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

Percentage Accurate: 99.4% → 99.8%

Time: 16.2s

Alternatives: 16

Speedup: 1.0×

• 44.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.7%

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

   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 99.8%

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

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 99.7%

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

   5. pow2 99.7%

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

6. Applied egg-rr 99.7%

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

   1. unpow1 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

8. Simplified 99.7%

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

   1. pow2 99.7%

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

   2. exp-prod 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.7%

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

   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 99.8%

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

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 99.7%

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

   5. pow2 99.7%

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

6. Applied egg-rr 99.7%

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

   1. unpow1 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.005 \lor \neg \left(t \cdot t \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \sqrt{2 \cdot z}\right) \cdot \left(-e^{\frac{t \cdot t}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* t t) 0.005) (not (<= (* t t) 5e+303)))
   (* (- (* x 0.5) y) (sqrt (* (* 2.0 z) (fma t t 1.0))))
   (* (* y (sqrt (* 2.0 z))) (- (exp (/ (* t t) 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((t * t) <= 0.005) || !((t * t) <= 5e+303)) {
		tmp = ((x * 0.5) - y) * sqrt(((2.0 * z) * fma(t, t, 1.0)));
	} else {
		tmp = (y * sqrt((2.0 * z))) * -exp(((t * t) / 2.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(t * t) <= 0.005) || !(Float64(t * t) <= 5e+303))
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))));
	else
		tmp = Float64(Float64(y * sqrt(Float64(2.0 * z))) * Float64(-exp(Float64(Float64(t * t) / 2.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(t * t), $MachinePrecision], 0.005], N[Not[LessEqual[N[(t * t), $MachinePrecision], 5e+303]], $MachinePrecision]], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 0.0050000000000000001 or 4.9999999999999997e303 < (*.f64 t t)

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. exp-sqrt 99.7%

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

      3. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 99.2%

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

       1. +-commutative 56.0%

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

       2. unpow2 56.0%

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

       3. fma-define 56.0%

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

   11. Simplified 99.2%

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

   if 0.0050000000000000001 < (*.f64 t t) < 4.9999999999999997e303

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 67.2%

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

      1. pow1 67.2%

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

      2. *-commutative 67.2%

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

      3. *-commutative 67.2%

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

      4. associate-*r* 67.2%

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

      5. sqrt-prod 67.2%

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

      6. *-commutative 67.2%

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

      7. *-commutative 67.2%

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

   5. Applied egg-rr 67.2%

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

      1. unpow1 67.2%

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

   7. Simplified 67.2%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0.005 \lor \neg \left(t \cdot t \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \sqrt{2 \cdot z}\right) \cdot \left(-e^{\frac{t \cdot t}{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 1:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+161} \lor \neg \left(t \leq 2.9 \cdot 10^{+215}\right):\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \end{array} \]

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)))
     (if (or (<= t 3.1e+161) (not (<= t 2.9e+215)))
       (* (* t (* t_1 (sqrt 2.0))) (sqrt z))
       (* (sqrt (* z (* 2.0 (fma t t 1.0)))) (- y))))))

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 if ((t <= 3.1e+161) || !(t <= 2.9e+215)) {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	} else {
		tmp = sqrt((z * (2.0 * fma(t, t, 1.0)))) * -y;
	}
	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)));
	elseif ((t <= 3.1e+161) || !(t <= 2.9e+215))
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	else
		tmp = Float64(sqrt(Float64(z * Float64(2.0 * fma(t, t, 1.0)))) * Float64(-y));
	end
	return 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], If[Or[LessEqual[t, 3.1e+161], N[Not[LessEqual[t, 2.9e+215]], $MachinePrecision]], N[(N[(t * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(z * N[(2.0 * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. exp-sqrt 99.7%

