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

Percentage Accurate: 99.4% → 99.8%

Time: 34.4s

Alternatives: 15

Speedup: 1.1×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. rem-square-sqrt N/A

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

   2. pow1/2 N/A

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

   3. lift-exp.f64 N/A

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

   4. exp-prod N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. pow1/2 N/A

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

   8. lift-exp.f64 N/A

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

   9. exp-prod N/A

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

   10. *-commutative N/A

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

   11. exp-prod N/A

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

   12. pow-sqr N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{if}\;t \cdot t \leq 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot t \leq 10^{+100}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{\left(e^{t \cdot t} \cdot 2\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (sqrt (* z 2.0)) (- (* x 0.5) y))))
   (if (<= (* t t) 1e-20)
     t_1
     (if (<= (* t t) 1e+100)
       (* (- y) (sqrt (* (* (exp (* t t)) 2.0) z)))
       (*
        (fma
         (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5)
         (* t t)
         1.0)
        t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0)) * ((x * 0.5) - y);
	double tmp;
	if ((t * t) <= 1e-20) {
		tmp = t_1;
	} else if ((t * t) <= 1e+100) {
		tmp = -y * sqrt(((exp((t * t)) * 2.0) * z));
	} else {
		tmp = fma(fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5), (t * t), 1.0) * t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y))
	tmp = 0.0
	if (Float64(t * t) <= 1e-20)
		tmp = t_1;
	elseif (Float64(t * t) <= 1e+100)
		tmp = Float64(Float64(-y) * sqrt(Float64(Float64(exp(Float64(t * t)) * 2.0) * z)));
	else
		tmp = Float64(fma(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0) * t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1e-20], t$95$1, If[LessEqual[N[(t * t), $MachinePrecision], 1e+100], N[((-y) * N[Sqrt[N[(N[(N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 9.99999999999999945e-21

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 9.99999999999999945e-21 < (*.f64 t t) < 1.00000000000000002e100

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.8

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

   7. Applied rewrites 88.8%

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

   if 1.00000000000000002e100 < (*.f64 t t)

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), {t}^{2}, 1\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{{t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}}, {t}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2}} + \frac{1}{2}, {t}^{2}, 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right)}, {t}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right)}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), \color{blue}{t \cdot t}, 1\right) \]

      14. lower-*.f64 99.1

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), \color{blue}{t \cdot t}, 1\right) \]

   5. Applied rewrites 99.1%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-20}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \cdot t \leq 10^{+100}:\\ \;\;\;\;\left(-y\right) \cdot \sqrt{\left(e^{t \cdot t} \cdot 2\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \cdot \left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 5: 94.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \cdot \left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (fma (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5) (* t t) 1.0)
  (* (sqrt (* z 2.0)) (- (* x 0.5) y))))

C

double code(double x, double y, double z, double t) {
	return fma(fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5), (t * t), 1.0) * (sqrt((z * 2.0)) * ((x * 0.5) - y));
}

Julia

function code(x, y, z, t)
	return Float64(fma(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0) * Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\color{blue}{\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2}} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), {t}^{2}, 1\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{{t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}}, {t}^{2}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2}} + \frac{1}{2}, {t}^{2}, 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right)}, {t}^{2}, 1\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right)}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{t \cdot t}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), \color{blue}{t \cdot t}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   13. unpow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), \color{blue}{t \cdot t}, 1\right) \]

   14. lower-*.f64 93.8

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), \color{blue}{t \cdot t}, 1\right) \]

5. Applied rewrites 93.8%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]

6. Final simplification 93.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \cdot \left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right) \]
7. Add Preprocessing

