Rosa's DopplerBench

Percentage Accurate: 72.7% → 97.8%

Time: 4.7s

Alternatives: 10

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

Alternative 2: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -3.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -3.5e+112)
     (/ (- v) (+ u t1))
     (if (<= t1 -1.15e-116)
       t_1
       (if (<= t1 2.5e-77)
         (/ (* (/ v u) t1) (- u))
         (if (<= t1 8.2e+91) t_1 (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -3.5e+112) {
		tmp = -v / (u + t1);
	} else if (t1 <= -1.15e-116) {
		tmp = t_1;
	} else if (t1 <= 2.5e-77) {
		tmp = ((v / u) * t1) / -u;
	} else if (t1 <= 8.2e+91) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
    if (t1 <= (-3.5d+112)) then
        tmp = -v / (u + t1)
    else if (t1 <= (-1.15d-116)) then
        tmp = t_1
    else if (t1 <= 2.5d-77) then
        tmp = ((v / u) * t1) / -u
    else if (t1 <= 8.2d+91) then
        tmp = t_1
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -3.5e+112) {
		tmp = -v / (u + t1);
	} else if (t1 <= -1.15e-116) {
		tmp = t_1;
	} else if (t1 <= 2.5e-77) {
		tmp = ((v / u) * t1) / -u;
	} else if (t1 <= 8.2e+91) {
		tmp = t_1;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
	tmp = 0
	if t1 <= -3.5e+112:
		tmp = -v / (u + t1)
	elif t1 <= -1.15e-116:
		tmp = t_1
	elif t1 <= 2.5e-77:
		tmp = ((v / u) * t1) / -u
	elif t1 <= 8.2e+91:
		tmp = t_1
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -3.5e+112)
		tmp = Float64(Float64(-v) / Float64(u + t1));
	elseif (t1 <= -1.15e-116)
		tmp = t_1;
	elseif (t1 <= 2.5e-77)
		tmp = Float64(Float64(Float64(v / u) * t1) / Float64(-u));
	elseif (t1 <= 8.2e+91)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	tmp = 0.0;
	if (t1 <= -3.5e+112)
		tmp = -v / (u + t1);
	elseif (t1 <= -1.15e-116)
		tmp = t_1;
	elseif (t1 <= 2.5e-77)
		tmp = ((v / u) * t1) / -u;
	elseif (t1 <= 8.2e+91)
		tmp = t_1;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.5e+112], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.15e-116], t$95$1, If[LessEqual[t1, 2.5e-77], N[(N[(N[(v / u), $MachinePrecision] * t1), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[t1, 8.2e+91], t$95$1, N[((-v) / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -3.49999999999999997e112

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.2

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

   7. Applied rewrites 84.2%

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

   if -3.49999999999999997e112 < t1 < -1.15000000000000001e-116 or 2.49999999999999982e-77 < t1 < 8.2000000000000005e91

   1. Initial program 93.1%

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

   if -1.15000000000000001e-116 < t1 < 2.49999999999999982e-77

   1. Initial program 82.2%

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

   3. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 89.5

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

   5. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 92.3

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

   7. Applied rewrites 92.3%

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

   if 8.2000000000000005e91 < t1

   1. Initial program 34.9%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{elif}\;t1 \leq 8.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.9e-90) (not (<= t1 1.3e+37)))
   (/ (- v) (+ u t1))
   (/ (* (/ v u) t1) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -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 <= (-2.9d-90)) .or. (.not. (t1 <= 1.3d+37))) then
        tmp = -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 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -v / (u + t1);
	} else {
		tmp = ((v / u) * t1) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.9e-90) or not (t1 <= 1.3e+37):
		tmp = -v / (u + t1)
	else:
		tmp = ((v / u) * t1) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(v / u) * t1) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.9e-90) || ~((t1 <= 1.3e+37)))
		tmp = -v / (u + t1);
	else
		tmp = ((v / u) * t1) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.9e-90], N[Not[LessEqual[t1, 1.3e+37]], $MachinePrecision]], N[((-v) / 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 < -2.89999999999999983e-90 or 1.3e37 < t1

   1. Initial program 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -2.89999999999999983e-90 < t1 < 1.3e37

   1. Initial program 86.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 88.0

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

   5. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-neg.f64 90.0

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

   7. Applied rewrites 90.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.9e-90) (not (<= t1 1.3e+37)))
   (/ (- v) (+ u t1))
   (/ (* v (/ (- t1) u)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (v * (-t1 / u)) / u;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.9d-90)) .or. (.not. (t1 <= 1.3d+37))) then
        tmp = -v / (u + t1)
    else
        tmp = (v * (-t1 / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (v * (-t1 / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.9e-90) or not (t1 <= 1.3e+37):
		tmp = -v / (u + t1)
	else:
		tmp = (v * (-t1 / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(v * Float64(Float64(-t1) / u)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.9e-90) || ~((t1 <= 1.3e+37)))
		tmp = -v / (u + t1);
	else
		tmp = (v * (-t1 / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.9e-90], N[Not[LessEqual[t1, 1.3e+37]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[((-t1) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.89999999999999983e-90 or 1.3e37 < t1

