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

Percentage Accurate: 99.3% → 99.7%

Time: 17.0s

Alternatives: 13

Speedup: 1.1×

• 44.0% 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: 99.3% 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: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

5. Final simplification 99.8%

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

Alternative 2: 99.7% 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 99.8%

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

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-*.f64 N/A

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

   5. lower-exp.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 75.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt(((pow((1.0 + 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(((((1.0d0 + 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((1.0 + t), t) * 2.0) * z)) * ((x * 0.5) - y);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 75.1

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

7. Applied rewrites 75.1%

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

8. Final simplification 75.1%

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

Alternative 4: 96.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
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 94.4

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

5. Applied rewrites 94.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

7. Applied rewrites 94.8%

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

8. Final simplification 94.8%

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

Alternative 5: 88.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot 0.5 - y\\ \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \left(\sqrt{z \cdot 2} \cdot t\_1\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x 0.5) y)))
   (if (<= (* t t) 2e+19)
     (* (fma (* t t) 0.5 1.0) (* (sqrt (* z 2.0)) t_1))
     (if (<= (* t t) 5e+307)
       (* (sqrt (* (fma (fma t t 2.0) (* t t) 2.0) z)) (- y))
       (* (sqrt (* (* (fma t t 1.0) 2.0) z)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * 0.5) - y;
	double tmp;
	if ((t * t) <= 2e+19) {
		tmp = fma((t * t), 0.5, 1.0) * (sqrt((z * 2.0)) * t_1);
	} else if ((t * t) <= 5e+307) {
		tmp = sqrt((fma(fma(t, t, 2.0), (t * t), 2.0) * z)) * -y;
	} else {
		tmp = sqrt(((fma(t, t, 1.0) * 2.0) * z)) * t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (Float64(t * t) <= 2e+19)
		tmp = Float64(fma(Float64(t * t), 0.5, 1.0) * Float64(sqrt(Float64(z * 2.0)) * t_1));
	elseif (Float64(t * t) <= 5e+307)
		tmp = Float64(sqrt(Float64(fma(fma(t, t, 2.0), Float64(t * t), 2.0) * z)) * Float64(-y));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(t, t, 1.0) * 2.0) * z)) * t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 2e+19], N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(t * t), $MachinePrecision], 5e+307], N[(N[Sqrt[N[(N[(N[(t * t + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(t * t + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 t t) < 2e19

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 2e19 < (*.f64 t t) < 5e307

   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%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{\color{blue}{2 \cdot z + {t}^{2} \cdot \left(2 \cdot z + {t}^{2} \cdot z\right)}} \]
   6. 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 72.3

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

   7. Applied rewrites 72.3%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 60.7

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

   10. Applied rewrites 60.7%

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

   if 5e307 < (*.f64 t 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%

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

   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 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \left(\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z} \cdot \left(x \cdot 0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{if}\;t \cdot t \leq 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (sqrt (* (* (fma t t 1.0) 2.0) z)) (- (* x 0.5) y))))
   (if (<= (* t t) 1e+145)
     t_1
     (if (<= (* t t) 5e+307)
       (* (sqrt (* (fma (fma t t 2.0) (* t t) 2.0) z)) (- y))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt(((fma(t, t, 1.0) * 2.0) * z)) * ((x * 0.5) - y);
	double tmp;
	if ((t * t) <= 1e+145) {
		tmp = t_1;
	} else if ((t * t) <= 5e+307) {
		tmp = sqrt((fma(fma(t, t, 2.0), (t * t), 2.0) * z)) * -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(Float64(fma(t, t, 1.0) * 2.0) * z)) * Float64(Float64(x * 0.5) - y))
	tmp = 0.0
	if (Float64(t * t) <= 1e+145)
		tmp = t_1;
	elseif (Float64(t * t) <= 5e+307)
		tmp = Float64(sqrt(Float64(fma(fma(t, t, 2.0), Float64(t * t), 2.0) * z)) * Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = 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]}, If[LessEqual[N[(t * t), $MachinePrecision], 1e+145], t$95$1, If[LessEqual[N[(t * t), $MachinePrecision], 5e+307], N[(N[Sqrt[N[(N[(N[(t * t + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 9.9999999999999999e144 or 5e307 < (*.f64 t t)

   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%

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

   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 87.9

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

   7. Applied rewrites 87.9%

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

   if 9.9999999999999999e144 < (*.f64 t t) < 5e307

   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%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{\color{blue}{2 \cdot z + {t}^{2} \cdot \left(2 \cdot z + {t}^{2} \cdot z\right)}} \]
   6. 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 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.5

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

   10. Applied rewrites 87.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 10^{+145}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{elif}\;t \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(t, t, 1\right) \cdot 2\right) \cdot z} \cdot \left(x \cdot 0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (fma (fma 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(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(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[(0.125 * 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 99.8%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 91.8

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

5. Applied rewrites 91.8%

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

6. Final simplification 91.8%

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

Alternative 8: 81.4% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 2e-10)
   (* (sqrt (* z 2.0)) (- (* x 0.5) y))
   (* (sqrt (* (fma (fma t t 2.0) (* t t) 2.0) z)) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 2e-10) {
		tmp = sqrt((z * 2.0)) * ((x * 0.5) - y);
	} else {
		tmp = sqrt((fma(fma(t, t, 2.0), (t * t), 2.0) * z)) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t * t) <= 2e-10)
		tmp = Float64(sqrt(Float64(z * 2.0)) * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(sqrt(Float64(fma(fma(t, t, 2.0), Float64(t * t), 2.0) * z)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t * t), $MachinePrecision], 2e-10], 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 + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 2.00000000000000007e-10

