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

Percentage Accurate: 99.4% → 99.6%

Time: 11.2s

Alternatives: 15

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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 * 0.5d0))
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(t), (t * 0.5));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(t) ^ (t * 0.5));
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[t], $MachinePrecision], N[(t * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. pow2 N/A

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

   5. exp-sqrt N/A

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

   6. pow1/2 N/A

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

   7. pow2 N/A

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

   8. exp-prod N/A

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

   9. pow-unpow N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-*.f64 99.4

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

3. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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))) * (sqrt(exp(t)) ** t)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (sqrt(exp(t)) ^ t);
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[Sqrt[N[Exp[t], $MachinePrecision]], $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

   10. exp-sqrt N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-exp.f64 99.4

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

3. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    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]

Derivation

1. Initial program 99.4%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.6%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.2%

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

Alternative 6: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z} \cdot \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \left(x \cdot 0.5 - y\right)\right)\\ \mathbf{if}\;t \leq 11.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+104}:\\ \;\;\;\;\left(-\sqrt{e^{t \cdot t} \cdot \left(z + z\right)}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (sqrt (+ z z)) (* (fma (* t t) 0.5 1.0) (- (* x 0.5) y)))))
   (if (<= t 11.2)
     t_1
     (if (<= t 3e+104) (* (- (sqrt (* (exp (* t t)) (+ z z)))) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z)) * (fma((t * t), 0.5, 1.0) * ((x * 0.5) - y));
	double tmp;
	if (t <= 11.2) {
		tmp = t_1;
	} else if (t <= 3e+104) {
		tmp = -sqrt((exp((t * t)) * (z + z))) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(z + z)) * Float64(fma(Float64(t * t), 0.5, 1.0) * Float64(Float64(x * 0.5) - y)))
	tmp = 0.0
	if (t <= 11.2)
		tmp = t_1;
	elseif (t <= 3e+104)
		tmp = Float64(Float64(-sqrt(Float64(exp(Float64(t * t)) * Float64(z + z)))) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 11.2], t$95$1, If[LessEqual[t, 3e+104], N[((-N[Sqrt[N[(N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < 11.199999999999999 or 2.99999999999999969e104 < t

   1. Initial program 99.4%

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

   2. Taylor expanded in t around 0

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. count-2-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

   if 11.199999999999999 < t < 2.99999999999999969e104

   1. Initial program 98.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 72.9%

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

Alternative 7: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{\mathsf{fma}\left(t, t + t, 2\right)}\right) \cdot \sqrt{z}\\ \mathbf{if}\;t \leq 2.8 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{2 \cdot 1} \cdot \left(\left(\sqrt{z} \cdot \mathsf{fma}\left(\frac{x}{y}, 0.5, -1\right)\right) \cdot y\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+141}:\\ \;\;\;\;\left(\sqrt{\left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot z} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* (- (* 0.5 x) y) (sqrt (fma t (+ t t) 2.0))) (sqrt z))))
   (if (<= t 2.8e+15)
     t_1
     (if (<= t 7e+56)
       (* (sqrt (* 2.0 1.0)) (* (* (sqrt z) (fma (/ x y) 0.5 -1.0)) y))
       (if (<= t 1.25e+141)
         (* (* (sqrt (* (* (* t t) (* t t)) z)) 0.5) x)
         t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((0.5 * x) - y) * sqrt(fma(t, (t + t), 2.0))) * sqrt(z);
	double tmp;
	if (t <= 2.8e+15) {
		tmp = t_1;
	} else if (t <= 7e+56) {
		tmp = sqrt((2.0 * 1.0)) * ((sqrt(z) * fma((x / y), 0.5, -1.0)) * y);
	} else if (t <= 1.25e+141) {
		tmp = (sqrt((((t * t) * (t * t)) * z)) * 0.5) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(0.5 * x) - y) * sqrt(fma(t, Float64(t + t), 2.0))) * sqrt(z))
	tmp = 0.0
	if (t <= 2.8e+15)
		tmp = t_1;
	elseif (t <= 7e+56)
		tmp = Float64(sqrt(Float64(2.0 * 1.0)) * Float64(Float64(sqrt(z) * fma(Float64(x / y), 0.5, -1.0)) * y));
	elseif (t <= 1.25e+141)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(t * t) * Float64(t * t)) * z)) * 0.5) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(t * N[(t + t), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 2.8e+15], t$95$1, If[LessEqual[t, 7e+56], N[(N[Sqrt[N[(2.0 * 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[z], $MachinePrecision] * N[(N[(x / y), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e+141], N[(N[(N[Sqrt[N[(N[(N[(t * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.8e15 or 1.25000000000000006e141 < t