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

      3. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 73.2%

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

   if 1 < t < 3.10000000000000007e161 or 2.8999999999999999e215 < t

   1. Initial program 97.9%

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

      1. associate-*l* 99.9%

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

      2. exp-sqrt 99.9%

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

      3. 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 64.6%

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

       1. +-commutative 42.4%

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

       2. unpow2 42.4%

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

       3. fma-define 42.4%

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

   11. Simplified 64.6%

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

   12. Taylor expanded in t around inf 55.3%

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

   if 3.10000000000000007e161 < t < 2.8999999999999999e215

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.0%

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

      1. associate-*l* 75.0%

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

      2. mul-1-neg 75.0%

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

      3. *-commutative 75.0%

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

      4. *-commutative 75.0%

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

      5. associate-*r* 75.0%

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

      6. sqrt-prod 75.0%

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

      7. *-commutative 75.0%

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

      8. pow2 75.0%

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

      9. div-inv 75.0%

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

      10. metadata-eval 75.0%

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

      11. *-commutative 75.0%

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

      12. *-un-lft-identity 75.0%

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

      13. pow-exp 75.0%

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

      14. associate-*l* 87.5%

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

      15. pow-exp 87.5%

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

      16. *-un-lft-identity 87.5%

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

      17. *-commutative 87.5%

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

      18. exp-prod 87.5%

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

      19. pow1/2 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. distribute-lft-neg-in 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*r* 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in t around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. unpow2 87.5%

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

      3. fma-define 87.5%

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

   10. Simplified 87.5%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(t, t, 1\right)} \cdot 2\right) \cdot z} \cdot \left(-y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+161} \lor \neg \left(t \leq 2.9 \cdot 10^{+215}\right):\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{+115}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+219} \lor \neg \left(t \leq 1.7 \cdot 10^{+230}\right):\\ \;\;\;\;\sqrt{z} \cdot \left(\left(-y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.8e+115)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (if (or (<= t 2.35e+219) (not (<= t 1.7e+230)))
     (* (sqrt z) (* (- y) (* t (sqrt 2.0))))
     (* y (sqrt (* (* 2.0 z) (fma t t 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.8e+115) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else if ((t <= 2.35e+219) || !(t <= 1.7e+230)) {
		tmp = sqrt(z) * (-y * (t * sqrt(2.0)));
	} else {
		tmp = y * sqrt(((2.0 * z) * fma(t, t, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.8e+115)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	elseif ((t <= 2.35e+219) || !(t <= 1.7e+230))
		tmp = Float64(sqrt(z) * Float64(Float64(-y) * Float64(t * sqrt(2.0))));
	else
		tmp = Float64(y * sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3.8e+115], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 2.35e+219], N[Not[LessEqual[t, 1.7e+230]], $MachinePrecision]], N[(N[Sqrt[z], $MachinePrecision] * N[((-y) * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.8000000000000001e115

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 67.2%

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

   if 3.8000000000000001e115 < t < 2.35000000000000007e219 or 1.69999999999999993e230 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 84.4%

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

      1. associate-*l* 84.4%

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

      2. mul-1-neg 84.4%

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

      3. *-commutative 84.4%

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

      4. *-commutative 84.4%

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

      5. associate-*r* 84.4%

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

      6. sqrt-prod 84.4%

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

      7. *-commutative 84.4%

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

      8. pow2 84.4%

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

      9. div-inv 84.4%

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

      10. metadata-eval 84.4%

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

      11. *-commutative 84.4%

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

      12. *-un-lft-identity 84.4%

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

      13. pow-exp 84.4%

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

      14. associate-*l* 87.5%

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

      15. pow-exp 87.5%

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

      16. *-un-lft-identity 87.5%

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

      17. *-commutative 87.5%

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

      18. exp-prod 87.5%

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

      19. pow1/2 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. distribute-lft-neg-in 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*r* 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in t around 0 78.5%

