Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.2%

Time: 3.5s

Alternatives: 14

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-t1}{u + t1} \cdot v}{u + t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (/ (- t1) (+ u t1)) v) (+ u t1)))

C

double code(double u, double v, double t1) {
	return ((-t1 / (u + t1)) * v) / (u + t1);
}

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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = ((-t1 / (u + t1)) * v) / (u + t1)
end function

Java

public static double code(double u, double v, double t1) {
	return ((-t1 / (u + t1)) * v) / (u + t1);
}

Python

def code(u, v, t1):
	return ((-t1 / (u + t1)) * v) / (u + t1)

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(Float64(-t1) / Float64(u + t1)) * v) / Float64(u + t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = ((-t1 / (u + t1)) * v) / (u + t1);
end

Wolfram

code[u_, v_, t1_] := N[(N[(N[((-t1) / N[(u + t1), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   7. times-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

   10. lift-neg.f64 N/A

       \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

   11. +-commutative N/A

       \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

   14. +-commutative N/A

       \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   15. lower-+.f64 98.1

       \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

3. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{u + t1} \cdot \frac{v}{u + t1} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{\color{blue}{u + t1}} \cdot \frac{v}{u + t1} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot \frac{v}{u + t1} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \color{blue}{\frac{v}{u + t1}} \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}}{u + t1} \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot v}{u + t1} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{-t1}}{u + t1} \cdot v}{u + t1} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{\frac{-t1}{\color{blue}{u + t1}} \cdot v}{u + t1} \]

   13. lift-+.f64 98.2

       \[\leadsto \frac{\frac{-t1}{u + t1} \cdot v}{\color{blue}{u + t1}} \]

5. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{\frac{-t1}{u + t1} \cdot v}{u + t1}} \]
6. Add Preprocessing

Alternative 2: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{-t1}{u + t1} \cdot \frac{v}{u + t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (* (/ (- t1) (+ u t1)) (/ v (+ u t1))))

C

double code(double u, double v, double t1) {
	return (-t1 / (u + t1)) * (v / (u + t1));
}

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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 / (u + t1)) * (v / (u + t1))
end function

Java

public static double code(double u, double v, double t1) {
	return (-t1 / (u + t1)) * (v / (u + t1));
}

Python

def code(u, v, t1):
	return (-t1 / (u + t1)) * (v / (u + t1))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) / Float64(u + t1)) * Float64(v / Float64(u + t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 / (u + t1)) * (v / (u + t1));
end

Wolfram

code[u_, v_, t1_] := N[(N[((-t1) / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   7. times-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

   10. lift-neg.f64 N/A

       \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

   11. +-commutative N/A

       \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

   14. +-commutative N/A

       \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   15. lower-+.f64 98.1

       \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

3. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
4. Add Preprocessing

Alternative 3: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{+148}:\\ \;\;\;\;\left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1}\\ \mathbf{elif}\;t1 \leq -2.7 \cdot 10^{-88}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{\mathsf{fma}\left(u + t1, u, \left(u + t1\right) \cdot t1\right)}\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-272}:\\ \;\;\;\;\left(v \cdot \frac{t1}{u}\right) \cdot \frac{-1}{u}\\ \mathbf{elif}\;t1 \leq 2.45 \cdot 10^{+150}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.9e+148)
   (* (- (/ u t1) 1.0) (/ v (+ u t1)))
   (if (<= t1 -2.7e-88)
     (* (- t1) (/ v (fma (+ u t1) u (* (+ u t1) t1))))
     (if (<= t1 1.42e-272)
       (* (* v (/ t1 u)) (/ -1.0 u))
       (if (<= t1 2.45e+150)
         (* (- t1) (/ v (* (+ u t1) (+ u t1))))
         (/ (fma u (/ v t1) (- v)) (+ u t1)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.9e+148) {
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	} else if (t1 <= -2.7e-88) {
		tmp = -t1 * (v / fma((u + t1), u, ((u + t1) * t1)));
	} else if (t1 <= 1.42e-272) {
		tmp = (v * (t1 / u)) * (-1.0 / u);
	} else if (t1 <= 2.45e+150) {
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	} else {
		tmp = fma(u, (v / t1), -v) / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.9e+148)
		tmp = Float64(Float64(Float64(u / t1) - 1.0) * Float64(v / Float64(u + t1)));
	elseif (t1 <= -2.7e-88)
		tmp = Float64(Float64(-t1) * Float64(v / fma(Float64(u + t1), u, Float64(Float64(u + t1) * t1))));
	elseif (t1 <= 1.42e-272)
		tmp = Float64(Float64(v * Float64(t1 / u)) * Float64(-1.0 / u));
	elseif (t1 <= 2.45e+150)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))));
	else
		tmp = Float64(fma(u, Float64(v / t1), Float64(-v)) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.9e+148], N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -2.7e-88], N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * u + N[(N[(u + t1), $MachinePrecision] * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.42e-272], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.45e+150], N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(u * N[(v / t1), $MachinePrecision] + (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t1 < -2.9e148

   1. Initial program 42.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 100.0

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

   4. Taylor expanded in u around 0

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{u + t1} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{u}{t1} - \color{blue}{1}\right) \cdot \frac{v}{u + t1} \]

      2. lower-/.f64 92.0

         \[\leadsto \left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1} \]

   6. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{u + t1} \]

   if -2.9e148 < t1 < -2.69999999999999995e-88

   1. Initial program 86.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      12. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      15. lower-+.f64 86.0

