Rosa's DopplerBench

Percentage Accurate: 72.8% → 96.4%

Time: 6.2s

Alternatives: 13

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: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

4. Applied rewrites 96.7%

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

5. Final simplification 96.7%

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

Alternative 2: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3700000000:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3700000000.0)
   (/ (* (/ (- v) u) t1) u)
   (if (<= u 9.5e-98)
     (/ (- v) t1)
     (if (<= u 6.2e+140)
       (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
       (/ (* (/ t1 u) (- v)) (- u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3700000000.0) {
		tmp = ((-v / u) * t1) / u;
	} else if (u <= 9.5e-98) {
		tmp = -v / t1;
	} else if (u <= 6.2e+140) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = ((t1 / u) * -v) / (u - t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3700000000.0d0)) then
        tmp = ((-v / u) * t1) / u
    else if (u <= 9.5d-98) then
        tmp = -v / t1
    else if (u <= 6.2d+140) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = ((t1 / u) * -v) / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3700000000.0) {
		tmp = ((-v / u) * t1) / u;
	} else if (u <= 9.5e-98) {
		tmp = -v / t1;
	} else if (u <= 6.2e+140) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = ((t1 / u) * -v) / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3700000000.0:
		tmp = ((-v / u) * t1) / u
	elif u <= 9.5e-98:
		tmp = -v / t1
	elif u <= 6.2e+140:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = ((t1 / u) * -v) / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3700000000.0)
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	elseif (u <= 9.5e-98)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 6.2e+140)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(Float64(t1 / u) * Float64(-v)) / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3700000000.0)
		tmp = ((-v / u) * t1) / u;
	elseif (u <= 9.5e-98)
		tmp = -v / t1;
	elseif (u <= 6.2e+140)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = ((t1 / u) * -v) / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3700000000.0], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 9.5e-98], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 6.2e+140], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if u < -3.7e9

   1. Initial program 80.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 88.6%

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

   if -3.7e9 < u < 9.5000000000000001e-98

   1. Initial program 57.2%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if 9.5000000000000001e-98 < u < 6.2000000000000001e140

   1. Initial program 84.0%

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

   if 6.2000000000000001e140 < u

   1. Initial program 74.2%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u around inf

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

      1. lower-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

Alternative 3: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3700000000:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3700000000.0)
   (/ (* (/ (- v) u) t1) u)
   (if (<= u 9.5e-98)
     (/ (- v) t1)
     (if (<= u 6.2e+140)
       (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
       (* (/ v u) (/ (- t1) u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3700000000.0) {
		tmp = ((-v / u) * t1) / u;
	} else if (u <= 9.5e-98) {
		tmp = -v / t1;
	} else if (u <= 6.2e+140) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} 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 (u <= (-3700000000.0d0)) then
        tmp = ((-v / u) * t1) / u
    else if (u <= 9.5d-98) then
        tmp = -v / t1
    else if (u <= 6.2d+140) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    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 (u <= -3700000000.0) {
		tmp = ((-v / u) * t1) / u;
	} else if (u <= 9.5e-98) {
		tmp = -v / t1;
	} else if (u <= 6.2e+140) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3700000000.0:
		tmp = ((-v / u) * t1) / u
	elif u <= 9.5e-98:
		tmp = -v / t1
	elif u <= 6.2e+140:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3700000000.0)
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	elseif (u <= 9.5e-98)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 6.2e+140)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3700000000.0)
		tmp = ((-v / u) * t1) / u;
	elseif (u <= 9.5e-98)
		tmp = -v / t1;
	elseif (u <= 6.2e+140)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3700000000.0], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 9.5e-98], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 6.2e+140], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if u < -3.7e9

   1. Initial program 80.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 88.6%

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

   if -3.7e9 < u < 9.5000000000000001e-98

   1. Initial program 57.2%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if 9.5000000000000001e-98 < u < 6.2000000000000001e140