Alternative 6: 66.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot 0.5\right) \cdot \sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z}\\ t_2 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 490000:\\ \;\;\;\;t\_2 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+225}:\\ \;\;\;\;\left(\left(-y\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* x 0.5) (sqrt (* (* (fma t t 1.0) 2.0) z))))
        (t_2 (sqrt (* z 2.0))))
   (if (<= t 490000.0)
     (* t_2 (- (* x 0.5) y))
     (if (<= t 1e+116)
       t_1
       (if (<= t 6e+225) (* (* (- y) (fma (* t t) 0.5 1.0)) t_2) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) * sqrt(((fma(t, t, 1.0) * 2.0) * z));
	double t_2 = sqrt((z * 2.0));
	double tmp;
	if (t <= 490000.0) {
		tmp = t_2 * ((x * 0.5) - y);
	} else if (t <= 1e+116) {
		tmp = t_1;
	} else if (t <= 6e+225) {
		tmp = (-y * fma((t * t), 0.5, 1.0)) * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) * sqrt(Float64(Float64(fma(t, t, 1.0) * 2.0) * z)))
	t_2 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (t <= 490000.0)
		tmp = Float64(t_2 * Float64(Float64(x * 0.5) - y));
	elseif (t <= 1e+116)
		tmp = t_1;
	elseif (t <= 6e+225)
		tmp = Float64(Float64(Float64(-y) * fma(Float64(t * t), 0.5, 1.0)) * t_2);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] * N[Sqrt[N[(N[(N[(t * t + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 490000.0], N[(t$95$2 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+116], t$95$1, If[LessEqual[t, 6e+225], N[(N[((-y) * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.9e5

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

   if 4.9e5 < t < 1.00000000000000002e116 or 6.000000000000001e225 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 58.0

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

   7. Applied rewrites 58.0%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.4

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

   10. Applied rewrites 50.4%

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

   if 1.00000000000000002e116 < t < 6.000000000000001e225

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 77.3%

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

   10. Applied rewrites 77.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 490000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \leq 10^{+116}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+225}:\\ \;\;\;\;\left(\left(-y\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := t\_1 \cdot \left(x \cdot 0.5\right)\\ \mathbf{if}\;x \cdot 0.5 \leq -8:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot 0.5 \leq 3.4 \cdot 10^{+42}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (* t_1 (* x 0.5))))
   (if (<= (* x 0.5) -8.0) t_2 (if (<= (* x 0.5) 3.4e+42) (* t_1 (- y)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double t_2 = t_1 * (x * 0.5);
	double tmp;
	if ((x * 0.5) <= -8.0) {
		tmp = t_2;
	} else if ((x * 0.5) <= 3.4e+42) {
		tmp = t_1 * -y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((z * 2.0d0))
    t_2 = t_1 * (x * 0.5d0)
    if ((x * 0.5d0) <= (-8.0d0)) then
        tmp = t_2
    else if ((x * 0.5d0) <= 3.4d+42) then
        tmp = t_1 * -y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z * 2.0));
	double t_2 = t_1 * (x * 0.5);
	double tmp;
	if ((x * 0.5) <= -8.0) {
		tmp = t_2;
	} else if ((x * 0.5) <= 3.4e+42) {
		tmp = t_1 * -y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z * 2.0))
	t_2 = t_1 * (x * 0.5)
	tmp = 0
	if (x * 0.5) <= -8.0:
		tmp = t_2
	elif (x * 0.5) <= 3.4e+42:
		tmp = t_1 * -y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	t_2 = Float64(t_1 * Float64(x * 0.5))
	tmp = 0.0
	if (Float64(x * 0.5) <= -8.0)
		tmp = t_2;
	elseif (Float64(x * 0.5) <= 3.4e+42)
		tmp = Float64(t_1 * Float64(-y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z * 2.0));
	t_2 = t_1 * (x * 0.5);
	tmp = 0.0;
	if ((x * 0.5) <= -8.0)
		tmp = t_2;
	elseif ((x * 0.5) <= 3.4e+42)
		tmp = t_1 * -y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -8.0], t$95$2, If[LessEqual[N[(x * 0.5), $MachinePrecision], 3.4e+42], N[(t$95$1 * (-y)), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -8 or 3.39999999999999975e42 < (*.f64 x #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. rem-square-sqrt N/A