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

   3. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(u + t1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right) \cdot u + \left(u + t1\right) \cdot t1}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\mathsf{fma}\left(u + t1, u, \left(u + t1\right) \cdot t1\right)}} \]

      6. lift-+.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\mathsf{fma}\left(\color{blue}{u + t1}, u, \left(u + t1\right) \cdot t1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\mathsf{fma}\left(u + t1, u, \color{blue}{\left(u + t1\right) \cdot t1}\right)} \]

      8. lift-+.f64 86.1

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\mathsf{fma}\left(u + t1, u, \color{blue}{\left(u + t1\right)} \cdot t1\right)} \]

   5. Applied rewrites 86.1%

      \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\mathsf{fma}\left(u + t1, u, \left(u + t1\right) \cdot t1\right)}} \]

   if -2.69999999999999995e-88 < t1 < 1.41999999999999997e-272

   1. Initial program 80.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t1 \cdot v\right) \cdot -1}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u}} \cdot \frac{-1}{t1 + u} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      15. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \color{blue}{\frac{-1}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

      19. lower-+.f64 87.1

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

   3. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u + t1} \cdot \frac{-1}{u + t1}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]
   5. Step-by-step derivation

   6. Applied rewrites 77.1%

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]

   7. Taylor expanded in u around inf

      \[\leadsto \color{blue}{\frac{t1 \cdot v}{u}} \cdot \frac{-1}{u} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t1 \cdot v}{\color{blue}{u}} \cdot \frac{-1}{u} \]

      2. *-commutative N/A

         \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

      3. lift-*.f64 79.4

         \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

   9. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u}} \cdot \frac{-1}{u} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u}} \cdot \frac{-1}{u} \]

       3. associate-/l* N/A

          \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

       5. lower-/.f64 82.7

          \[\leadsto \left(v \cdot \frac{t1}{\color{blue}{u}}\right) \cdot \frac{-1}{u} \]

   11. Applied rewrites 82.7%

       \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

   if 1.41999999999999997e-272 < t1 < 2.45000000000000003e150

   1. Initial program 84.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      12. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      15. lower-+.f64 83.0

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

   3. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]

   if 2.45000000000000003e150 < t1

   1. Initial program 39.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{u + t1} \cdot \frac{v}{u + t1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{\color{blue}{u + t1}} \cdot \frac{v}{u + t1} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot \frac{v}{u + t1} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \color{blue}{\frac{v}{u + t1}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}}{u + t1} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot v}{u + t1} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{-t1}}{u + t1} \cdot v}{u + t1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-t1}{\color{blue}{u + t1}} \cdot v}{u + t1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{-t1}{u + t1} \cdot v}{\color{blue}{u + t1}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{u + t1} \cdot v}{u + t1}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v + \frac{u \cdot v}{t1}}}{u + t1} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{u \cdot v}{t1} + \color{blue}{-1 \cdot v}}{u + t1} \]

      2. associate-/l* N/A

         \[\leadsto \frac{u \cdot \frac{v}{t1} + \color{blue}{-1} \cdot v}{u + t1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(u, \color{blue}{\frac{v}{t1}}, -1 \cdot v\right)}{u + t1} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(u, \frac{v}{\color{blue}{t1}}, -1 \cdot v\right)}{u + t1} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(u, \frac{v}{t1}, \mathsf{neg}\left(v\right)\right)}{u + t1} \]

      6. lower-neg.f64 91.6

         \[\leadsto \frac{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}{u + t1} \]

   8. Applied rewrites 91.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}}{u + t1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{if}\;t1 \leq -2.9 \cdot 10^{+148}:\\ \;\;\;\;\left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1}\\ \mathbf{elif}\;t1 \leq -2.7 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-272}:\\ \;\;\;\;\left(v \cdot \frac{t1}{u}\right) \cdot \frac{-1}{u}\\ \mathbf{elif}\;t1 \leq 2.45 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) (/ v (* (+ u t1) (+ u t1))))))
   (if (<= t1 -2.9e+148)
     (* (- (/ u t1) 1.0) (/ v (+ u t1)))
     (if (<= t1 -2.7e-88)
       t_1
       (if (<= t1 1.42e-272)
         (* (* v (/ t1 u)) (/ -1.0 u))
         (if (<= t1 2.45e+150) t_1 (/ (fma u (/ v t1) (- v)) (+ u t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / ((u + t1) * (u + t1)));
	double tmp;
	if (t1 <= -2.9e+148) {
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	} else if (t1 <= -2.7e-88) {
		tmp = t_1;
	} else if (t1 <= 1.42e-272) {
		tmp = (v * (t1 / u)) * (-1.0 / u);
	} else if (t1 <= 2.45e+150) {
		tmp = t_1;
	} else {
		tmp = fma(u, (v / t1), -v) / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))))
	tmp = 0.0
	if (t1 <= -2.9e+148)
		tmp = Float64(Float64(Float64(u / t1) - 1.0) * Float64(v / Float64(u + t1)));
	elseif (t1 <= -2.7e-88)
		tmp = t_1;
	elseif (t1 <= 1.42e-272)
		tmp = Float64(Float64(v * Float64(t1 / u)) * Float64(-1.0 / u));
	elseif (t1 <= 2.45e+150)
		tmp = t_1;
	else
		tmp = Float64(fma(u, Float64(v / t1), Float64(-v)) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.9e+148], N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -2.7e-88], t$95$1, If[LessEqual[t1, 1.42e-272], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.45e+150], t$95$1, N[(N[(u * N[(v / t1), $MachinePrecision] + (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.9e148