   1. Initial program 99.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. exp-sqrt N/A

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

      6. pow1/2 N/A

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

      7. pow2 N/A

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

      8. exp-prod N/A

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

      9. pow-unpow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-*.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. count-2-rev N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 88.7

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

   8. Applied rewrites 88.7%

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

   if 2.8e15 < t < 6.99999999999999999e56

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 27.3%

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

   if 6.99999999999999999e56 < t < 1.25000000000000006e141

   1. Initial program 98.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f64 53.0

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

   7. Applied rewrites 53.0%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{{t}^{4} \cdot z} \cdot \frac{1}{2}\right) \cdot x \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.6

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

   10. Applied rewrites 68.6%

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

Alternative 8: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{\mathsf{fma}\left(t, t + t, 2\right)}\right) \cdot \sqrt{z}\\ \mathbf{if}\;t \leq 32500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(\frac{1}{\sqrt{z}} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot z\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+141}:\\ \;\;\;\;\left(\sqrt{\left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot z} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* (- (* 0.5 x) y) (sqrt (fma t (+ t t) 2.0))) (sqrt z))))
   (if (<= t 32500.0)
     t_1
     (if (<= t 3.2e+40)
       (* (sqrt 2.0) (* (* (/ 1.0 (sqrt z)) (- (* x 0.5) y)) z))
       (if (<= t 1.25e+141)
         (* (* (sqrt (* (* (* t t) (* t t)) z)) 0.5) x)
         t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((0.5 * x) - y) * sqrt(fma(t, (t + t), 2.0))) * sqrt(z);
	double tmp;
	if (t <= 32500.0) {
		tmp = t_1;
	} else if (t <= 3.2e+40) {
		tmp = sqrt(2.0) * (((1.0 / sqrt(z)) * ((x * 0.5) - y)) * z);
	} else if (t <= 1.25e+141) {
		tmp = (sqrt((((t * t) * (t * t)) * z)) * 0.5) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(0.5 * x) - y) * sqrt(fma(t, Float64(t + t), 2.0))) * sqrt(z))
	tmp = 0.0
	if (t <= 32500.0)
		tmp = t_1;
	elseif (t <= 3.2e+40)
		tmp = Float64(sqrt(2.0) * Float64(Float64(Float64(1.0 / sqrt(z)) * Float64(Float64(x * 0.5) - y)) * z));
	elseif (t <= 1.25e+141)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(t * t) * Float64(t * t)) * z)) * 0.5) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(t * N[(t + t), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 32500.0], t$95$1, If[LessEqual[t, 3.2e+40], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e+141], N[(N[(N[Sqrt[N[(N[(N[(t * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < 32500 or 1.25000000000000006e141 < t

   1. Initial program 99.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. exp-sqrt N/A

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

      6. pow1/2 N/A

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

      7. pow2 N/A

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

      8. exp-prod N/A

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

      9. pow-unpow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-*.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. count-2-rev N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 89.3

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

   8. Applied rewrites 89.3%

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

   if 32500 < t < 3.19999999999999981e40

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 15.2%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 28.4

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

   9. Applied rewrites 28.4%

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

   if 3.19999999999999981e40 < t < 1.25000000000000006e141

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f64 49.1

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{{t}^{4} \cdot z} \cdot \frac{1}{2}\right) \cdot x \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 61.9

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

   10. Applied rewrites 61.9%

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

Alternative 9: 85.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft1-in N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 85.3%

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

5. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. sqrt-unprod N/A

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

   4. *-commutative N/A

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

   5. count-2-rev N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. +-commutative N/A