   1. Initial program 84.0%

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

   if 6.2000000000000001e140 < u

   1. Initial program 74.2%

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 95.0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3700000000:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3700000000 \lor \neg \left(u \leq 1.15 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3700000000.0) (not (<= u 1.15e+76)))
   (* (/ v u) (/ (- t1) u))
   (/ (* -1.0 v) (+ u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3700000000.0) || !(u <= 1.15e+76)) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = (-1.0 * v) / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3700000000.0d0)) .or. (.not. (u <= 1.15d+76))) then
        tmp = (v / u) * (-t1 / u)
    else
        tmp = ((-1.0d0) * v) / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3700000000.0) || !(u <= 1.15e+76)) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = (-1.0 * v) / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3700000000.0) or not (u <= 1.15e+76):
		tmp = (v / u) * (-t1 / u)
	else:
		tmp = (-1.0 * v) / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3700000000.0) || !(u <= 1.15e+76))
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(Float64(-1.0 * v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3700000000.0) || ~((u <= 1.15e+76)))
		tmp = (v / u) * (-t1 / u);
	else
		tmp = (-1.0 * v) / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3700000000.0], N[Not[LessEqual[u, 1.15e+76]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.7e9 or 1.15000000000000001e76 < u

   1. Initial program 79.3%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -3.7e9 < u < 1.15000000000000001e76

   1. Initial program 65.1%

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 78.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 78.2

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

   9. Applied rewrites 78.2%

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

       1. lift-fabs.f64 N/A

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

       2. rem-sqrt-square-rev N/A

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

       3. sqr-neg-rev N/A

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

       4. sqrt-prod N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-neg.f64 78.9

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

   11. Applied rewrites 78.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3700000000 \lor \neg \left(u \leq 1.15 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3700000000:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+76}:\\ \;\;\;\;\frac{-1 \cdot v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3700000000.0)
   (/ (* (/ (- v) u) t1) u)
   (if (<= u 1.15e+76) (/ (* -1.0 v) (+ u t1)) (* (/ v u) (/ (- t1) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3700000000.0) {
		tmp = ((-v / u) * t1) / u;
	} else if (u <= 1.15e+76) {
		tmp = (-1.0 * v) / (u + t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3700000000.0d0)) then
        tmp = ((-v / u) * t1) / u
    else if (u <= 1.15d+76) then
        tmp = ((-1.0d0) * v) / (u + t1)
    else
        tmp = (v / u) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3700000000.0) {
		tmp = ((-v / u) * t1) / u;
	} else if (u <= 1.15e+76) {
		tmp = (-1.0 * v) / (u + t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3700000000.0:
		tmp = ((-v / u) * t1) / u
	elif u <= 1.15e+76:
		tmp = (-1.0 * v) / (u + t1)
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3700000000.0)
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	elseif (u <= 1.15e+76)
		tmp = Float64(Float64(-1.0 * v) / Float64(u + t1));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3700000000.0)
		tmp = ((-v / u) * t1) / u;
	elseif (u <= 1.15e+76)
		tmp = (-1.0 * v) / (u + t1);
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3700000000.0], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 1.15e+76], N[(N[(-1.0 * v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.7e9

   1. Initial program 80.7%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 88.6%

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

   if -3.7e9 < u < 1.15000000000000001e76

   1. Initial program 65.1%

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 78.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 78.2

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

   9. Applied rewrites 78.2%

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

       1. lift-fabs.f64 N/A

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

       2. rem-sqrt-square-rev N/A

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

       3. sqr-neg-rev N/A

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

       4. sqrt-prod N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-neg.f64 78.9

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

   11. Applied rewrites 78.9%

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

   if 1.15000000000000001e76 < u

   1. Initial program 77.6%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3700000000:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+76}:\\ \;\;\;\;\frac{-1 \cdot v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.3e-80)
   (/ v (- (fabs u) t1))
   (if (<= t1 4.1e+24) (* (- v) (/ (/ t1 u) u)) (/ (* -1.0 v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.3e-80) {
		tmp = v / (fabs(u) - t1);
	} else if (t1 <= 4.1e+24) {
		tmp = -v * ((t1 / u) / u);
	} else {
		tmp = (-1.0 * v) / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.3d-80)) then
        tmp = v / (abs(u) - t1)
    else if (t1 <= 4.1d+24) then
        tmp = -v * ((t1 / u) / u)
    else
        tmp = ((-1.0d0) * v) / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.3e-80) {
		tmp = v / (Math.abs(u) - t1);
	} else if (t1 <= 4.1e+24) {
		tmp = -v * ((t1 / u) / u);
	} else {
		tmp = (-1.0 * v) / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.3e-80:
		tmp = v / (math.fabs(u) - t1)
	elif t1 <= 4.1e+24:
		tmp = -v * ((t1 / u) / u)
	else:
		tmp = (-1.0 * v) / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.3e-80)
		tmp = Float64(v / Float64(abs(u) - t1));
	elseif (t1 <= 4.1e+24)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(Float64(-1.0 * v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.3e-80)
		tmp = v / (abs(u) - t1);
	elseif (t1 <= 4.1e+24)
		tmp = -v * ((t1 / u) / u);
	else
		tmp = (-1.0 * v) / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.3e-80], N[(v / N[(N[Abs[u], $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.1e+24], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.3e-80

