Rosa's DopplerBench

Percentage Accurate: 73.3% → 98.4%

Time: 3.5s

Alternatives: 8

Speedup: N/A×

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: 73.3% 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: 98.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{t1}{u + t1} \cdot v}{-1 \cdot \left(u + t1\right)} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (/ (* (/ t1 (+ u t1)) v) (* -1.0 (+ u t1))))

C

double code(double u, double v, double t1) {
	return ((t1 / (u + t1)) * v) / (-1.0 * (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) / ((-1.0d0) * (u + t1))
end function

Java

public static double code(double u, double v, double t1) {
	return ((t1 / (u + t1)) * v) / (-1.0 * (u + t1));
}

Python

def code(u, v, t1):
	return ((t1 / (u + t1)) * v) / (-1.0 * (u + t1))

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(t1 / Float64(u + t1)) * v) / Float64(-1.0 * Float64(u + t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = ((t1 / (u + t1)) * v) / (-1.0 * (u + t1));
end

Wolfram

code[u_, v_, t1_] := N[(N[(N[(t1 / N[(u + t1), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision] / N[(-1.0 * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing
3. 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. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 98.3

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

6. Applied rewrites 98.3%

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

7. Final simplification 98.3%

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

Alternative 2: 98.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1 \cdot t1}{\left(\frac{u}{t1} + 1\right) \cdot t1} \cdot v}{u + t1} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (/ (* (/ (* -1.0 t1) (* (+ (/ u t1) 1.0) t1)) v) (+ u t1)))

C

double code(double u, double v, double t1) {
	return (((-1.0 * t1) / (((u / t1) + 1.0) * 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 = ((((-1.0d0) * t1) / (((u / t1) + 1.0d0) * t1)) * v) / (u + t1)
end function

Java

public static double code(double u, double v, double t1) {
	return (((-1.0 * t1) / (((u / t1) + 1.0) * t1)) * v) / (u + t1);
}

Python

def code(u, v, t1):
	return (((-1.0 * t1) / (((u / t1) + 1.0) * t1)) * v) / (u + t1)

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(Float64(-1.0 * t1) / Float64(Float64(Float64(u / t1) + 1.0) * t1)) * v) / Float64(u + t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (((-1.0 * t1) / (((u / t1) + 1.0) * t1)) * v) / (u + t1);
end

Wolfram

code[u_, v_, t1_] := N[(N[(N[(N[(-1.0 * t1), $MachinePrecision] / N[(N[(N[(u / t1), $MachinePrecision] + 1.0), $MachinePrecision] * t1), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing
3. 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. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. lift-+.f64 98.3

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in t1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 98.3

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

9. Applied rewrites 98.3%

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

10. Final simplification 98.3%

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

Alternative 3: 98.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{t1}{u + t1} \cdot \frac{-1 \cdot v}{\left(\frac{u}{t1} + 1\right) \cdot t1} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (* (/ t1 (+ u t1)) (/ (* -1.0 v) (* (+ (/ u t1) 1.0) t1))))

C

double code(double u, double v, double t1) {
	return (t1 / (u + t1)) * ((-1.0 * v) / (((u / t1) + 1.0) * 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)) * (((-1.0d0) * v) / (((u / t1) + 1.0d0) * t1))
end function

Java

public static double code(double u, double v, double t1) {
	return (t1 / (u + t1)) * ((-1.0 * v) / (((u / t1) + 1.0) * t1));
}

Python

def code(u, v, t1):
	return (t1 / (u + t1)) * ((-1.0 * v) / (((u / t1) + 1.0) * t1))

Julia

function code(u, v, t1)
	return Float64(Float64(t1 / Float64(u + t1)) * Float64(Float64(-1.0 * v) / Float64(Float64(Float64(u / t1) + 1.0) * t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (t1 / (u + t1)) * ((-1.0 * v) / (((u / t1) + 1.0) * t1));
end

Wolfram

code[u_, v_, t1_] := N[(N[(t1 / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 * v), $MachinePrecision] / N[(N[(N[(u / t1), $MachinePrecision] + 1.0), $MachinePrecision] * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing
3. 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. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in t1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 98.0