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

      1. +-commutative 78.5%

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

      2. unpow2 78.5%

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

      3. fma-define 78.5%

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

   10. Simplified 78.5%

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

   11. Taylor expanded in t around inf 54.6%

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

       1. mul-1-neg 54.6%

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

       2. *-commutative 54.6%

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

       3. distribute-rgt-neg-in 54.6%

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

       4. *-commutative 54.6%

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

       5. associate-*r* 54.6%

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

   13. Simplified 54.6%

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

   if 2.35000000000000007e219 < t < 1.69999999999999993e230

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 20.0%

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

      1. associate-*l* 20.0%

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

      2. mul-1-neg 20.0%

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

      3. *-commutative 20.0%

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

      4. *-commutative 20.0%

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

      5. associate-*r* 20.0%

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

      6. sqrt-prod 20.0%

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

      7. *-commutative 20.0%

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

      8. pow2 20.0%

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

      9. div-inv 20.0%

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

      10. metadata-eval 20.0%

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

      11. *-commutative 20.0%

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

      12. *-un-lft-identity 20.0%

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

      13. pow-exp 20.0%

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

      14. associate-*l* 20.0%

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

      15. pow-exp 20.0%

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

      16. *-un-lft-identity 20.0%

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

      17. *-commutative 20.0%

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

      18. exp-prod 20.0%

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

      19. pow1/2 20.0%

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

   5. Applied egg-rr 20.0%

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

      1. distribute-lft-neg-in 20.0%

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

      2. *-commutative 20.0%

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

      3. associate-*r* 20.0%

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

   7. Simplified 20.0%

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

   8. Taylor expanded in t around 0 20.0%

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

      1. +-commutative 20.0%

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

      2. unpow2 20.0%

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

      3. fma-define 20.0%

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

   10. Simplified 20.0%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 20.0%

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

       3. sqr-neg 20.0%

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

       4. sqrt-unprod 60.0%

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

       5. add-sqr-sqrt 80.0%

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

       6. pow1 80.0%

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

       7. *-commutative 80.0%

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

       8. associate-*l* 80.0%

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

       9. *-commutative 80.0%

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

       10. associate-*l* 80.0%

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

   12. Applied egg-rr 80.0%

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

       1. unpow1 80.0%

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

       2. associate-*r* 80.0%

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

       3. *-commutative 80.0%

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

   14. Simplified 80.0%

       \[\leadsto \color{blue}{y \cdot \sqrt{\left(z \cdot 2\right) \cdot \mathsf{fma}\left(t, t, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{+115}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+219} \lor \neg \left(t \leq 1.7 \cdot 10^{+230}\right):\\ \;\;\;\;\sqrt{z} \cdot \left(\left(-y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ t_2 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 0.14:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+215}:\\ \;\;\;\;\left(y \cdot t\_2\right) \cdot \left(-e^{\frac{t \cdot t}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)) (t_2 (sqrt (* 2.0 z))))
   (if (<= t 0.14)
     (* t_1 (* t_2 (hypot 1.0 t)))
     (if (<= t 3e+215)
       (* (* y t_2) (- (exp (/ (* t t) 2.0))))
       (* (* 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 t_2 = sqrt((2.0 * z));
	double tmp;
	if (t <= 0.14) {
		tmp = t_1 * (t_2 * hypot(1.0, t));
	} else if (t <= 3e+215) {
		tmp = (y * t_2) * -exp(((t * t) / 2.0));
	} else {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	}
	return tmp;
}

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 <= 0.14) {
		tmp = t_1 * (t_2 * Math.hypot(1.0, t));
	} else if (t <= 3e+215) {
		tmp = (y * t_2) * -Math.exp(((t * t) / 2.0));
	} 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
	t_2 = math.sqrt((2.0 * z))
	tmp = 0
	if t <= 0.14:
		tmp = t_1 * (t_2 * math.hypot(1.0, t))
	elif t <= 3e+215:
		tmp = (y * t_2) * -math.exp(((t * t) / 2.0))
	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)
	t_2 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 0.14)
		tmp = Float64(t_1 * Float64(t_2 * hypot(1.0, t)));
	elseif (t <= 3e+215)
		tmp = Float64(Float64(y * t_2) * Float64(-exp(Float64(Float64(t * t) / 2.0))));
	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;
	t_2 = sqrt((2.0 * z));
	tmp = 0.0;
	if (t <= 0.14)
		tmp = t_1 * (t_2 * hypot(1.0, t));
	elseif (t <= 3e+215)
		tmp = (y * t_2) * -exp(((t * t) / 2.0));
	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]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.14], N[(t$95$1 * N[(t$95$2 * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e+215], N[(N[(y * t$95$2), $MachinePrecision] * (-N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $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 3 regimes