   1. Initial program 64.8%

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 76.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 76.6

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

   9. Applied rewrites 76.6%

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

   10. Taylor expanded in u around 0

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left|u\right| - t1}} \]

       3. lower-fabs.f64 76.6

          \[\leadsto \frac{v}{\color{blue}{\left|u\right|} - t1} \]

   12. Applied rewrites 76.6%

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

   if -1.3e-80 < t1 < 4.1000000000000001e24

   1. Initial program 81.6%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 86.2

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

   5. Applied rewrites 86.2%

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

   7. Applied rewrites 82.7%

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

   if 4.1000000000000001e24 < t1

   1. Initial program 66.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.6%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 85.4

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

   9. Applied rewrites 85.4%

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

       1. lift-fabs.f64 N/A

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

       2. rem-sqrt-square-rev N/A

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

       3. sqr-neg-rev N/A

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

       4. sqrt-prod N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-neg.f64 85.6

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

   11. Applied rewrites 85.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 2.8 \cdot 10^{+172}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u - t1}}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 2.8e+172)
   (* (- v) (/ (/ t1 (- u t1)) (- u t1)))
   (/ (* (/ t1 u) (- v)) (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 2.8e+172) {
		tmp = -v * ((t1 / (u - t1)) / (u - t1));
	} else {
		tmp = ((t1 / u) * -v) / (u - t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 2.8d+172) then
        tmp = -v * ((t1 / (u - t1)) / (u - t1))
    else
        tmp = ((t1 / u) * -v) / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 2.8e+172) {
		tmp = -v * ((t1 / (u - t1)) / (u - t1));
	} else {
		tmp = ((t1 / u) * -v) / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 2.8e+172:
		tmp = -v * ((t1 / (u - t1)) / (u - t1))
	else:
		tmp = ((t1 / u) * -v) / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 2.8e+172)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / Float64(u - t1)) / Float64(u - t1)));
	else
		tmp = Float64(Float64(Float64(t1 / u) * Float64(-v)) / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 2.8e+172)
		tmp = -v * ((t1 / (u - t1)) / (u - t1));
	else
		tmp = ((t1 / u) * -v) / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 2.8e+172], N[((-v) * N[(N[(t1 / N[(u - t1), $MachinePrecision]), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 2.8e172

   1. Initial program 70.9%

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

   4. Applied rewrites 96.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.9

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

   6. Applied rewrites 94.9%

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

   if 2.8e172 < u

   1. Initial program 75.3%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in u around inf

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

      1. lower-/.f64 96.8

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

   7. Applied rewrites 96.8%

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

Alternative 8: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.3e-80)
   (/ v (- (fabs u) t1))
   (if (<= t1 4.1e+24) (* (/ (- v) (* u u)) t1) (/ (* -1.0 v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.3e-80) {
		tmp = v / (fabs(u) - t1);
	} else if (t1 <= 4.1e+24) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = (-1.0 * v) / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.3d-80)) then
        tmp = v / (abs(u) - t1)
    else if (t1 <= 4.1d+24) then
        tmp = (-v / (u * u)) * t1
    else
        tmp = ((-1.0d0) * v) / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.3e-80) {
		tmp = v / (Math.abs(u) - t1);
	} else if (t1 <= 4.1e+24) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = (-1.0 * v) / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.3e-80:
		tmp = v / (math.fabs(u) - t1)
	elif t1 <= 4.1e+24:
		tmp = (-v / (u * u)) * t1
	else:
		tmp = (-1.0 * v) / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.3e-80)
		tmp = Float64(v / Float64(abs(u) - t1));
	elseif (t1 <= 4.1e+24)
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	else
		tmp = Float64(Float64(-1.0 * v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.3e-80)
		tmp = v / (abs(u) - t1);
	elseif (t1 <= 4.1e+24)
		tmp = (-v / (u * u)) * t1;
	else
		tmp = (-1.0 * v) / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.3e-80], N[(v / N[(N[Abs[u], $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.1e+24], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], N[(N[(-1.0 * v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.3e-80