   1. Initial program 42.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 100.0

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

   4. Taylor expanded in u around 0

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{u + t1} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{u}{t1} - \color{blue}{1}\right) \cdot \frac{v}{u + t1} \]

      2. lower-/.f64 92.0

         \[\leadsto \left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1} \]

   6. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{u + t1} \]

   if -2.9e148 < t1 < -2.69999999999999995e-88 or 1.41999999999999997e-272 < t1 < 2.45000000000000003e150

   1. Initial program 84.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      12. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      15. lower-+.f64 84.1

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

   3. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]

   if -2.69999999999999995e-88 < t1 < 1.41999999999999997e-272

   1. Initial program 80.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t1 \cdot v\right) \cdot -1}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u}} \cdot \frac{-1}{t1 + u} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      15. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \color{blue}{\frac{-1}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

      19. lower-+.f64 87.1

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

   3. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u + t1} \cdot \frac{-1}{u + t1}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]
   5. Step-by-step derivation

   6. Applied rewrites 77.1%

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]

   7. Taylor expanded in u around inf

      \[\leadsto \color{blue}{\frac{t1 \cdot v}{u}} \cdot \frac{-1}{u} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t1 \cdot v}{\color{blue}{u}} \cdot \frac{-1}{u} \]

      2. *-commutative N/A

         \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

      3. lift-*.f64 79.4

         \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

   9. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u}} \cdot \frac{-1}{u} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u}} \cdot \frac{-1}{u} \]

       3. associate-/l* N/A

          \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

       5. lower-/.f64 82.7

          \[\leadsto \left(v \cdot \frac{t1}{\color{blue}{u}}\right) \cdot \frac{-1}{u} \]

   11. Applied rewrites 82.7%

       \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

   if 2.45000000000000003e150 < t1

   1. Initial program 39.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{u + t1} \cdot \frac{v}{u + t1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{\color{blue}{u + t1}} \cdot \frac{v}{u + t1} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot \frac{v}{u + t1} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \color{blue}{\frac{v}{u + t1}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}}{u + t1} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot v}{u + t1} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{-t1}}{u + t1} \cdot v}{u + t1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-t1}{\color{blue}{u + t1}} \cdot v}{u + t1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{-t1}{u + t1} \cdot v}{\color{blue}{u + t1}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{u + t1} \cdot v}{u + t1}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v + \frac{u \cdot v}{t1}}}{u + t1} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{u \cdot v}{t1} + \color{blue}{-1 \cdot v}}{u + t1} \]

      2. associate-/l* N/A

         \[\leadsto \frac{u \cdot \frac{v}{t1} + \color{blue}{-1} \cdot v}{u + t1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(u, \color{blue}{\frac{v}{t1}}, -1 \cdot v\right)}{u + t1} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(u, \frac{v}{\color{blue}{t1}}, -1 \cdot v\right)}{u + t1} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(u, \frac{v}{t1}, \mathsf{neg}\left(v\right)\right)}{u + t1} \]

      6. lower-neg.f64 91.6

         \[\leadsto \frac{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}{u + t1} \]

   8. Applied rewrites 91.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}}{u + t1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ t_2 := \left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1}\\ \mathbf{if}\;t1 \leq -2.9 \cdot 10^{+148}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -2.7 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-272}:\\ \;\;\;\;\left(v \cdot \frac{t1}{u}\right) \cdot \frac{-1}{u}\\ \mathbf{elif}\;t1 \leq 1.95 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) (/ v (* (+ u t1) (+ u t1)))))
        (t_2 (* (- (/ u t1) 1.0) (/ v (+ u t1)))))
   (if (<= t1 -2.9e+148)
     t_2
     (if (<= t1 -2.7e-88)
       t_1
       (if (<= t1 1.42e-272)
         (* (* v (/ t1 u)) (/ -1.0 u))
         (if (<= t1 1.95e+130) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / ((u + t1) * (u + t1)));
	double t_2 = ((u / t1) - 1.0) * (v / (u + t1));
	double tmp;
	if (t1 <= -2.9e+148) {
		tmp = t_2;
	} else if (t1 <= -2.7e-88) {
		tmp = t_1;
	} else if (t1 <= 1.42e-272) {
		tmp = (v * (t1 / u)) * (-1.0 / u);
	} else if (t1 <= 1.95e+130) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -t1 * (v / ((u + t1) * (u + t1)))
    t_2 = ((u / t1) - 1.0d0) * (v / (u + t1))
    if (t1 <= (-2.9d+148)) then
        tmp = t_2
    else if (t1 <= (-2.7d-88)) then
        tmp = t_1
    else if (t1 <= 1.42d-272) then
        tmp = (v * (t1 / u)) * ((-1.0d0) / u)
    else if (t1 <= 1.95d+130) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / ((u + t1) * (u + t1)));
	double t_2 = ((u / t1) - 1.0) * (v / (u + t1));
	double tmp;
	if (t1 <= -2.9e+148) {
		tmp = t_2;
	} else if (t1 <= -2.7e-88) {
		tmp = t_1;
	} else if (t1 <= 1.42e-272) {
		tmp = (v * (t1 / u)) * (-1.0 / u);
	} else if (t1 <= 1.95e+130) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 * (v / ((u + t1) * (u + t1)))
	t_2 = ((u / t1) - 1.0) * (v / (u + t1))
	tmp = 0
	if t1 <= -2.9e+148:
		tmp = t_2
	elif t1 <= -2.7e-88:
		tmp = t_1
	elif t1 <= 1.42e-272:
		tmp = (v * (t1 / u)) * (-1.0 / u)
	elif t1 <= 1.95e+130:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))))
	t_2 = Float64(Float64(Float64(u / t1) - 1.0) * Float64(v / Float64(u + t1)))
	tmp = 0.0
	if (t1 <= -2.9e+148)
		tmp = t_2;
	elseif (t1 <= -2.7e-88)
		tmp = t_1;
	elseif (t1 <= 1.42e-272)
		tmp = Float64(Float64(v * Float64(t1 / u)) * Float64(-1.0 / u));
	elseif (t1 <= 1.95e+130)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 * (v / ((u + t1) * (u + t1)));
	t_2 = ((u / t1) - 1.0) * (v / (u + t1));
	tmp = 0.0;
	if (t1 <= -2.9e+148)
		tmp = t_2;
	elseif (t1 <= -2.7e-88)
		tmp = t_1;
	elseif (t1 <= 1.42e-272)
		tmp = (v * (t1 / u)) * (-1.0 / u);
	elseif (t1 <= 1.95e+130)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.9e+148], t$95$2, If[LessEqual[t1, -2.7e-88], t$95$1, If[LessEqual[t1, 1.42e-272], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.95e+130], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.9e148 or 1.9500000000000001e130 < t1