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

   12. lift-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift--.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 87.5

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

7. Applied rewrites 87.5%

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

Alternative 10: 69.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(\frac{1}{\sqrt{z}} \cdot \left(x \cdot 0.5 - y\right)\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot z} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 5.6e-6)
   (* (sqrt (+ z z)) (- (* 0.5 x) y))
   (if (<= t 3.2e+40)
     (* (sqrt 2.0) (* (* (/ 1.0 (sqrt z)) (- (* x 0.5) y)) z))
     (* (* (sqrt (* (* (* t t) (* t t)) z)) 0.5) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.6e-6) {
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	} else if (t <= 3.2e+40) {
		tmp = sqrt(2.0) * (((1.0 / sqrt(z)) * ((x * 0.5) - y)) * z);
	} else {
		tmp = (sqrt((((t * t) * (t * t)) * z)) * 0.5) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 5.6d-6) then
        tmp = sqrt((z + z)) * ((0.5d0 * x) - y)
    else if (t <= 3.2d+40) then
        tmp = sqrt(2.0d0) * (((1.0d0 / sqrt(z)) * ((x * 0.5d0) - y)) * z)
    else
        tmp = (sqrt((((t * t) * (t * t)) * z)) * 0.5d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.6e-6) {
		tmp = Math.sqrt((z + z)) * ((0.5 * x) - y);
	} else if (t <= 3.2e+40) {
		tmp = Math.sqrt(2.0) * (((1.0 / Math.sqrt(z)) * ((x * 0.5) - y)) * z);
	} else {
		tmp = (Math.sqrt((((t * t) * (t * t)) * z)) * 0.5) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 5.6e-6:
		tmp = math.sqrt((z + z)) * ((0.5 * x) - y)
	elif t <= 3.2e+40:
		tmp = math.sqrt(2.0) * (((1.0 / math.sqrt(z)) * ((x * 0.5) - y)) * z)
	else:
		tmp = (math.sqrt((((t * t) * (t * t)) * z)) * 0.5) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 5.6e-6)
		tmp = Float64(sqrt(Float64(z + z)) * Float64(Float64(0.5 * x) - y));
	elseif (t <= 3.2e+40)
		tmp = Float64(sqrt(2.0) * Float64(Float64(Float64(1.0 / sqrt(z)) * Float64(Float64(x * 0.5) - y)) * z));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(t * t) * Float64(t * t)) * z)) * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 5.6e-6)
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	elseif (t <= 3.2e+40)
		tmp = sqrt(2.0) * (((1.0 / sqrt(z)) * ((x * 0.5) - y)) * z);
	else
		tmp = (sqrt((((t * t) * (t * t)) * z)) * 0.5) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 5.6e-6], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+40], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(1.0 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[(t * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.59999999999999975e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 71.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 71.8

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

   4. Applied rewrites 71.8%

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

   if 5.59999999999999975e-6 < t < 3.19999999999999981e40

   1. Initial program 99.4%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 21.3%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 31.3

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

   9. Applied rewrites 31.3%

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

   if 3.19999999999999981e40 < t

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f64 63.7

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

   7. Applied rewrites 63.7%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{{t}^{4} \cdot z} \cdot \frac{1}{2}\right) \cdot x \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.8

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

   10. Applied rewrites 68.8%

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

Alternative 11: 69.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(\left(t \cdot t\right) \cdot \left(t \cdot t\right)\right) \cdot z} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 5.5e+38)
   (* (sqrt (+ z z)) (- (* 0.5 x) y))
   (* (* (sqrt (* (* (* t t) (* t t)) z)) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.5e+38) {
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	} else {
		tmp = (sqrt((((t * t) * (t * t)) * z)) * 0.5) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 5.5d+38) then
        tmp = sqrt((z + z)) * ((0.5d0 * x) - y)
    else
        tmp = (sqrt((((t * t) * (t * t)) * z)) * 0.5d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.5e+38) {
		tmp = Math.sqrt((z + z)) * ((0.5 * x) - y);
	} else {
		tmp = (Math.sqrt((((t * t) * (t * t)) * z)) * 0.5) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 5.5e+38:
		tmp = math.sqrt((z + z)) * ((0.5 * x) - y)
	else:
		tmp = (math.sqrt((((t * t) * (t * t)) * z)) * 0.5) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 5.5e+38)
		tmp = Float64(sqrt(Float64(z + z)) * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(t * t) * Float64(t * t)) * z)) * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 5.5e+38)
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	else
		tmp = (sqrt((((t * t) * (t * t)) * z)) * 0.5) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 5.5e+38], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[(t * t), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.5000000000000003e38