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

7. Applied rewrites 98.0%

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

8. Final simplification 98.0%

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

Alternative 4: 91.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq 4.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{t1}{u + t1} \cdot \frac{-1 \cdot v}{\left(\frac{t1}{u} + 1\right) \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 4.2e+194)
   (* (/ t1 (+ u t1)) (/ (* -1.0 v) (* (+ (/ t1 u) 1.0) u)))
   (fma (/ 2.0 t1) (* u (/ v t1)) (* (/ v t1) -1.0))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= 4.2e+194) {
		tmp = (t1 / (u + t1)) * ((-1.0 * v) / (((t1 / u) + 1.0) * u));
	} else {
		tmp = fma((2.0 / t1), (u * (v / t1)), ((v / t1) * -1.0));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= 4.2e+194)
		tmp = Float64(Float64(t1 / Float64(u + t1)) * Float64(Float64(-1.0 * v) / Float64(Float64(Float64(t1 / u) + 1.0) * u)));
	else
		tmp = fma(Float64(2.0 / t1), Float64(u * Float64(v / t1)), Float64(Float64(v / t1) * -1.0));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, 4.2e+194], N[(N[(t1 / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 * v), $MachinePrecision] / N[(N[(N[(t1 / u), $MachinePrecision] + 1.0), $MachinePrecision] * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t1), $MachinePrecision] * N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] + N[(N[(v / t1), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < 4.20000000000000032e194

   1. Initial program 75.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 93.1

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

   7. Applied rewrites 93.1%

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

   if 4.20000000000000032e194 < t1

   1. Initial program 38.9%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \frac{u \cdot v}{{t1}^{2}} + \color{blue}{-1 \cdot \frac{v}{t1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{{t1}^{2}} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{t1 \cdot t1} + -1 \cdot \frac{v}{t1} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{t1} \cdot \frac{u \cdot v}{t1} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \color{blue}{\frac{u \cdot v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \frac{\color{blue}{u \cdot v}}{t1}, -1 \cdot \frac{v}{t1}\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{\color{blue}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      12. lower-/.f64 96.3

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq 4.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{t1}{u + t1} \cdot \frac{-1 \cdot v}{\left(\frac{t1}{u} + 1\right) \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.2 \cdot 10^{+138}:\\ \;\;\;\;\left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v\\ \mathbf{elif}\;t1 \leq 7.5 \cdot 10^{+113}:\\ \;\;\;\;\left(-1 \cdot t1\right) \cdot \frac{v}{{\left(u + t1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -7.2e+138)
   (* (- (* (/ u (* t1 t1)) 2.0) (pow t1 -1.0)) v)
   (if (<= t1 7.5e+113)
     (* (* -1.0 t1) (/ v (pow (+ u t1) 2.0)))
     (fma (/ 2.0 t1) (* u (/ v t1)) (* (/ v t1) -1.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -7.2e+138) {
		tmp = (((u / (t1 * t1)) * 2.0) - pow(t1, -1.0)) * v;
	} else if (t1 <= 7.5e+113) {
		tmp = (-1.0 * t1) * (v / pow((u + t1), 2.0));
	} else {
		tmp = fma((2.0 / t1), (u * (v / t1)), ((v / t1) * -1.0));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -7.2e+138)
		tmp = Float64(Float64(Float64(Float64(u / Float64(t1 * t1)) * 2.0) - (t1 ^ -1.0)) * v);
	elseif (t1 <= 7.5e+113)
		tmp = Float64(Float64(-1.0 * t1) * Float64(v / (Float64(u + t1) ^ 2.0)));
	else
		tmp = fma(Float64(2.0 / t1), Float64(u * Float64(v / t1)), Float64(Float64(v / t1) * -1.0));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -7.2e+138], N[(N[(N[(N[(u / N[(t1 * t1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - N[Power[t1, -1.0], $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], If[LessEqual[t1, 7.5e+113], N[(N[(-1.0 * t1), $MachinePrecision] * N[(v / N[Power[N[(u + t1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t1), $MachinePrecision] * N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] + N[(N[(v / t1), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -7.2000000000000002e138

   1. Initial program 47.8%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \frac{u \cdot v}{{t1}^{2}} + \color{blue}{-1 \cdot \frac{v}{t1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{{t1}^{2}} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{t1 \cdot t1} + -1 \cdot \frac{v}{t1} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{t1} \cdot \frac{u \cdot v}{t1} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \color{blue}{\frac{u \cdot v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \frac{\color{blue}{u \cdot v}}{t1}, -1 \cdot \frac{v}{t1}\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{\color{blue}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      12. lower-/.f64 81.3

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)} \]

   6. Taylor expanded in v around 0

      \[\leadsto v \cdot \color{blue}{\left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

      3. lower--.f64 N/A

         \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      9. inv-pow N/A

         \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v \]

      10. lower-pow.f64 83.2

          \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v \]

   8. Applied rewrites 83.2%

      \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot \color{blue}{v} \]

   if -7.2000000000000002e138 < t1 < 7.5000000000000001e113

   1. Initial program 82.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. mul-1-neg N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

          \[\leadsto \left(-1 \cdot t1\right) \cdot \frac{v}{\color{blue}{{\left(t1 + u\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \left(-1 \cdot t1\right) \cdot \frac{v}{\color{blue}{{\left(t1 + u\right)}^{2}}} \]