   1. Initial program 43.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

   4. Taylor expanded in u around 0

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{u + t1} \]
   5. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{u}{t1} - \color{blue}{1}\right) \cdot \frac{v}{u + t1} \]

      2. lower-/.f64 91.6

         \[\leadsto \left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1} \]

   6. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{u + t1} \]

   if -2.9e148 < t1 < -2.69999999999999995e-88 or 1.41999999999999997e-272 < t1 < 1.9500000000000001e130

   1. Initial program 85.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      12. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      15. lower-+.f64 84.7

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]

   if -2.69999999999999995e-88 < t1 < 1.41999999999999997e-272

   1. Initial program 80.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t1 \cdot v\right) \cdot -1}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u}} \cdot \frac{-1}{t1 + u} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      15. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \color{blue}{\frac{-1}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

      19. lower-+.f64 87.1

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

   3. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u + t1} \cdot \frac{-1}{u + t1}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]
   5. Step-by-step derivation

   6. Applied rewrites 77.1%

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]

   7. Taylor expanded in u around inf

      \[\leadsto \color{blue}{\frac{t1 \cdot v}{u}} \cdot \frac{-1}{u} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t1 \cdot v}{\color{blue}{u}} \cdot \frac{-1}{u} \]

      2. *-commutative N/A

         \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

      3. lift-*.f64 79.4

         \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

   9. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u}} \cdot \frac{-1}{u} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u}} \cdot \frac{-1}{u} \]

       3. associate-/l* N/A

          \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

       5. lower-/.f64 82.7

          \[\leadsto \left(v \cdot \frac{t1}{\color{blue}{u}}\right) \cdot \frac{-1}{u} \]

   11. Applied rewrites 82.7%

       \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ t_2 := \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -2.7 \cdot 10^{-88}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{-272}:\\ \;\;\;\;\left(v \cdot \frac{t1}{u}\right) \cdot \frac{-1}{u}\\ \mathbf{elif}\;t1 \leq 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))) (t_2 (* (- t1) (/ v (* (+ u t1) (+ u t1))))))
   (if (<= t1 -2e+148)
     t_1
     (if (<= t1 -2.7e-88)
       t_2
       (if (<= t1 1.42e-272)
         (* (* v (/ t1 u)) (/ -1.0 u))
         (if (<= t1 1e+129) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double t_2 = -t1 * (v / ((u + t1) * (u + t1)));
	double tmp;
	if (t1 <= -2e+148) {
		tmp = t_1;
	} else if (t1 <= -2.7e-88) {
		tmp = t_2;
	} else if (t1 <= 1.42e-272) {
		tmp = (v * (t1 / u)) * (-1.0 / u);
	} else if (t1 <= 1e+129) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (u + t1)
    t_2 = -t1 * (v / ((u + t1) * (u + t1)))
    if (t1 <= (-2d+148)) then
        tmp = t_1
    else if (t1 <= (-2.7d-88)) then
        tmp = t_2
    else if (t1 <= 1.42d-272) then
        tmp = (v * (t1 / u)) * ((-1.0d0) / u)
    else if (t1 <= 1d+129) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double t_2 = -t1 * (v / ((u + t1) * (u + t1)));
	double tmp;
	if (t1 <= -2e+148) {
		tmp = t_1;
	} else if (t1 <= -2.7e-88) {
		tmp = t_2;
	} else if (t1 <= 1.42e-272) {
		tmp = (v * (t1 / u)) * (-1.0 / u);
	} else if (t1 <= 1e+129) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	t_2 = -t1 * (v / ((u + t1) * (u + t1)))
	tmp = 0
	if t1 <= -2e+148:
		tmp = t_1
	elif t1 <= -2.7e-88:
		tmp = t_2
	elif t1 <= 1.42e-272:
		tmp = (v * (t1 / u)) * (-1.0 / u)
	elif t1 <= 1e+129:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	t_2 = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))))
	tmp = 0.0
	if (t1 <= -2e+148)
		tmp = t_1;
	elseif (t1 <= -2.7e-88)
		tmp = t_2;
	elseif (t1 <= 1.42e-272)
		tmp = Float64(Float64(v * Float64(t1 / u)) * Float64(-1.0 / u));
	elseif (t1 <= 1e+129)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	t_2 = -t1 * (v / ((u + t1) * (u + t1)));
	tmp = 0.0;
	if (t1 <= -2e+148)
		tmp = t_1;
	elseif (t1 <= -2.7e-88)
		tmp = t_2;
	elseif (t1 <= 1.42e-272)
		tmp = (v * (t1 / u)) * (-1.0 / u);
	elseif (t1 <= 1e+129)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2e+148], t$95$1, If[LessEqual[t1, -2.7e-88], t$95$2, If[LessEqual[t1, 1.42e-272], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1e+129], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.0000000000000001e148 or 1e129 < t1