   1. Initial program 64.8%

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 76.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 76.6

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

   9. Applied rewrites 76.6%

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

   10. Taylor expanded in u around 0

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left|u\right| - t1}} \]

       3. lower-fabs.f64 76.6

          \[\leadsto \frac{v}{\color{blue}{\left|u\right|} - t1} \]

   12. Applied rewrites 76.6%

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

   if -1.3e-80 < t1 < 4.1000000000000001e24

   1. Initial program 81.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if 4.1000000000000001e24 < t1

   1. Initial program 66.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.6%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 85.4

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

   9. Applied rewrites 85.4%

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

       1. lift-fabs.f64 N/A

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

       2. rem-sqrt-square-rev N/A

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

       3. sqr-neg-rev N/A

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

       4. sqrt-prod N/A

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

       5. rem-square-sqrt N/A

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

       6. lower-neg.f64 85.6

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

   11. Applied rewrites 85.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.3e-80)
   (/ v (- (fabs u) t1))
   (if (<= t1 4.1e+24) (* (/ (- v) (* u u)) t1) (/ (* -1.0 v) (+ (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.3e-80) {
		tmp = v / (fabs(u) - t1);
	} else if (t1 <= 4.1e+24) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = (-1.0 * v) / (-u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.3d-80)) then
        tmp = v / (abs(u) - t1)
    else if (t1 <= 4.1d+24) then
        tmp = (-v / (u * u)) * t1
    else
        tmp = ((-1.0d0) * v) / (-u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.3e-80) {
		tmp = v / (Math.abs(u) - t1);
	} else if (t1 <= 4.1e+24) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = (-1.0 * v) / (-u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.3e-80:
		tmp = v / (math.fabs(u) - t1)
	elif t1 <= 4.1e+24:
		tmp = (-v / (u * u)) * t1
	else:
		tmp = (-1.0 * v) / (-u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.3e-80)
		tmp = Float64(v / Float64(abs(u) - t1));
	elseif (t1 <= 4.1e+24)
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	else
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.3e-80)
		tmp = v / (abs(u) - t1);
	elseif (t1 <= 4.1e+24)
		tmp = (-v / (u * u)) * t1;
	else
		tmp = (-1.0 * v) / (-u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.3e-80], N[(v / N[(N[Abs[u], $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.1e+24], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.3e-80

   1. Initial program 64.8%

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 76.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 76.6

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

   9. Applied rewrites 76.6%

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

   10. Taylor expanded in u around 0

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left|u\right| - t1}} \]

       3. lower-fabs.f64 76.6

          \[\leadsto \frac{v}{\color{blue}{\left|u\right|} - t1} \]

   12. Applied rewrites 76.6%

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

   if -1.3e-80 < t1 < 4.1000000000000001e24

   1. Initial program 81.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if 4.1000000000000001e24 < t1

   1. Initial program 66.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.6%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.3e-80)
   (/ v (- (fabs u) t1))
   (if (<= t1 4.1e+24) (* v (/ (- t1) (* u u))) (/ (* -1.0 v) (+ (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.3e-80) {
		tmp = v / (fabs(u) - t1);
	} else if (t1 <= 4.1e+24) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = (-1.0 * v) / (-u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.3d-80)) then
        tmp = v / (abs(u) - t1)
    else if (t1 <= 4.1d+24) then
        tmp = v * (-t1 / (u * u))
    else
        tmp = ((-1.0d0) * v) / (-u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.3e-80) {
		tmp = v / (Math.abs(u) - t1);
	} else if (t1 <= 4.1e+24) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = (-1.0 * v) / (-u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.3e-80:
		tmp = v / (math.fabs(u) - t1)
	elif t1 <= 4.1e+24:
		tmp = v * (-t1 / (u * u))
	else:
		tmp = (-1.0 * v) / (-u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.3e-80)
		tmp = Float64(v / Float64(abs(u) - t1));
	elseif (t1 <= 4.1e+24)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	else
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.3e-80)
		tmp = v / (abs(u) - t1);
	elseif (t1 <= 4.1e+24)
		tmp = v * (-t1 / (u * u));
	else
		tmp = (-1.0 * v) / (-u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.3e-80], N[(v / N[(N[Abs[u], $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.1e+24], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.3e-80