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

      2. pow1/2 N/A

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

      3. lift-exp.f64 N/A

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

      4. exp-prod N/A

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

      5. *-commutative N/A

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

      6. exp-prod N/A

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

      7. pow1/2 N/A

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

      8. lift-exp.f64 N/A

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

      9. exp-prod N/A

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

      10. *-commutative N/A

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

      11. exp-prod N/A

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

      12. pow-sqr N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 53.8

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

   9. Applied rewrites 53.8%

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

   10. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 44.2

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

   12. Applied rewrites 44.2%

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

   if -8 < (*.f64 x #s(literal 1/2 binary64)) < 3.39999999999999975e42

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. rem-square-sqrt N/A

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

      2. pow1/2 N/A

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

      3. lift-exp.f64 N/A

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

      4. exp-prod N/A

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

      5. *-commutative N/A

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

      6. exp-prod N/A

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

      7. pow1/2 N/A

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

      8. lift-exp.f64 N/A

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

      9. exp-prod N/A

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

      10. *-commutative N/A

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

      11. exp-prod N/A

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

      12. pow-sqr N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 54.2

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

   9. Applied rewrites 54.2%

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

   10. Taylor expanded in y around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 46.3

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

   12. Applied rewrites 46.3%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -8:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 3.4 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 92.6

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

9. Applied rewrites 92.6%

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

10. Final simplification 92.6%

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

Alternative 9: 74.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-20}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 1e-20)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (* (sqrt (* (* (fma t t 1.0) 2.0) z)) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 1e-20) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = sqrt(((fma(t, t, 1.0) * 2.0) * z)) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 1e-20)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(t, t, 1.0) * 2.0) * z)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 1e-20], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(t * t + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 9.99999999999999945e-21

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 9.99999999999999945e-21 < (*.f64 t t)

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.4

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.2

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

   10. Applied rewrites 48.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{-20}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

5. Applied rewrites 88.7%

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

7. Applied rewrites 88.9%

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

Alternative 11: 65.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 490000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 490000.0)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (* (* x 0.5) (sqrt (* (* (fma t t 1.0) 2.0) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 490000.0) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = (x * 0.5) * sqrt(((fma(t, t, 1.0) * 2.0) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 490000.0)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(Float64(x * 0.5) * sqrt(Float64(Float64(fma(t, t, 1.0) * 2.0) * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 490000.0], N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] * N[Sqrt[N[(N[(N[(t * t + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.9e5

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

   if 4.9e5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. pow1/2 N/A

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

      15. lift-exp.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. exp-sqrt N/A

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

      18. sqrt-unprod N/A

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

      19. pow1/2 N/A

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

      20. sqrt-unprod N/A

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

      21. lower-sqrt.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 67.2

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

   7. Applied rewrites 67.2%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.4

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

   10. Applied rewrites 56.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 490000:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 85.6

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

7. Applied rewrites 85.6%

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

8. Final simplification 85.6%

   \[\leadsto \sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z} \cdot \left(x \cdot 0.5 - y\right) \]
9. Add Preprocessing

Alternative 13: 84.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in t around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 85.6

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

7. Applied rewrites 85.6%

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

8. Final simplification 85.6%

   \[\leadsto \sqrt{\mathsf{fma}\left(t \cdot t, z, z\right) \cdot 2} \cdot \left(x \cdot 0.5 - y\right) \]
9. Add Preprocessing

Alternative 14: 57.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in t around 0

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

   1. lower-*.f64 54.0

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

7. Applied rewrites 54.0%

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

8. Final simplification 54.0%

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

Alternative 15: 30.2% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. pow1/2 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. exp-sqrt N/A

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

   18. sqrt-unprod N/A

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

   19. pow1/2 N/A

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

   20. sqrt-unprod N/A

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

   21. lower-sqrt.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. rem-square-sqrt N/A

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

   2. pow1/2 N/A

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

   3. lift-exp.f64 N/A

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

   4. exp-prod N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. pow1/2 N/A

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

   8. lift-exp.f64 N/A

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

   9. exp-prod N/A

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

   10. *-commutative N/A

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

   11. exp-prod N/A

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

   12. pow-sqr N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in t around 0

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

   1. lower-*.f64 54.0

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

9. Applied rewrites 54.0%

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

10. Taylor expanded in y around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 30.0

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

12. Applied rewrites 30.0%

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

13. Final simplification 30.0%

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

Developer Target 1: 99.4% accurate, 0.6× 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 2024235 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))