   1. Initial program 43.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{u + t1} \cdot \frac{v}{u + t1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{\color{blue}{u + t1}} \cdot \frac{v}{u + t1} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot \frac{v}{u + t1} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \color{blue}{\frac{v}{u + t1}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}}{u + t1} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot v}{u + t1} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{-t1}}{u + t1} \cdot v}{u + t1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-t1}{\color{blue}{u + t1}} \cdot v}{u + t1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{-t1}{u + t1} \cdot v}{\color{blue}{u + t1}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{u + t1} \cdot v}{u + t1}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{u + t1} \]

      2. lower-neg.f64 91.5

         \[\leadsto \frac{-v}{u + t1} \]

   8. Applied rewrites 91.5%

      \[\leadsto \frac{\color{blue}{-v}}{u + t1} \]

   if -2.0000000000000001e148 < t1 < -2.69999999999999995e-88 or 1.41999999999999997e-272 < t1 < 1e129

   1. Initial program 85.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      12. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      15. lower-+.f64 84.7

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]

   if -2.69999999999999995e-88 < t1 < 1.41999999999999997e-272

   1. Initial program 80.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t1 \cdot v\right) \cdot -1}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u}} \cdot \frac{-1}{t1 + u} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      15. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \color{blue}{\frac{-1}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

      19. lower-+.f64 87.1

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

   3. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u + t1} \cdot \frac{-1}{u + t1}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]
   5. Step-by-step derivation

   6. Applied rewrites 77.1%

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]

   7. Taylor expanded in u around inf

      \[\leadsto \color{blue}{\frac{t1 \cdot v}{u}} \cdot \frac{-1}{u} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t1 \cdot v}{\color{blue}{u}} \cdot \frac{-1}{u} \]

      2. *-commutative N/A

         \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

      3. lift-*.f64 79.4

         \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

   9. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u}} \cdot \frac{-1}{u} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u} \cdot \frac{-1}{u} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u}} \cdot \frac{-1}{u} \]

       3. associate-/l* N/A

          \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]

       5. lower-/.f64 82.7

          \[\leadsto \left(v \cdot \frac{t1}{\color{blue}{u}}\right) \cdot \frac{-1}{u} \]

   11. Applied rewrites 82.7%

       \[\leadsto \left(v \cdot \color{blue}{\frac{t1}{u}}\right) \cdot \frac{-1}{u} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 10^{+129}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2e+148)
     t_1
     (if (<= t1 1e+129) (* (- t1) (/ v (* (+ u t1) (+ u t1)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2e+148) {
		tmp = t_1;
	} else if (t1 <= 1e+129) {
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	} else {
		tmp = t_1;
	}
	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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-2d+148)) then
        tmp = t_1
    else if (t1 <= 1d+129) then
        tmp = -t1 * (v / ((u + t1) * (u + t1)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2e+148) {
		tmp = t_1;
	} else if (t1 <= 1e+129) {
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -2e+148:
		tmp = t_1
	elif t1 <= 1e+129:
		tmp = -t1 * (v / ((u + t1) * (u + t1)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -2e+148)
		tmp = t_1;
	elseif (t1 <= 1e+129)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -2e+148)
		tmp = t_1;
	elseif (t1 <= 1e+129)
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2e+148], t$95$1, If[LessEqual[t1, 1e+129], N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.0000000000000001e148 or 1e129 < t1

   1. Initial program 43.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{u + t1} \cdot \frac{v}{u + t1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{\color{blue}{u + t1}} \cdot \frac{v}{u + t1} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot \frac{v}{u + t1} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \color{blue}{\frac{v}{u + t1}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}}{u + t1} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot v}{u + t1} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{-t1}}{u + t1} \cdot v}{u + t1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-t1}{\color{blue}{u + t1}} \cdot v}{u + t1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{-t1}{u + t1} \cdot v}{\color{blue}{u + t1}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{u + t1} \cdot v}{u + t1}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{u + t1} \]

      2. lower-neg.f64 91.5

         \[\leadsto \frac{-v}{u + t1} \]

   8. Applied rewrites 91.5%

      \[\leadsto \frac{\color{blue}{-v}}{u + t1} \]

   if -2.0000000000000001e148 < t1 < 1e129

   1. Initial program 83.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      12. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      15. lower-+.f64 84.0