   1. Initial program 64.8%

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 76.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 76.6

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

   9. Applied rewrites 76.6%

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

   10. Taylor expanded in u around 0

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left|u\right| - t1}} \]

       3. lower-fabs.f64 76.6

          \[\leadsto \frac{v}{\color{blue}{\left|u\right|} - t1} \]

   12. Applied rewrites 76.6%

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

   if -1.3e-80 < t1 < 4.1000000000000001e24

   1. Initial program 81.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if 4.1000000000000001e24 < t1

   1. Initial program 66.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -105000000000 \lor \neg \left(u \leq 4.5 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{v \cdot t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -105000000000.0) (not (<= u 4.5e+162)))
   (/ (* v t1) (* u u))
   (/ v (- (fabs u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -105000000000.0) || !(u <= 4.5e+162)) {
		tmp = (v * t1) / (u * u);
	} else {
		tmp = v / (fabs(u) - t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-105000000000.0d0)) .or. (.not. (u <= 4.5d+162))) then
        tmp = (v * t1) / (u * u)
    else
        tmp = v / (abs(u) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -105000000000.0) || !(u <= 4.5e+162)) {
		tmp = (v * t1) / (u * u);
	} else {
		tmp = v / (Math.abs(u) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -105000000000.0) or not (u <= 4.5e+162):
		tmp = (v * t1) / (u * u)
	else:
		tmp = v / (math.fabs(u) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -105000000000.0) || !(u <= 4.5e+162))
		tmp = Float64(Float64(v * t1) / Float64(u * u));
	else
		tmp = Float64(v / Float64(abs(u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -105000000000.0) || ~((u <= 4.5e+162)))
		tmp = (v * t1) / (u * u);
	else
		tmp = v / (abs(u) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -105000000000.0], N[Not[LessEqual[u, 4.5e+162]], $MachinePrecision]], N[(N[(v * t1), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[Abs[u], $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.05e11 or 4.49999999999999972e162 < u

   1. Initial program 78.4%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 65.3%

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

   if -1.05e11 < u < 4.49999999999999972e162

   1. Initial program 67.0%

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 75.7%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. rem-sqrt-square-rev N/A

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

      6. lower-fabs.f64 75.8

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

   9. Applied rewrites 75.8%

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

   10. Taylor expanded in u around 0

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{v}{\color{blue}{\left|u\right| - t1}} \]

       3. lower-fabs.f64 75.8

          \[\leadsto \frac{v}{\color{blue}{\left|u\right|} - t1} \]

   12. Applied rewrites 75.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -105000000000 \lor \neg \left(u \leq 4.5 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{v \cdot t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left|u\right| - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left|u\right| - t1} \end{array} \]

FPCore

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

C

double code(double u, double v, double t1) {
	return v / (fabs(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 / (abs(u) - t1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 63.0%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. pow2 N/A

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

   5. rem-sqrt-square-rev N/A

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

   6. lower-fabs.f64 63.0

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

9. Applied rewrites 63.0%

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

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]

    2. lower--.f64 N/A

       \[\leadsto \frac{v}{\color{blue}{\left|u\right| - t1}} \]

    3. lower-fabs.f64 63.0

       \[\leadsto \frac{v}{\color{blue}{\left|u\right|} - t1} \]

12. Applied rewrites 63.0%

    \[\leadsto \color{blue}{\frac{v}{\left|u\right| - t1}} \]
13. Add Preprocessing

Alternative 13: 53.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 55.3

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

5. Applied rewrites 55.3%

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

Reproduce

herbie shell --seed 2024351 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))