2. if t < 0.14000000000000001

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. exp-sqrt 99.7%

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

      3. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 89.0%

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

       1. +-commutative 47.8%

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

       2. unpow2 47.8%

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

       3. fma-define 47.8%

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

   11. Simplified 89.0%

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

       1. sqrt-prod 88.6%

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

   13. Applied egg-rr 88.6%

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

       1. *-commutative 88.6%

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

       2. fma-undefine 88.6%

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

       3. unpow2 88.6%

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

       4. +-commutative 88.6%

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

       5. unpow2 88.6%

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

       6. hypot-1-def 85.2%

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

   15. Simplified 85.2%

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

   if 0.14000000000000001 < t < 2.9999999999999999e215

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0 65.6%

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

      1. pow1 65.6%

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

      2. *-commutative 65.6%

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

      3. *-commutative 65.6%

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

      4. associate-*r* 65.6%

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

      5. sqrt-prod 65.6%

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

      6. *-commutative 65.6%

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

      7. *-commutative 65.6%

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

   5. Applied egg-rr 65.6%

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

      1. unpow1 65.6%

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

   7. Simplified 65.6%

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

   if 2.9999999999999999e215 < t

   1. Initial program 100.0%

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

      1. 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 0 100.0%

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

       1. +-commutative 70.8%

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

       2. unpow2 70.8%

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

       3. fma-define 70.8%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 84.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.14:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+215}:\\ \;\;\;\;\left(y \cdot \sqrt{2 \cdot z}\right) \cdot \left(-e^{\frac{t \cdot t}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 4.1 \cdot 10^{+73}:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 4.1e+73)
     (* t_1 (sqrt (* 2.0 z)))
     (if (<= t 2.8e+215)
       (* (sqrt (* z (* 2.0 (fma t t 1.0)))) (- y))
       (* 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 <= 4.1e+73) {
		tmp = t_1 * sqrt((2.0 * z));
	} else if (t <= 2.8e+215) {
		tmp = sqrt((z * (2.0 * fma(t, t, 1.0)))) * -y;
	} else {
		tmp = t * (t_1 * (sqrt(2.0) * sqrt(z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 4.1e+73)
		tmp = Float64(t_1 * sqrt(Float64(2.0 * z)));
	elseif (t <= 2.8e+215)
		tmp = Float64(sqrt(Float64(z * Float64(2.0 * fma(t, t, 1.0)))) * Float64(-y));
	else
		tmp = Float64(t * Float64(t_1 * Float64(sqrt(2.0) * sqrt(z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 4.1e+73], N[(t$95$1 * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+215], N[(N[Sqrt[N[(z * N[(2.0 * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[(t * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.0999999999999998e73

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 68.9%

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

   if 4.0999999999999998e73 < t < 2.8e215

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 70.0%

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

      1. associate-*l* 70.0%

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

      2. mul-1-neg 70.0%

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

      3. *-commutative 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-*r* 70.0%

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

      6. sqrt-prod 70.0%

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

      7. *-commutative 70.0%

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

      8. pow2 70.0%

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

      9. div-inv 70.0%

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

      10. metadata-eval 70.0%

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

      11. *-commutative 70.0%

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

      12. *-un-lft-identity 70.0%

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

      13. pow-exp 70.0%

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

      14. associate-*l* 75.0%

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

      15. pow-exp 75.0%

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

      16. *-un-lft-identity 75.0%

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

      17. *-commutative 75.0%

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

      18. exp-prod 75.0%

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

      19. pow1/2 75.0%

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

   5. Applied egg-rr 75.0%

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

      1. distribute-lft-neg-in 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-*r* 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in t around 0 51.2%