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 69.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   if 5.5000000000000003e38 < t

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f64 63.5

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{{t}^{4} \cdot z} \cdot \frac{1}{2}\right) \cdot x \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.5

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

   10. Applied rewrites 68.5%

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

Alternative 12: 66.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;t \leq 10^{+39}:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t \cdot t\right) \cdot t\_1\right) \cdot 0.5\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (<= t 1e+39)
     (* t_1 (- (* 0.5 x) y))
     (* (* (* (* (* t t) t_1) 0.5) 0.5) x))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if (t <= 1e+39) {
		tmp = t_1 * ((0.5 * x) - y);
	} else {
		tmp = ((((t * t) * t_1) * 0.5) * 0.5) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((z + z))
    if (t <= 1d+39) then
        tmp = t_1 * ((0.5d0 * x) - y)
    else
        tmp = ((((t * t) * t_1) * 0.5d0) * 0.5d0) * x
    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 + z));
	double tmp;
	if (t <= 1e+39) {
		tmp = t_1 * ((0.5 * x) - y);
	} else {
		tmp = ((((t * t) * t_1) * 0.5) * 0.5) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if t <= 1e+39:
		tmp = t_1 * ((0.5 * x) - y)
	else:
		tmp = ((((t * t) * t_1) * 0.5) * 0.5) * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if (t <= 1e+39)
		tmp = Float64(t_1 * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t * t) * t_1) * 0.5) * 0.5) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if (t <= 1e+39)
		tmp = t_1 * ((0.5 * x) - y);
	else
		tmp = ((((t * t) * t_1) * 0.5) * 0.5) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 1e+39], N[(t$95$1 * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(t * t), $MachinePrecision] * t$95$1), $MachinePrecision] * 0.5), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999994e38

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 69.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   if 9.9999999999999994e38 < t

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-sqrt.f64 56.0

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

   7. Applied rewrites 56.0%

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

   8. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. *-commutative N/A

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

      10. lower-sqrt.f64 N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 56.0

          \[\leadsto \left(\left(\left(\left(t \cdot t\right) \cdot \sqrt{z + z}\right) \cdot 0.5\right) \cdot 0.5\right) \cdot x \]

   10. Applied rewrites 56.0%

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

Alternative 13: 65.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(0.5 \cdot x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(t, t, 1\right) \cdot \left(z + z\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 8e+39)
   (* (sqrt (+ z z)) (- (* 0.5 x) y))
   (* (* (sqrt (* (fma t t 1.0) (+ z z))) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 8e+39) {
		tmp = sqrt((z + z)) * ((0.5 * x) - y);
	} else {
		tmp = (sqrt((fma(t, t, 1.0) * (z + z))) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 8e+39)
		tmp = Float64(sqrt(Float64(z + z)) * Float64(Float64(0.5 * x) - y));
	else
		tmp = Float64(Float64(sqrt(Float64(fma(t, t, 1.0) * Float64(z + z))) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 8e+39], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(t * t + 1.0), $MachinePrecision] * N[(z + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.99999999999999952e39

   1. Initial program 99.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 69.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   if 7.99999999999999952e39 < t

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 54.0

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

   7. Applied rewrites 54.0%

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

Alternative 14: 57.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lift-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 57.0

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 57.0

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

4. Applied rewrites 57.0%

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

Alternative 15: 29.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 62.4%

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

5. Taylor expanded in t around 0

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

   1. count-2-rev N/A

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

   2. lift-+.f64 29.9

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

7. Applied rewrites 29.9%

   \[\leadsto \left(\sqrt{z + z} \cdot 0.5\right) \cdot x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))