r* 11.1%

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

      7. *-commutative 11.1%

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

      8. *-commutative 11.1%

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

      9. swap-sqr 11.1%

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

      10. swap-sqr 11.1%

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

      11. metadata-eval 11.1%

          \[\leadsto \sqrt{z \cdot \left(\left(\color{blue}{0.25} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right) \cdot \left(x \cdot x\right)\right)} \]

      12. rem-square-sqrt 11.1%

          \[\leadsto \sqrt{z \cdot \left(\left(0.25 \cdot \color{blue}{2}\right) \cdot \left(x \cdot x\right)\right)} \]

      13. metadata-eval 11.1%

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

   10. Simplified 11.1%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \sqrt{0.5 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.9% 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 99.0%

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

3. Taylor expanded in t around 0 53.3%

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

4. Taylor expanded in x around inf 29.2%

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

   1. associate-*r* 29.2%

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

   2. *-commutative 29.2%

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

   3. *-commutative 29.2%

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

   4. associate-*r* 29.2%

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

   5. *-commutative 29.2%

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

6. Simplified 29.2%

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

   1. add-sqr-sqrt 13.2%

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

   2. sqrt-unprod 15.7%

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

   3. *-commutative 15.7%

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

   4. *-commutative 15.7%

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

   5. swap-sqr 14.4%

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

   6. add-sqr-sqrt 14.4%

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

   7. *-commutative 14.4%

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

   8. *-commutative 14.4%

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

   9. swap-sqr 14.4%

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

   10. rem-square-sqrt 14.4%

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

   11. pow2 14.4%

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

8. Applied egg-rr 14.4%

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

   1. *-commutative 14.4%

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

   2. unpow2 14.4%

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

   3. rem-square-sqrt 14.4%

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

   4. swap-sqr 14.4%

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

   5. associate-*r* 14.4%

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

   6. associate-*r* 14.4%

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

   7. *-commutative 14.4%

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

   8. *-commutative 14.4%

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

   9. swap-sqr 14.4%

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

   10. swap-sqr 14.4%

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

   11. metadata-eval 14.4%

       \[\leadsto \sqrt{z \cdot \left(\left(\color{blue}{0.25} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right) \cdot \left(x \cdot x\right)\right)} \]

   12. rem-square-sqrt 14.4%

       \[\leadsto \sqrt{z \cdot \left(\left(0.25 \cdot \color{blue}{2}\right) \cdot \left(x \cdot x\right)\right)} \]

   13. metadata-eval 14.4%

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

10. Simplified 14.4%

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

    1. associate-*r* 14.4%

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

    2. sqrt-prod 12.9%

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

    3. *-commutative 12.9%

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

    4. sqrt-prod 13.2%

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

    5. add-sqr-sqrt 29.3%

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

12. Applied egg-rr 29.3%

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

13. Final simplification 29.3%

    \[\leadsto x \cdot \sqrt{0.5 \cdot z} \]
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 2024095 
(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))))