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

      1. +-commutative 51.2%

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

      2. unpow2 51.2%

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

      3. fma-define 51.2%

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

   10. Simplified 51.2%

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

   if 2.8e215 < t

   1. Initial program 100.0%

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

      1. 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 0 100.0%

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

       1. +-commutative 70.8%

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

       2. unpow2 70.8%

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

       3. fma-define 70.8%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 84.2%

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

       1. associate-*l* 73.0%

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

       2. *-commutative 73.0%

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

       3. associate-*r* 73.0%

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

       4. *-commutative 73.0%

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

   14. Simplified 73.0%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{+73}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 5.1 \cdot 10^{+160}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= t 5.1e+160)
     (* t_1 (* (sqrt (* 2.0 z)) (hypot 1.0 t)))
     (if (<= t 2.05e+215)
       (* (sqrt (* z (* 2.0 (fma t t 1.0)))) (- y))
       (* (* 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 <= 5.1e+160) {
		tmp = t_1 * (sqrt((2.0 * z)) * hypot(1.0, t));
	} else if (t <= 2.05e+215) {
		tmp = sqrt((z * (2.0 * fma(t, t, 1.0)))) * -y;
	} else {
		tmp = (t * (t_1 * sqrt(2.0))) * sqrt(z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 5.1e+160)
		tmp = Float64(t_1 * Float64(sqrt(Float64(2.0 * z)) * hypot(1.0, t)));
	elseif (t <= 2.05e+215)
		tmp = Float64(sqrt(Float64(z * Float64(2.0 * fma(t, t, 1.0)))) * Float64(-y));
	else
		tmp = Float64(Float64(t * Float64(t_1 * sqrt(2.0))) * sqrt(z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 5.1e+160], N[(t$95$1 * N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e+215], N[(N[Sqrt[N[(z * N[(2.0 * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $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 3 regimes

2. if t < 5.1000000000000001e160

   1. Initial program 99.3%

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

      1. associate-*l* 99.7%

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

      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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 82.3%

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

       1. +-commutative 44.1%

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

       2. unpow2 44.1%

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

       3. fma-define 44.1%

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

   11. Simplified 82.3%

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

       1. sqrt-prod 81.0%

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

   13. Applied egg-rr 81.0%

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

       1. *-commutative 81.0%

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

       2. fma-undefine 81.0%

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

       3. unpow2 81.0%

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

       4. +-commutative 81.0%

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

       5. unpow2 81.0%

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

       6. hypot-1-def 77.7%

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

   15. Simplified 77.7%

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

   if 5.1000000000000001e160 < t < 2.0500000000000002e215

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.0%

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

      1. associate-*l* 75.0%

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

      2. mul-1-neg 75.0%

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

      3. *-commutative 75.0%

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

      4. *-commutative 75.0%

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

      5. associate-*r* 75.0%

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

      6. sqrt-prod 75.0%

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

      7. *-commutative 75.0%

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

      8. pow2 75.0%

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

      9. div-inv 75.0%

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

      10. metadata-eval 75.0%

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

      11. *-commutative 75.0%

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

      12. *-un-lft-identity 75.0%

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

      13. pow-exp 75.0%

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

      14. associate-*l* 87.5%

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

      15. pow-exp 87.5%

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

      16. *-un-lft-identity 87.5%

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

      17. *-commutative 87.5%

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

      18. exp-prod 87.5%

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

      19. pow1/2 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. distribute-lft-neg-in 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*r* 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in t around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. unpow2 87.5%

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

      3. fma-define 87.5%

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

   10. Simplified 87.5%

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

   if 2.0500000000000002e215 < t

   1. Initial program 100.0%

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

      1. 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 0 100.0%

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

       1. +-commutative 70.8%

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

       2. unpow2 70.8%

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

       3. fma-define 70.8%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in t around inf 84.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{+160}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{hypot}\left(1, t\right)\right)\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+73}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 7.2e+73)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* (sqrt (* z (* 2.0 (fma t t 1.0)))) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 7.2e+73) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt((z * (2.0 * fma(t, t, 1.0)))) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 7.2e+73)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(sqrt(Float64(z * Float64(2.0 * fma(t, t, 1.0)))) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 7.2e+73], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(z * N[(2.0 * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.1999999999999998e73

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 68.9%

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

   if 7.1999999999999998e73 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 70.5%

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

      1. associate-*l* 70.5%

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

      2. mul-1-neg 70.5%

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

      3. *-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. associate-*r* 70.5%

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

      6. sqrt-prod 70.5%

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

      7. *-commutative 70.5%

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

      8. pow2 70.5%

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

      9. div-inv 70.5%

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

      10. metadata-eval 70.5%

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

      11. *-commutative 70.5%

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

      12. *-un-lft-identity 70.5%

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

      13. pow-exp 70.5%

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

      14. associate-*l* 72.7%

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

      15. pow-exp 72.7%

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

      16. *-un-lft-identity 72.7%

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

      17. *-commutative 72.7%

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

      18. exp-prod 72.7%

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

      19. pow1/2 72.7%

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

   5. Applied egg-rr 72.7%

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

      1. distribute-lft-neg-in 72.7%

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

      2. *-commutative 72.7%

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

      3. associate-*r* 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in t around 0 61.9%

