Rosa's DopplerBench

Percentage Accurate: 73.1% → 96.6%

Time: 6.5s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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: 96.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

4. Applied rewrites 97.5%

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

5. Final simplification 97.5%

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

Alternative 2: 91.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq 4 \cdot 10^{+281}:\\ \;\;\;\;\frac{\frac{v \cdot t1}{t1 - u}}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) 4e+281)
   (/ (/ (* v t1) (- t1 u)) (- u t1))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (((-t1 * v) / ((t1 + u) * (t1 + u))) <= 4e+281) {
		tmp = ((v * t1) / (t1 - u)) / (u - t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (((-t1 * v) / ((t1 + u) * (t1 + u))) <= 4d+281) then
        tmp = ((v * t1) / (t1 - u)) / (u - t1)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (((-t1 * v) / ((t1 + u) * (t1 + u))) <= 4e+281) {
		tmp = ((v * t1) / (t1 - u)) / (u - t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if ((-t1 * v) / ((t1 + u) * (t1 + u))) <= 4e+281:
		tmp = ((v * t1) / (t1 - u)) / (u - t1)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u))) <= 4e+281)
		tmp = Float64(Float64(Float64(v * t1) / Float64(t1 - u)) / Float64(u - t1));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (((-t1 * v) / ((t1 + u) * (t1 + u))) <= 4e+281)
		tmp = ((v * t1) / (t1 - u)) / (u - t1);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+281], N[(N[(N[(v * t1), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u))) < 4.0000000000000001e281

   1. Initial program 86.3%

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. flip-- N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-frac2 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. +-commutative N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. sqr-neg-rev N/A

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

      20. distribute-rgt-neg-out N/A

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

      21. distribute-neg-in N/A

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

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

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

   6. Applied rewrites 97.7%

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

   if 4.0000000000000001e281 < (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u)))

   1. Initial program 9.8%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.9

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

   5. Applied rewrites 76.9%

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

Alternative 3: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1e-41) (not (<= t1 9e-5)))
   (/ (* -1.0 v) (+ (- u) t1))
   (/ (* (/ (- v) u) t1) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((-v / u) * t1) / u;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1d-41)) .or. (.not. (t1 <= 9d-5))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = ((-v / u) * t1) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((-v / u) * t1) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1e-41) or not (t1 <= 9e-5):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = ((-v / u) * t1) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1e-41) || !(t1 <= 9e-5))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1e-41) || ~((t1 <= 9e-5)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = ((-v / u) * t1) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1e-41], N[Not[LessEqual[t1, 9e-5]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.00000000000000001e-41 or 9.00000000000000057e-5 < t1

   1. Initial program 59.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 81.8%

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

   if -1.00000000000000001e-41 < t1 < 9.00000000000000057e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 85.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1e-41) (not (<= t1 9e-5)))
   (/ (* -1.0 v) (+ (- u) t1))
   (/ (* (/ t1 u) v) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((t1 / u) * v) / -u;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1d-41)) .or. (.not. (t1 <= 9d-5))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = ((t1 / u) * v) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((t1 / u) * v) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1e-41) or not (t1 <= 9e-5):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = ((t1 / u) * v) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1e-41) || !(t1 <= 9e-5))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(t1 / u) * v) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1e-41) || ~((t1 <= 9e-5)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = ((t1 / u) * v) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1e-41], N[Not[LessEqual[t1, 9e-5]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * v), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.00000000000000001e-41 or 9.00000000000000057e-5 < t1

   1. Initial program 59.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 81.8%

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

   if -1.00000000000000001e-41 < t1 < 9.00000000000000057e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 85.6%

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

   9. Applied rewrites 85.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.12e-41) (not (<= t1 9e-5)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.12d-41)) .or. (.not. (t1 <= 9d-5))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = (v / u) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.12e-41) or not (t1 <= 9e-5):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.12e-41) || ~((t1 <= 9e-5)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.12e-41], N[Not[LessEqual[t1, 9e-5]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.11999999999999999e-41 or 9.00000000000000057e-5 < t1

   1. Initial program 59.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 81.8%

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

   if -1.11999999999999999e-41 < t1 < 9.00000000000000057e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.2e+90)
   (/ (* -1.0 v) (+ (- u) t1))
   (if (<= t1 1.2e+117) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.2e+90) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.2e+117) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-4.2d+90)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= 1.2d+117) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.2e+90) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.2e+117) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4.2e+90:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= 1.2e+117:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.2e+90)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= 1.2e+117)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4.2e+90)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= 1.2e+117)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.2e+90], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.2e+117], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.19999999999999961e90

   1. Initial program 58.0%

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 87.3%

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

   if -4.19999999999999961e90 < t1 < 1.1999999999999999e117

   1. Initial program 88.5%

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

   if 1.1999999999999999e117 < t1

   1. Initial program 29.8%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1e-41) (not (<= t1 9e-5)))
   (/ (* -1.0 v) (+ (- u) t1))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1d-41)) .or. (.not. (t1 <= 9d-5))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = (-t1 * v) / (u * u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1e-41) or not (t1 <= 9e-5):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (-t1 * v) / (u * u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1e-41) || !(t1 <= 9e-5))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1e-41) || ~((t1 <= 9e-5)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (-t1 * v) / (u * u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1e-41], N[Not[LessEqual[t1, 9e-5]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.00000000000000001e-41 or 9.00000000000000057e-5 < t1

   1. Initial program 59.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 81.8%

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

   if -1.00000000000000001e-41 < t1 < 9.00000000000000057e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.12e-41) (not (<= t1 9e-5)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (/ (- t1) (* u u)) v)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-t1 / (u * u)) * v;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.12d-41)) .or. (.not. (t1 <= 9d-5))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = (-t1 / (u * u)) * v
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-t1 / (u * u)) * v;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.12e-41) or not (t1 <= 9e-5):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (-t1 / (u * u)) * v
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(u * u)) * v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.12e-41) || ~((t1 <= 9e-5)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (-t1 / (u * u)) * v;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.12e-41], N[Not[LessEqual[t1, 9e-5]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.11999999999999999e-41 or 9.00000000000000057e-5 < t1

   1. Initial program 59.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 81.8%

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

   if -1.11999999999999999e-41 < t1 < 9.00000000000000057e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 76.4

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

   7. Applied rewrites 76.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.12e-41) (not (<= t1 9e-5)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (/ (- v) (* u u)) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-v / (u * u)) * t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.12d-41)) .or. (.not. (t1 <= 9d-5))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = (-v / (u * u)) * t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-v / (u * u)) * t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.12e-41) or not (t1 <= 9e-5):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (-v / (u * u)) * t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.12e-41) || !(t1 <= 9e-5))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.12e-41) || ~((t1 <= 9e-5)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (-v / (u * u)) * t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.12e-41], N[Not[LessEqual[t1, 9e-5]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.11999999999999999e-41 or 9.00000000000000057e-5 < t1

   1. Initial program 59.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 81.8%

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

   if -1.11999999999999999e-41 < t1 < 9.00000000000000057e-5

   1. Initial program 87.4%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.6

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

   7. Applied rewrites 75.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-41} \lor \neg \left(t1 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 59.0%

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

8. Final simplification 59.0%

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

Alternative 11: 61.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 59.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 58.9

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

9. Applied rewrites 58.9%

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

Alternative 12: 54.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 50.8

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

5. Applied rewrites 50.8%

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

Reproduce

herbie shell --seed 2024364 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))