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

   3. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -5.2 \cdot 10^{+26}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot t1}\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2e+148)
     t_1
     (if (<= t1 -5.2e+26)
       (* (- t1) (/ v (* (+ u t1) t1)))
       (if (<= t1 4.4e-57) (/ (* (- t1) v) (* u u)) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2e+148) {
		tmp = t_1;
	} else if (t1 <= -5.2e+26) {
		tmp = -t1 * (v / ((u + t1) * t1));
	} else if (t1 <= 4.4e-57) {
		tmp = (-t1 * v) / (u * u);
	} else {
		tmp = t_1;
	}
	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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-2d+148)) then
        tmp = t_1
    else if (t1 <= (-5.2d+26)) then
        tmp = -t1 * (v / ((u + t1) * t1))
    else if (t1 <= 4.4d-57) then
        tmp = (-t1 * v) / (u * u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2e+148) {
		tmp = t_1;
	} else if (t1 <= -5.2e+26) {
		tmp = -t1 * (v / ((u + t1) * t1));
	} else if (t1 <= 4.4e-57) {
		tmp = (-t1 * v) / (u * u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -2e+148:
		tmp = t_1
	elif t1 <= -5.2e+26:
		tmp = -t1 * (v / ((u + t1) * t1))
	elif t1 <= 4.4e-57:
		tmp = (-t1 * v) / (u * u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -2e+148)
		tmp = t_1;
	elseif (t1 <= -5.2e+26)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * t1)));
	elseif (t1 <= 4.4e-57)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -2e+148)
		tmp = t_1;
	elseif (t1 <= -5.2e+26)
		tmp = -t1 * (v / ((u + t1) * t1));
	elseif (t1 <= 4.4e-57)
		tmp = (-t1 * v) / (u * u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2e+148], t$95$1, If[LessEqual[t1, -5.2e+26], N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.4e-57], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.0000000000000001e148 or 4.39999999999999997e-57 < t1

   1. Initial program 58.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{u + t1} \cdot \frac{v}{u + t1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{\color{blue}{u + t1}} \cdot \frac{v}{u + t1} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot \frac{v}{u + t1} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \color{blue}{\frac{v}{u + t1}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}}{u + t1} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot v}{u + t1} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{-t1}}{u + t1} \cdot v}{u + t1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-t1}{\color{blue}{u + t1}} \cdot v}{u + t1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{-t1}{u + t1} \cdot v}{\color{blue}{u + t1}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{u + t1} \cdot v}{u + t1}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{u + t1} \]

      2. lower-neg.f64 82.8

         \[\leadsto \frac{-v}{u + t1} \]

   8. Applied rewrites 82.8%

      \[\leadsto \frac{\color{blue}{-v}}{u + t1} \]

   if -2.0000000000000001e148 < t1 < -5.20000000000000004e26

   1. Initial program 80.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   2. Taylor expanded in u around 0

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 71.0%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot t1}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot t1} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot t1} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot t1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot t1}} \]

      6. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot t1} \]

      7. lower-/.f64 71.8

         \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot t1}} \]

      8. lift-+.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right)} \cdot t1} \]

      9. +-commutative N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot t1} \]

      10. lift-+.f64 71.8

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot t1} \]

      11. +-commutative 71.8

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot t1} \]

   6. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot t1}} \]

   if -5.20000000000000004e26 < t1 < 4.39999999999999997e-57

   1. Initial program 83.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   2. Taylor expanded in u around inf

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{{u}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\left(-t1\right) \cdot v}{u \cdot \color{blue}{u}} \]

      2. lower-*.f64 71.1

         \[\leadsto \frac{\left(-t1\right) \cdot v}{u \cdot \color{blue}{u}} \]

   4. Applied rewrites 71.1%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -8.6e-8)
   (* (- t1) (/ v (* u u)))
   (if (<= u 1.55e-6) (/ (- v) t1) (/ (* (- t1) v) (* (+ t1 u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.6e-8) {
		tmp = -t1 * (v / (u * u));
	} else if (u <= 1.55e-6) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 * v) / ((t1 + u) * u);
	}
	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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-8.6d-8)) then
        tmp = -t1 * (v / (u * u))
    else if (u <= 1.55d-6) then
        tmp = -v / t1
    else
        tmp = (-t1 * v) / ((t1 + u) * u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.6e-8) {
		tmp = -t1 * (v / (u * u));
	} else if (u <= 1.55e-6) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 * v) / ((t1 + u) * u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -8.6e-8:
		tmp = -t1 * (v / (u * u))
	elif u <= 1.55e-6:
		tmp = -v / t1
	else:
		tmp = (-t1 * v) / ((t1 + u) * u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -8.6e-8)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	elseif (u <= 1.55e-6)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -8.6e-8)
		tmp = -t1 * (v / (u * u));
	elseif (u <= 1.55e-6)
		tmp = -v / t1;
	else
		tmp = (-t1 * v) / ((t1 + u) * u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -8.6e-8], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.55e-6], N[((-v) / t1), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.6000000000000002e-8

   1. Initial program 77.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      12. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      15. lower-+.f64 78.3

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

   3. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{{u}^{2}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{{u}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{u \cdot \color{blue}{u}} \]

      3. lower-*.f64 70.4

         \[\leadsto \left(-t1\right) \cdot \frac{v}{u \cdot \color{blue}{u}} \]

   6. Applied rewrites 70.4%

      \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{u \cdot u}} \]

   if -8.6000000000000002e-8 < u < 1.55e-6

   1. Initial program 67.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   2. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot v}{\color{blue}{t1}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot v}{\color{blue}{t1}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{t1} \]

      4. lower-neg.f64 78.0

         \[\leadsto \frac{-v}{t1} \]