   1. Initial program 99.6%

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

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

   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 99.2

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

   7. Applied rewrites 99.2%

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

   if 2.00000000000000007e-10 < (*.f64 t t)

   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 100.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{\color{blue}{2 \cdot z + {t}^{2} \cdot \left(2 \cdot z + {t}^{2} \cdot z\right)}} \]
   6. 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 82.7

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

   7. Applied rewrites 82.7%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.6

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

   10. Applied rewrites 62.6%

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

4. Final simplification 80.5%

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

Alternative 9: 75.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \cdot t \leq 1.26 \cdot 10^{-6}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (<= (* t t) 1.26e-6)
     (* t_1 (- (* x 0.5) y))
     (* (- y) (* (fma (* t t) 0.5 1.0) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t * t) <= 1.26e-6) {
		tmp = t_1 * ((x * 0.5) - y);
	} else {
		tmp = -y * (fma((t * t), 0.5, 1.0) * t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if (Float64(t * t) <= 1.26e-6)
		tmp = Float64(t_1 * Float64(Float64(x * 0.5) - y));
	else
		tmp = Float64(Float64(-y) * Float64(fma(Float64(t * t), 0.5, 1.0) * t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t * t), $MachinePrecision], 1.26e-6], N[(t$95$1 * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t t) < 1.26000000000000001e-6

   1. Initial program 99.6%

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

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

   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 99.2

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

   7. Applied rewrites 99.2%

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

   if 1.26000000000000001e-6 < (*.f64 t t)

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 67.2

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

   8. Applied rewrites 67.2%

      \[\leadsto \left(\color{blue}{\left(-y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 67.9%

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

   11. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 48.7

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

   13. Applied rewrites 48.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 1.26 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(x \cdot 0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \sqrt{z \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ t_2 := \left(x \cdot 0.5\right) \cdot t\_1\\ \mathbf{if}\;x \cdot 0.5 \leq -2.7 \cdot 10^{+166}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot 0.5 \leq 7.6:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))) (t_2 (* (* x 0.5) t_1)))
   (if (<= (* x 0.5) -2.7e+166)
     t_2
     (if (<= (* x 0.5) 7.6) (* 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 = (x * 0.5) * t_1;
	double tmp;
	if ((x * 0.5) <= -2.7e+166) {
		tmp = t_2;
	} else if ((x * 0.5) <= 7.6) {
		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 = (x * 0.5d0) * t_1
    if ((x * 0.5d0) <= (-2.7d+166)) then
        tmp = t_2
    else if ((x * 0.5d0) <= 7.6d0) 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 = (x * 0.5) * t_1;
	double tmp;
	if ((x * 0.5) <= -2.7e+166) {
		tmp = t_2;
	} else if ((x * 0.5) <= 7.6) {
		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 = (x * 0.5) * t_1
	tmp = 0
	if (x * 0.5) <= -2.7e+166:
		tmp = t_2
	elif (x * 0.5) <= 7.6:
		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(Float64(x * 0.5) * t_1)
	tmp = 0.0
	if (Float64(x * 0.5) <= -2.7e+166)
		tmp = t_2;
	elseif (Float64(x * 0.5) <= 7.6)
		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 = (x * 0.5) * t_1;
	tmp = 0.0;
	if ((x * 0.5) <= -2.7e+166)
		tmp = t_2;
	elseif ((x * 0.5) <= 7.6)
		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[(N[(x * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -2.7e+166], t$95$2, If[LessEqual[N[(x * 0.5), $MachinePrecision], 7.6], 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)) < -2.70000000000000012e166 or 7.5999999999999996 < (*.f64 x #s(literal 1/2 binary64))

   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%

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

   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 53.1

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

   7. Applied rewrites 53.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \sqrt{2 \cdot z} \]
   9. 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 45.4

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

   10. Applied rewrites 45.4%

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

   if -2.70000000000000012e166 < (*.f64 x #s(literal 1/2 binary64)) < 7.5999999999999996

   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%

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

   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 55.6

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

   7. Applied rewrites 55.6%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 41.2

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

   10. Applied rewrites 41.2%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2.7 \cdot 10^{+166}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\\ \mathbf{elif}\;x \cdot 0.5 \leq 7.6:\\ \;\;\;\;\sqrt{z \cdot 2} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \sqrt{z \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.7% accurate, 3.3× speedup

Math

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

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(Float64(fma(t, t, 2.0) * t), t, 2.0) * z)) * Float64(Float64(x * 0.5) - y))
end

Wolfram

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

Derivation

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%

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

5. Taylor expanded in t around 0

   \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{\color{blue}{2 \cdot z + {t}^{2} \cdot \left(2 \cdot z + {t}^{2} \cdot z\right)}} \]
6. 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 91.0

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

7. Applied rewrites 91.0%

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

9. Applied rewrites 91.0%

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

10. Final simplification 91.0%

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

Alternative 12: 57.7% 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 99.8%

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

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

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 54.7

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

7. Applied rewrites 54.7%

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

8. Final simplification 54.7%

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

Alternative 13: 30.4% 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 99.8%

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

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

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 54.7

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

7. Applied rewrites 54.7%

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

8. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 29.8

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

10. Applied rewrites 29.8%

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

11. Final simplification 29.8%

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

Developer Target 1: 99.3% 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 2024270 
(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))))