   1. Initial program 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -2.89999999999999983e-90 < t1 < 1.3e37

   1. Initial program 86.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 88.0

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

   5. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 88.7

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

   7. Applied rewrites 88.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.9e-90) (not (<= t1 1.3e+37)))
   (/ (- v) (+ u t1))
   (* (/ (- v) u) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -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 <= (-2.9d-90)) .or. (.not. (t1 <= 1.3d+37))) then
        tmp = -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 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-v / u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.9e-90) or not (t1 <= 1.3e+37):
		tmp = -v / (u + t1)
	else:
		tmp = (-v / u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.9e-90) || ~((t1 <= 1.3e+37)))
		tmp = -v / (u + t1);
	else
		tmp = (-v / u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.9e-90], N[Not[LessEqual[t1, 1.3e+37]], $MachinePrecision]], N[((-v) / 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 < -2.89999999999999983e-90 or 1.3e37 < t1

   1. Initial program 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -2.89999999999999983e-90 < t1 < 1.3e37

   1. Initial program 86.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.9e-90) (not (<= t1 1.3e+37)))
   (/ (- v) (+ u t1))
   (* v (/ (/ (- t1) u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -v / (u + t1);
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.9d-90)) .or. (.not. (t1 <= 1.3d+37))) then
        tmp = -v / (u + t1)
    else
        tmp = v * ((-t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -v / (u + t1);
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.9e-90) or not (t1 <= 1.3e+37):
		tmp = -v / (u + t1)
	else:
		tmp = v * ((-t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.9e-90) || ~((t1 <= 1.3e+37)))
		tmp = -v / (u + t1);
	else
		tmp = v * ((-t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.9e-90], N[Not[LessEqual[t1, 1.3e+37]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.89999999999999983e-90 or 1.3e37 < t1

   1. Initial program 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -2.89999999999999983e-90 < t1 < 1.3e37

   1. Initial program 86.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 88.0

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

   5. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 88.7

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

   7. Applied rewrites 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 82.2

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

   9. Applied rewrites 82.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.9e-90) (not (<= t1 1.3e+37)))
   (/ (- v) (+ u t1))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -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 <= (-2.9d-90)) .or. (.not. (t1 <= 1.3d+37))) then
        tmp = -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 <= -2.9e-90) || !(t1 <= 1.3e+37)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.9e-90) or not (t1 <= 1.3e+37):
		tmp = -v / (u + t1)
	else:
		tmp = (-t1 * v) / (u * u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.9e-90) || !(t1 <= 1.3e+37))
		tmp = Float64(Float64(-v) / 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 <= -2.9e-90) || ~((t1 <= 1.3e+37)))
		tmp = -v / (u + t1);
	else
		tmp = (-t1 * v) / (u * u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.9e-90], N[Not[LessEqual[t1, 1.3e+37]], $MachinePrecision]], N[((-v) / 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 < -2.89999999999999983e-90 or 1.3e37 < t1

   1. Initial program 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -2.89999999999999983e-90 < t1 < 1.3e37

   1. Initial program 86.1%

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

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

      2. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.9 \cdot 10^{-90} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{-91} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7e-91) (not (<= t1 1.3e+37)))
   (/ (- v) (+ u t1))
   (* (- t1) (/ v (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7e-91) || !(t1 <= 1.3e+37)) {
		tmp = -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 <= (-7d-91)) .or. (.not. (t1 <= 1.3d+37))) then
        tmp = -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 <= -7e-91) || !(t1 <= 1.3e+37)) {
		tmp = -v / (u + t1);
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7e-91) or not (t1 <= 1.3e+37):
		tmp = -v / (u + t1)
	else:
		tmp = -t1 * (v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7e-91) || !(t1 <= 1.3e+37))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -7e-91) || ~((t1 <= 1.3e+37)))
		tmp = -v / (u + t1);
	else
		tmp = -t1 * (v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7e-91], N[Not[LessEqual[t1, 1.3e+37]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.9999999999999997e-91 or 1.3e37 < t1

   1. Initial program 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.9

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

   7. Applied rewrites 79.9%

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

   if -6.9999999999999997e-91 < t1 < 1.3e37

   1. Initial program 86.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 88.0

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

   5. Applied rewrites 88.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 77.2

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

   7. Applied rewrites 77.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{-91} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. mul-1-neg N/A

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

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

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 85.5

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

4. Applied rewrites 85.5%

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

5. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 58.5

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

7. Applied rewrites 58.5%

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

Alternative 10: 54.1% 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 74.0%

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

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 47.4

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

5. Applied rewrites 47.4%

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

Reproduce

herbie shell --seed 2025064 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))