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 1.55e-6 < u

   1. Initial program 76.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   2. Taylor expanded in u around inf

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{u}} \]
   3. Step-by-step derivation

   4. Applied rewrites 70.1%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{u}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -1.9 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -1.9e+28)
     t_1
     (if (<= t1 4.4e-57) (/ (* (- t1) v) (* u u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -1.9e+28) {
		tmp = t_1;
	} else if (t1 <= 4.4e-57) {
		tmp = (-t1 * v) / (u * u);
	} else {
		tmp = t_1;
	}
	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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-1.9d+28)) then
        tmp = t_1
    else if (t1 <= 4.4d-57) then
        tmp = (-t1 * v) / (u * u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -1.9e+28) {
		tmp = t_1;
	} else if (t1 <= 4.4e-57) {
		tmp = (-t1 * v) / (u * u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -1.9e+28:
		tmp = t_1
	elif t1 <= 4.4e-57:
		tmp = (-t1 * v) / (u * u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -1.9e+28)
		tmp = t_1;
	elseif (t1 <= 4.4e-57)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -1.9e+28)
		tmp = t_1;
	elseif (t1 <= 4.4e-57)
		tmp = (-t1 * v) / (u * u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.9e+28], t$95$1, If[LessEqual[t1, 4.4e-57], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.8999999999999999e28 or 4.39999999999999997e-57 < t1

   1. Initial program 62.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

      11. +-commutative N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

      14. +-commutative N/A

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{u + t1} \cdot \frac{v}{u + t1} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{\color{blue}{u + t1}} \cdot \frac{v}{u + t1} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot \frac{v}{u + t1} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \color{blue}{\frac{v}{u + t1}} \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}}{u + t1} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot v}{u + t1} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{-t1}}{u + t1} \cdot v}{u + t1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-t1}{\color{blue}{u + t1}} \cdot v}{u + t1} \]

      13. lift-+.f64 99.9

          \[\leadsto \frac{\frac{-t1}{u + t1} \cdot v}{\color{blue}{u + t1}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{u + t1} \cdot v}{u + t1}} \]

   6. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{u + t1} \]

      2. lower-neg.f64 81.6

         \[\leadsto \frac{-v}{u + t1} \]

   8. Applied rewrites 81.6%

      \[\leadsto \frac{\color{blue}{-v}}{u + t1} \]

   if -1.8999999999999999e28 < t1 < 4.39999999999999997e-57

   1. Initial program 83.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   2. Taylor expanded in u around inf

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{{u}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\left(-t1\right) \cdot v}{u \cdot \color{blue}{u}} \]

      2. lower-*.f64 71.0

         \[\leadsto \frac{\left(-t1\right) \cdot v}{u \cdot \color{blue}{u}} \]

   4. Applied rewrites 71.0%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 8 \cdot 10^{-20}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) (/ v (* u u)))))
   (if (<= u -8.6e-8) t_1 (if (<= u 8e-20) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / (u * u));
	double tmp;
	if (u <= -8.6e-8) {
		tmp = t_1;
	} else if (u <= 8e-20) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t1 * (v / (u * u))
    if (u <= (-8.6d-8)) then
        tmp = t_1
    else if (u <= 8d-20) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / (u * u));
	double tmp;
	if (u <= -8.6e-8) {
		tmp = t_1;
	} else if (u <= 8e-20) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 * (v / (u * u))
	tmp = 0
	if u <= -8.6e-8:
		tmp = t_1
	elif u <= 8e-20:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * Float64(v / Float64(u * u)))
	tmp = 0.0
	if (u <= -8.6e-8)
		tmp = t_1;
	elseif (u <= 8e-20)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 * (v / (u * u));
	tmp = 0.0;
	if (u <= -8.6e-8)
		tmp = t_1;
	elseif (u <= 8e-20)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -8.6e-8], t$95$1, If[LessEqual[u, 8e-20], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -8.6000000000000002e-8 or 7.99999999999999956e-20 < u

   1. Initial program 77.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-t1\right)} \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      12. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{\left(u + t1\right)} \cdot \left(t1 + u\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

      15. lower-+.f64 78.1

          \[\leadsto \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \color{blue}{\left(u + t1\right)}} \]

   3. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{{u}^{2}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{\color{blue}{{u}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \left(-t1\right) \cdot \frac{v}{u \cdot \color{blue}{u}} \]

      3. lower-*.f64 69.9

         \[\leadsto \left(-t1\right) \cdot \frac{v}{u \cdot \color{blue}{u}} \]

   6. Applied rewrites 69.9%

      \[\leadsto \left(-t1\right) \cdot \color{blue}{\frac{v}{u \cdot u}} \]

   if -8.6000000000000002e-8 < u < 7.99999999999999956e-20

   1. Initial program 67.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   2. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot v}{\color{blue}{t1}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot v}{\color{blue}{t1}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{t1} \]

      4. lower-neg.f64 78.5

         \[\leadsto \frac{-v}{t1} \]

   4. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{-v}{u + t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (- v) (+ u t1)))

C

double code(double u, double v, double t1) {
	return -v / (u + t1);
}

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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = -v / (u + t1)
end function

Java

public static double code(double u, double v, double t1) {
	return -v / (u + t1);
}

Python

def code(u, v, t1):
	return -v / (u + t1)

Julia

function code(u, v, t1)
	return Float64(Float64(-v) / Float64(u + t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = -v / (u + t1);
end