      14. +-commutative N/A

          \[\leadsto \left(-1 \cdot t1\right) \cdot \frac{v}{{\color{blue}{\left(u + t1\right)}}^{2}} \]

      15. lower-+.f64 89.3

          \[\leadsto \left(-1 \cdot t1\right) \cdot \frac{v}{{\color{blue}{\left(u + t1\right)}}^{2}} \]

   4. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\left(-1 \cdot t1\right) \cdot \frac{v}{{\left(u + t1\right)}^{2}}} \]

   if 7.5000000000000001e113 < t1

   1. Initial program 48.3%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \frac{u \cdot v}{{t1}^{2}} + \color{blue}{-1 \cdot \frac{v}{t1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{{t1}^{2}} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{t1 \cdot t1} + -1 \cdot \frac{v}{t1} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{t1} \cdot \frac{u \cdot v}{t1} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \color{blue}{\frac{u \cdot v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \frac{\color{blue}{u \cdot v}}{t1}, -1 \cdot \frac{v}{t1}\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{\color{blue}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      12. lower-/.f64 87.0

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   5. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v\\ \mathbf{elif}\;t1 \leq 1.32 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(\frac{t1}{u \cdot u} - {u}^{-1}\right) \cdot t1\right) \cdot \frac{v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.9e-15)
   (* (- (* (/ u (* t1 t1)) 2.0) (pow t1 -1.0)) v)
   (if (<= t1 1.32e+16)
     (* (* (- (/ t1 (* u u)) (pow u -1.0)) t1) (/ v (+ u t1)))
     (fma (/ 2.0 t1) (* u (/ v t1)) (* (/ v t1) -1.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.9e-15) {
		tmp = (((u / (t1 * t1)) * 2.0) - pow(t1, -1.0)) * v;
	} else if (t1 <= 1.32e+16) {
		tmp = (((t1 / (u * u)) - pow(u, -1.0)) * t1) * (v / (u + t1));
	} else {
		tmp = fma((2.0 / t1), (u * (v / t1)), ((v / t1) * -1.0));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.9e-15)
		tmp = Float64(Float64(Float64(Float64(u / Float64(t1 * t1)) * 2.0) - (t1 ^ -1.0)) * v);
	elseif (t1 <= 1.32e+16)
		tmp = Float64(Float64(Float64(Float64(t1 / Float64(u * u)) - (u ^ -1.0)) * t1) * Float64(v / Float64(u + t1)));
	else
		tmp = fma(Float64(2.0 / t1), Float64(u * Float64(v / t1)), Float64(Float64(v / t1) * -1.0));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.9e-15], N[(N[(N[(N[(u / N[(t1 * t1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - N[Power[t1, -1.0], $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], If[LessEqual[t1, 1.32e+16], N[(N[(N[(N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision] - N[Power[u, -1.0], $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t1), $MachinePrecision] * N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] + N[(N[(v / t1), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.9000000000000001e-15

   1. Initial program 58.2%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \frac{u \cdot v}{{t1}^{2}} + \color{blue}{-1 \cdot \frac{v}{t1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{{t1}^{2}} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{t1 \cdot t1} + -1 \cdot \frac{v}{t1} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{t1} \cdot \frac{u \cdot v}{t1} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \color{blue}{\frac{u \cdot v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \frac{\color{blue}{u \cdot v}}{t1}, -1 \cdot \frac{v}{t1}\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{\color{blue}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      12. lower-/.f64 80.0

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)} \]

   6. Taylor expanded in v around 0

      \[\leadsto v \cdot \color{blue}{\left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

      3. lower--.f64 N/A

         \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

      9. inv-pow N/A

         \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v \]

      10. lower-pow.f64 80.9

          \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v \]

   8. Applied rewrites 80.9%

      \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot \color{blue}{v} \]

   if -1.9000000000000001e-15 < t1 < 1.32e16

   1. Initial program 84.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{\left(t1 \cdot \left(\frac{t1}{{u}^{2}} - \frac{1}{u}\right)\right)} \cdot \frac{v}{u + t1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{t1}{{u}^{2}} - \frac{1}{u}\right) \cdot \color{blue}{t1}\right) \cdot \frac{v}{u + t1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{t1}{{u}^{2}} - \frac{1}{u}\right) \cdot \color{blue}{t1}\right) \cdot \frac{v}{u + t1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(\frac{t1}{{u}^{2}} - \frac{1}{u}\right) \cdot t1\right) \cdot \frac{v}{u + t1} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{t1}{{u}^{2}} - \frac{1}{u}\right) \cdot t1\right) \cdot \frac{v}{u + t1} \]

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 78.5

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

   7. Applied rewrites 78.5%

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

   if 1.32e16 < t1

   1. Initial program 56.4%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 2 \cdot \frac{u \cdot v}{{t1}^{2}} + \color{blue}{-1 \cdot \frac{v}{t1}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{{t1}^{2}} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{t1 \cdot t1} + -1 \cdot \frac{v}{t1} \]

