Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.0%

Time: 2.5s

Alternatives: 5

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: 72.4% 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: 98.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.4%

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

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

3. Applied rewrites 98.0%

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

Alternative 2: 98.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.4%

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

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

3. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{-1 \cdot t1}{u + t1} \cdot \frac{v}{u + t1}} \]
4. 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. frac-times N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. frac-times N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. *-commutative N/A

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

   19. +-commutative N/A

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

   20. associate-/l* N/A

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

   21. +-commutative N/A

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

   22. lower-*.f64 N/A

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

   23. lift-/.f64 N/A

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

   24. lift-+.f64 98.0

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

5. Applied rewrites 98.0%

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

Alternative 3: 85.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(u \cdot \frac{v}{t1}, 2, -1 \cdot v\right)}{t1}\\ \mathbf{if}\;t1 \leq -5.8 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+132}:\\ \;\;\;\;\left(-1 \cdot t1\right) \cdot \frac{v}{{\left(u + t1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (fma (* u (/ v t1)) 2.0 (* -1.0 v)) t1)))
   (if (<= t1 -5.8e+138)
     t_1
     (if (<= t1 6e+132) (* (* -1.0 t1) (/ v (pow (+ u t1) 2.0))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = fma((u * (v / t1)), 2.0, (-1.0 * v)) / t1;
	double tmp;
	if (t1 <= -5.8e+138) {
		tmp = t_1;
	} else if (t1 <= 6e+132) {
		tmp = (-1.0 * t1) * (v / pow((u + t1), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(fma(Float64(u * Float64(v / t1)), 2.0, Float64(-1.0 * v)) / t1)
	tmp = 0.0
	if (t1 <= -5.8e+138)
		tmp = t_1;
	elseif (t1 <= 6e+132)
		tmp = Float64(Float64(-1.0 * t1) * Float64(v / (Float64(u + t1) ^ 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(-1.0 * v), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision]}, If[LessEqual[t1, -5.8e+138], t$95$1, If[LessEqual[t1, 6e+132], N[(N[(-1.0 * t1), $MachinePrecision] * N[(v / N[Power[N[(u + t1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.80000000000000019e138 or 5.9999999999999996e132 < t1

   1. Initial program 43.0%

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

   2. Taylor expanded in t1 around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -5.80000000000000019e138 < t1 < 5.9999999999999996e132

   1. Initial program 83.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 84.1

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

   3. Applied rewrites 84.1%

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

Alternative 4: 85.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(u \cdot \frac{v}{t1}, 2, -1 \cdot v\right)}{t1}\\ \mathbf{if}\;t1 \leq -5.8 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+132}:\\ \;\;\;\;\left(-1 \cdot t1\right) \cdot \frac{v}{\frac{1}{{\left(u + t1\right)}^{-2}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (fma (* u (/ v t1)) 2.0 (* -1.0 v)) t1)))
   (if (<= t1 -5.8e+138)
     t_1
     (if (<= t1 6e+132)
       (* (* -1.0 t1) (/ v (/ 1.0 (pow (+ u t1) -2.0))))
       t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = fma((u * (v / t1)), 2.0, (-1.0 * v)) / t1;
	double tmp;
	if (t1 <= -5.8e+138) {
		tmp = t_1;
	} else if (t1 <= 6e+132) {
		tmp = (-1.0 * t1) * (v / (1.0 / pow((u + t1), -2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(fma(Float64(u * Float64(v / t1)), 2.0, Float64(-1.0 * v)) / t1)
	tmp = 0.0
	if (t1 <= -5.8e+138)
		tmp = t_1;
	elseif (t1 <= 6e+132)
		tmp = Float64(Float64(-1.0 * t1) * Float64(v / Float64(1.0 / (Float64(u + t1) ^ -2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(-1.0 * v), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision]}, If[LessEqual[t1, -5.8e+138], t$95$1, If[LessEqual[t1, 6e+132], N[(N[(-1.0 * t1), $MachinePrecision] * N[(v / N[(1.0 / N[Power[N[(u + t1), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.80000000000000019e138 or 5.9999999999999996e132 < t1

   1. Initial program 43.0%

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

   2. Taylor expanded in t1 around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -5.80000000000000019e138 < t1 < 5.9999999999999996e132

   1. Initial program 83.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 84.1

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

   3. Applied rewrites 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

Alternative 5: 51.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.4%

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

2. Taylor expanded in t1 around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 51.8

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

4. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))