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

      1. +-commutative 61.9%

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

      2. unpow2 61.9%

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

      3. fma-define 61.9%

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

   10. Simplified 61.9%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(t, t, 1\right)} \cdot 2\right) \cdot z} \cdot \left(-y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+73}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot \left(2 \cdot \mathsf{fma}\left(t, t, 1\right)\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 11: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+112}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{z} \cdot \left(y \cdot \left(-\sqrt{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.3e+112)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* t (* (sqrt z) (* y (- (sqrt 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.3e+112) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = t * (sqrt(z) * (y * -sqrt(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) :: tmp
    if (t <= 1.3d+112) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = t * (sqrt(z) * (y * -sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.3e+112) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = t * (Math.sqrt(z) * (y * -Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.3e+112:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = t * (math.sqrt(z) * (y * -math.sqrt(2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.3e+112)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(t * Float64(sqrt(z) * Float64(y * Float64(-sqrt(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.3e+112)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = t * (sqrt(z) * (y * -sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.3e+112], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[z], $MachinePrecision] * N[(y * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.3e112

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 67.2%

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

   if 1.3e112 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.7%

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

      1. associate-*l* 75.7%

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

      2. mul-1-neg 75.7%

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

      3. *-commutative 75.7%

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

      4. *-commutative 75.7%

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

      5. associate-*r* 75.7%

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

      6. sqrt-prod 75.7%

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

      7. *-commutative 75.7%

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

      8. pow2 75.7%

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

      9. div-inv 75.7%

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

      10. metadata-eval 75.7%

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

      11. *-commutative 75.7%

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

      12. *-un-lft-identity 75.7%

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

      13. pow-exp 75.7%

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

      14. associate-*l* 78.4%

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

      15. pow-exp 78.4%

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

      16. *-un-lft-identity 78.4%

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

      17. *-commutative 78.4%

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

      18. exp-prod 78.4%

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

      19. pow1/2 78.4%

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

   5. Applied egg-rr 78.4%

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

      1. distribute-lft-neg-in 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in t around 0 70.6%

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

      1. +-commutative 70.6%

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

      2. unpow2 70.6%

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

      3. fma-define 70.6%

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

   10. Simplified 70.6%

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

   11. Taylor expanded in t around inf 47.4%

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

       1. mul-1-neg 47.4%

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

       2. associate-*l* 37.4%

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

       3. *-commutative 37.4%

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

       4. *-commutative 37.4%

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

   13. Simplified 37.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+112}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{z} \cdot \left(y \cdot \left(-\sqrt{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{+114}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(-y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.35e+114)
   (* (- (* x 0.5) y) (sqrt (* 2.0 z)))
   (* (sqrt z) (* (- y) (* t (sqrt 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.35e+114) {
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	} else {
		tmp = sqrt(z) * (-y * (t * sqrt(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) :: tmp
    if (t <= 1.35d+114) then
        tmp = ((x * 0.5d0) - y) * sqrt((2.0d0 * z))
    else
        tmp = sqrt(z) * (-y * (t * sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.35e+114) {
		tmp = ((x * 0.5) - y) * Math.sqrt((2.0 * z));
	} else {
		tmp = Math.sqrt(z) * (-y * (t * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.35e+114:
		tmp = ((x * 0.5) - y) * math.sqrt((2.0 * z))
	else:
		tmp = math.sqrt(z) * (-y * (t * math.sqrt(2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.35e+114)
		tmp = Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(2.0 * z)));
	else
		tmp = Float64(sqrt(z) * Float64(Float64(-y) * Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.35e+114)
		tmp = ((x * 0.5) - y) * sqrt((2.0 * z));
	else
		tmp = sqrt(z) * (-y * (t * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.35e+114], N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[((-y) * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.35e114