      4. times-frac N/A

         \[\leadsto \frac{2}{t1} \cdot \frac{u \cdot v}{t1} + \color{blue}{-1} \cdot \frac{v}{t1} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \color{blue}{\frac{u \cdot v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \frac{\color{blue}{u \cdot v}}{t1}, -1 \cdot \frac{v}{t1}\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{\color{blue}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

      12. lower-/.f64 83.4

          \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (fma (/ 2.0 t1) (* u (/ v t1)) (* (/ v t1) -1.0)))

C

double code(double u, double v, double t1) {
	return fma((2.0 / t1), (u * (v / t1)), ((v / t1) * -1.0));
}

Julia

function code(u, v, t1)
	return fma(Float64(2.0 / t1), Float64(u * Float64(v / t1)), Float64(Float64(v / t1) * -1.0))
end

Wolfram

code[u_, v_, t1_] := N[(N[(2.0 / t1), $MachinePrecision] * N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] + N[(N[(v / t1), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot \frac{u \cdot v}{{t1}^{2}} + \color{blue}{-1 \cdot \frac{v}{t1}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{{t1}^{2}} + \color{blue}{-1} \cdot \frac{v}{t1} \]

   3. unpow2 N/A

      \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{t1 \cdot t1} + -1 \cdot \frac{v}{t1} \]

   4. times-frac N/A

      \[\leadsto \frac{2}{t1} \cdot \frac{u \cdot v}{t1} + \color{blue}{-1} \cdot \frac{v}{t1} \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \color{blue}{\frac{u \cdot v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \frac{\color{blue}{u \cdot v}}{t1}, -1 \cdot \frac{v}{t1}\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{\color{blue}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   12. lower-/.f64 47.1

       \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

5. Applied rewrites 47.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)} \]
6. Add Preprocessing

Alternative 8: 50.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (* (- (* (/ u (* t1 t1)) 2.0) (pow t1 -1.0)) v))

C

double code(double u, double v, double t1) {
	return (((u / (t1 * t1)) * 2.0) - pow(t1, -1.0)) * v;
}

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 = (((u / (t1 * t1)) * 2.0d0) - (t1 ** (-1.0d0))) * v
end function

Java

public static double code(double u, double v, double t1) {
	return (((u / (t1 * t1)) * 2.0) - Math.pow(t1, -1.0)) * v;
}

Python

def code(u, v, t1):
	return (((u / (t1 * t1)) * 2.0) - math.pow(t1, -1.0)) * v

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(Float64(u / Float64(t1 * t1)) * 2.0) - (t1 ^ -1.0)) * v)
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (((u / (t1 * t1)) * 2.0) - (t1 ^ -1.0)) * v;
end

Wolfram

code[u_, v_, t1_] := N[(N[(N[(N[(u / N[(t1 * t1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] - N[Power[t1, -1.0], $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot \frac{u \cdot v}{{t1}^{2}} + \color{blue}{-1 \cdot \frac{v}{t1}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{{t1}^{2}} + \color{blue}{-1} \cdot \frac{v}{t1} \]

   3. unpow2 N/A

      \[\leadsto \frac{2 \cdot \left(u \cdot v\right)}{t1 \cdot t1} + -1 \cdot \frac{v}{t1} \]

   4. times-frac N/A

      \[\leadsto \frac{2}{t1} \cdot \frac{u \cdot v}{t1} + \color{blue}{-1} \cdot \frac{v}{t1} \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \color{blue}{\frac{u \cdot v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, \frac{\color{blue}{u \cdot v}}{t1}, -1 \cdot \frac{v}{t1}\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \color{blue}{\frac{v}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{\color{blue}{t1}}, -1 \cdot \frac{v}{t1}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

   12. lower-/.f64 47.1

       \[\leadsto \mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right) \]

5. Applied rewrites 47.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{t1}, u \cdot \frac{v}{t1}, \frac{v}{t1} \cdot -1\right)} \]

6. Taylor expanded in v around 0

   \[\leadsto v \cdot \color{blue}{\left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

   2. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

   3. lower--.f64 N/A

      \[\leadsto \left(2 \cdot \frac{u}{{t1}^{2}} - \frac{1}{t1}\right) \cdot v \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\frac{u}{{t1}^{2}} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - \frac{1}{t1}\right) \cdot v \]

   9. inv-pow N/A

      \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v \]

   10. lower-pow.f64 46.3

       \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot v \]

8. Applied rewrites 46.3%

   \[\leadsto \left(\frac{u}{t1 \cdot t1} \cdot 2 - {t1}^{-1}\right) \cdot \color{blue}{v} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))