Wolfram

code[u_, v_, t1_] := N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   7. times-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{t1 + u}} \cdot \frac{v}{t1 + u} \]

   10. lift-neg.f64 N/A

       \[\leadsto \frac{\color{blue}{-t1}}{t1 + u} \cdot \frac{v}{t1 + u} \]

   11. +-commutative N/A

       \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{-t1}{\color{blue}{u + t1}} \cdot \frac{v}{t1 + u} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{-t1}{u + t1} \cdot \color{blue}{\frac{v}{t1 + u}} \]

   14. +-commutative N/A

       \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   15. lower-+.f64 98.1

       \[\leadsto \frac{-t1}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

3. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1\right)}}{u + t1} \cdot \frac{v}{u + t1} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{\color{blue}{u + t1}} \cdot \frac{v}{u + t1} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot \frac{v}{u + t1} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \frac{v}{\color{blue}{u + t1}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot \color{blue}{\frac{v}{u + t1}} \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}{u + t1}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1} \cdot v}}{u + t1} \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(t1\right)}{u + t1}} \cdot v}{u + t1} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{-t1}}{u + t1} \cdot v}{u + t1} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{\frac{-t1}{\color{blue}{u + t1}} \cdot v}{u + t1} \]

   13. lift-+.f64 98.2

       \[\leadsto \frac{\frac{-t1}{u + t1} \cdot v}{\color{blue}{u + t1}} \]

5. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{\frac{-t1}{u + t1} \cdot v}{u + t1}} \]

6. Taylor expanded in u around 0

   \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{u + t1} \]

   2. lower-neg.f64 61.5

      \[\leadsto \frac{-v}{u + t1} \]

8. Applied rewrites 61.5%

   \[\leadsto \frac{\color{blue}{-v}}{u + t1} \]
9. Add Preprocessing

Alternative 13: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-1}{u}\\ \mathbf{if}\;u \leq -7.6 \cdot 10^{+238}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ -1.0 u))))
   (if (<= u -7.6e+238) t_1 (if (<= u 3.2e+151) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-1.0 / u);
	double tmp;
	if (u <= -7.6e+238) {
		tmp = t_1;
	} else if (u <= 3.2e+151) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v * ((-1.0d0) / u)
    if (u <= (-7.6d+238)) then
        tmp = t_1
    else if (u <= 3.2d+151) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-1.0 / u);
	double tmp;
	if (u <= -7.6e+238) {
		tmp = t_1;
	} else if (u <= 3.2e+151) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-1.0 / u)
	tmp = 0
	if u <= -7.6e+238:
		tmp = t_1
	elif u <= 3.2e+151:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(-1.0 / u))
	tmp = 0.0
	if (u <= -7.6e+238)
		tmp = t_1;
	elseif (u <= 3.2e+151)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-1.0 / u);
	tmp = 0.0;
	if (u <= -7.6e+238)
		tmp = t_1;
	elseif (u <= 3.2e+151)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -7.6e+238], t$95$1, If[LessEqual[u, 3.2e+151], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -7.60000000000000049e238 or 3.19999999999999994e151 < u

   1. Initial program 76.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right)} \cdot \left(t1 + u\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(t1 + u\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(t1\right)\right) \cdot v}{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(t1 \cdot v\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(t1 \cdot v\right) \cdot -1}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u} \cdot \frac{-1}{t1 + u}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 + u}} \cdot \frac{-1}{t1 + u} \]

      13. *-commutative N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 + u} \cdot \frac{-1}{t1 + u} \]

      15. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{v \cdot t1}{\color{blue}{u + t1}} \cdot \frac{-1}{t1 + u} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \color{blue}{\frac{-1}{t1 + u}} \]

      18. +-commutative N/A

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

      19. lower-+.f64 87.4

          \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u + t1}} \]

   3. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\frac{v \cdot t1}{u + t1} \cdot \frac{-1}{u + t1}} \]

   4. Taylor expanded in u around inf

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]
   5. Step-by-step derivation

   6. Applied rewrites 85.0%

      \[\leadsto \frac{v \cdot t1}{u + t1} \cdot \frac{-1}{\color{blue}{u}} \]

   7. Taylor expanded in u around 0

      \[\leadsto \color{blue}{v} \cdot \frac{-1}{u} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.6%

      \[\leadsto \color{blue}{v} \cdot \frac{-1}{u} \]

   if -7.60000000000000049e238 < u < 3.19999999999999994e151

   1. Initial program 71.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   2. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot v}{\color{blue}{t1}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot v}{\color{blue}{t1}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{t1} \]

      4. lower-neg.f64 61.4

         \[\leadsto \frac{-v}{t1} \]

   4. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{-v}{t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (- v) t1))

C

double code(double u, double v, double t1) {
	return -v / t1;
}

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(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = -v / t1
end function

Java

public static double code(double u, double v, double t1) {
	return -v / t1;
}

Python

def code(u, v, t1):
	return -v / t1

Julia

function code(u, v, t1)
	return Float64(Float64(-v) / t1)
end

MATLAB

function tmp = code(u, v, t1)
	tmp = -v / t1;
end

Wolfram

code[u_, v_, t1_] := N[((-v) / t1), $MachinePrecision]

Derivation

1. Initial program 72.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot v}{\color{blue}{t1}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{-1 \cdot v}{\color{blue}{t1}} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{t1} \]

   4. lower-neg.f64 53.7

      \[\leadsto \frac{-v}{t1} \]

4. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\frac{-v}{t1}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))