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 67.2%

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

   if 1.35e114 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.7%

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

      1. associate-*l* 75.7%

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

      2. mul-1-neg 75.7%

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

      3. *-commutative 75.7%

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

      4. *-commutative 75.7%

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

      5. associate-*r* 75.7%

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

      6. sqrt-prod 75.7%

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

      7. *-commutative 75.7%

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

      8. pow2 75.7%

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

      9. div-inv 75.7%

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

      10. metadata-eval 75.7%

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

      11. *-commutative 75.7%

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

      12. *-un-lft-identity 75.7%

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

      13. pow-exp 75.7%

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

      14. associate-*l* 78.4%

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

      15. pow-exp 78.4%

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

      16. *-un-lft-identity 78.4%

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

      17. *-commutative 78.4%

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

      18. exp-prod 78.4%

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

      19. pow1/2 78.4%

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

   5. Applied egg-rr 78.4%

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

      1. distribute-lft-neg-in 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in t around 0 70.6%

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

      1. +-commutative 70.6%

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

      2. unpow2 70.6%

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

      3. fma-define 70.6%

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

   10. Simplified 70.6%

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

   11. Taylor expanded in t around inf 47.4%

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

       1. mul-1-neg 47.4%

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

       2. *-commutative 47.4%

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

       3. distribute-rgt-neg-in 47.4%

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

       4. *-commutative 47.4%

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

       5. associate-*r* 47.4%

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

   13. Simplified 47.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{+114}:\\ \;\;\;\;\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(-y\right) \cdot \left(t \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+53} \lor \neg \left(x \leq 5.5 \cdot 10^{-100}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (or (<= x -4.5e+53) (not (<= x 5.5e-100)))
     (* 0.5 (* x t_1))
     (* y (- t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if ((x <= -4.5e+53) || !(x <= 5.5e-100)) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 * z))
    if ((x <= (-4.5d+53)) .or. (.not. (x <= 5.5d-100))) then
        tmp = 0.5d0 * (x * t_1)
    else
        tmp = y * -t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((2.0 * z));
	double tmp;
	if ((x <= -4.5e+53) || !(x <= 5.5e-100)) {
		tmp = 0.5 * (x * t_1);
	} else {
		tmp = y * -t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((2.0 * z))
	tmp = 0
	if (x <= -4.5e+53) or not (x <= 5.5e-100):
		tmp = 0.5 * (x * t_1)
	else:
		tmp = y * -t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if ((x <= -4.5e+53) || !(x <= 5.5e-100))
		tmp = Float64(0.5 * Float64(x * t_1));
	else
		tmp = Float64(y * Float64(-t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((2.0 * z));
	tmp = 0.0;
	if ((x <= -4.5e+53) || ~((x <= 5.5e-100)))
		tmp = 0.5 * (x * t_1);
	else
		tmp = y * -t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -4.5e+53], N[Not[LessEqual[x, 5.5e-100]], $MachinePrecision]], N[(0.5 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * (-t$95$1)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5000000000000002e53 or 5.50000000000000011e-100 < x

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. exp-sqrt 99.7%

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

      3. 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 99.7%

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

      3. associate-*l* 99.7%

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

      4. pow-exp 99.7%

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

      5. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t around 0 61.6%

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

   10. Taylor expanded in x around inf 49.7%

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

       1. pow1 49.7%

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

       2. associate-*l* 49.7%

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

       3. sqrt-prod 49.7%

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

   12. Applied egg-rr 49.7%

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

       1. unpow1 49.7%

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

       2. *-commutative 49.7%

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

   14. Simplified 49.7%

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

   if -4.5000000000000002e53 < x < 5.50000000000000011e-100

   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 x around 0 79.8%

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

      1. associate-*l* 79.8%

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

      2. mul-1-neg 79.8%

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

      3. *-commutative 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-*r* 79.9%

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

      6. sqrt-prod 79.9%

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

      7. *-commutative 79.9%

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

      8. pow2 79.9%

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

      9. div-inv 79.9%

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

      10. metadata-eval 79.9%

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

      11. *-commutative 79.9%

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

      12. *-un-lft-identity 79.9%

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

      13. pow-exp 79.9%

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

      14. associate-*l* 82.5%

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

      15. pow-exp 82.5%

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

      16. *-un-lft-identity 82.5%

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

      17. *-commutative 82.5%

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

      18. exp-prod 82.5%

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

      19. pow1/2 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. distribute-lft-neg-in 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r* 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in t around 0 47.6%

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

      1. distribute-rgt-neg-out 47.6%

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

      2. neg-sub0 47.6%

         \[\leadsto \color{blue}{0 - \sqrt{\left(1 \cdot 2\right) \cdot z} \cdot y} \]

      3. metadata-eval 47.6%

         \[\leadsto 0 - \sqrt{\color{blue}{2} \cdot z} \cdot y \]

   10. Applied egg-rr 47.6%

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

       1. neg-sub0 47.6%

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

       2. distribute-rgt-neg-in 47.6%

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

       3. *-commutative 47.6%

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

   12. Simplified 47.6%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+53} \lor \neg \left(x \leq 5.5 \cdot 10^{-100}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \sqrt{2 \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-\sqrt{2 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. 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 99.8%

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

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 99.7%

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

   5. pow2 99.7%

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

6. Applied egg-rr 99.7%

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

   1. unpow1 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in t around 0 60.0%

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

10. Final simplification 60.0%

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

Alternative 15: 29.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0 57.7%

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

   1. associate-*l* 57.7%

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

   2. mul-1-neg 57.7%

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

   3. *-commutative 57.7%

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

   4. *-commutative 57.7%

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

   5. associate-*r* 57.7%

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

   6. sqrt-prod 57.8%

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

   7. *-commutative 57.8%

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

   8. pow2 57.8%

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

   9. div-inv 57.8%

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

   10. metadata-eval 57.8%

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

   11. *-commutative 57.8%

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

   12. *-un-lft-identity 57.8%

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

   13. pow-exp 57.8%

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

   14. associate-*l* 59.3%

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

   15. pow-exp 59.3%

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

   16. *-un-lft-identity 59.3%

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

   17. *-commutative 59.3%

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

   18. exp-prod 59.3%

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

   19. pow1/2 59.3%

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

5. Applied egg-rr 59.3%

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

   1. distribute-lft-neg-in 59.3%

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

   2. *-commutative 59.3%

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

   3. associate-*r* 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in t around 0 29.1%

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

   1. distribute-rgt-neg-out 29.1%

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

   2. neg-sub0 29.1%

      \[\leadsto \color{blue}{0 - \sqrt{\left(1 \cdot 2\right) \cdot z} \cdot y} \]

   3. metadata-eval 29.1%

      \[\leadsto 0 - \sqrt{\color{blue}{2} \cdot z} \cdot y \]

10. Applied egg-rr 29.1%

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

    1. neg-sub0 29.1%

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

    2. distribute-rgt-neg-in 29.1%

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

    3. *-commutative 29.1%

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

12. Simplified 29.1%

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

13. Final simplification 29.1%

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

Alternative 16: 2.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0 57.7%

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

   1. associate-*l* 57.7%

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

   2. mul-1-neg 57.7%

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

   3. *-commutative 57.7%

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

   4. *-commutative 57.7%

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

   5. associate-*r* 57.7%

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

   6. sqrt-prod 57.8%

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

   7. *-commutative 57.8%

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

   8. pow2 57.8%

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

   9. div-inv 57.8%

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

   10. metadata-eval 57.8%

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

   11. *-commutative 57.8%

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

   12. *-un-lft-identity 57.8%

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

   13. pow-exp 57.8%

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

   14. associate-*l* 59.3%

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

   15. pow-exp 59.3%

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

   16. *-un-lft-identity 59.3%

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

   17. *-commutative 59.3%

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

   18. exp-prod 59.3%

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

   19. pow1/2 59.3%

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

5. Applied egg-rr 59.3%

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

   1. distribute-lft-neg-in 59.3%

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

   2. *-commutative 59.3%

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

   3. associate-*r* 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in t around 0 29.1%

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

   1. add-sqr-sqrt 16.6%

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

   2. sqrt-unprod 15.3%

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

   3. sqr-neg 15.3%

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

   4. sqrt-unprod 2.1%

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

   5. add-sqr-sqrt 3.3%

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

   6. pow1 3.3%

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

   7. metadata-eval 3.3%

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

10. Applied egg-rr 3.3%

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

    1. unpow1 3.3%

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

    2. *-commutative 3.3%

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

    3. *-commutative 3.3%

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

12. Simplified 3.3%

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

13. Final simplification 3.3%

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

Developer target: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024067 
(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))))