Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.0%

Time: 3.5s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 98.0

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

3. Applied rewrites 98.0%

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

Alternative 2: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.65 \cdot 10^{+175}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{+167}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u + t1}}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.65e+175)
   (/ (- v) (+ u t1))
   (if (<= t1 1.4e+167) (* (- t1) (/ (/ v (+ u t1)) (+ u t1))) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.65e+175) {
		tmp = -v / (u + t1);
	} else if (t1 <= 1.4e+167) {
		tmp = -t1 * ((v / (u + t1)) / (u + t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.65d+175)) then
        tmp = -v / (u + t1)
    else if (t1 <= 1.4d+167) then
        tmp = -t1 * ((v / (u + t1)) / (u + t1))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.65e+175) {
		tmp = -v / (u + t1);
	} else if (t1 <= 1.4e+167) {
		tmp = -t1 * ((v / (u + t1)) / (u + t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.65e+175:
		tmp = -v / (u + t1)
	elif t1 <= 1.4e+167:
		tmp = -t1 * ((v / (u + t1)) / (u + t1))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.65e+175)
		tmp = Float64(Float64(-v) / Float64(u + t1));
	elseif (t1 <= 1.4e+167)
		tmp = Float64(Float64(-t1) * Float64(Float64(v / Float64(u + t1)) / Float64(u + t1)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.65e+175)
		tmp = -v / (u + t1);
	elseif (t1 <= 1.4e+167)
		tmp = -t1 * ((v / (u + t1)) / (u + t1));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.65e+175], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.4e+167], N[((-t1) * N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.65000000000000006e175

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.0

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

   3. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if -2.65000000000000006e175 < t1 < 1.3999999999999999e167

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 72.7

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

   3. Applied rewrites 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-+.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if 1.3999999999999999e167 < t1

   1. Initial program 72.7%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 54.9

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

   4. Applied rewrites 54.9%

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

Alternative 3: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u \cdot \frac{v}{t1}, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq 4.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u, -v\right)}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.1e+98)
   (/ (fma (* u (/ v t1)) 2.0 (- v)) t1)
   (if (<= t1 4.5e+106)
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (fma (/ v t1) u (- v)) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.1e+98) {
		tmp = fma((u * (v / t1)), 2.0, -v) / t1;
	} else if (t1 <= 4.5e+106) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = fma((v / t1), u, -v) / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.1e+98)
		tmp = Float64(fma(Float64(u * Float64(v / t1)), 2.0, Float64(-v)) / t1);
	elseif (t1 <= 4.5e+106)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(fma(Float64(v / t1), u, Float64(-v)) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.1e+98], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, 4.5e+106], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v / t1), $MachinePrecision] * u + (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.10000000000000004e98

   1. Initial program 72.7%

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

   2. Taylor expanded in t1 around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 52.5

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

   4. Applied rewrites 52.5%

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

   if -1.10000000000000004e98 < t1 < 4.4999999999999997e106

   1. Initial program 72.7%

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

   if 4.4999999999999997e106 < t1

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 83.2

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

   3. Applied rewrites 83.2%

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

   4. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 52.1

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

   6. Applied rewrites 52.1%

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

Alternative 4: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+116}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 5.7 \cdot 10^{+97}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u, -v\right)}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.4e+116)
   (/ (- v) t1)
   (if (<= t1 5.7e+97)
     (* (- t1) (/ v (* (+ u t1) (+ u t1))))
     (/ (fma (/ v t1) u (- v)) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.4e+116) {
		tmp = -v / t1;
	} else if (t1 <= 5.7e+97) {
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	} else {
		tmp = fma((v / t1), u, -v) / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.4e+116)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 5.7e+97)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))));
	else
		tmp = Float64(fma(Float64(v / t1), u, Float64(-v)) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.4e+116], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 5.7e+97], N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v / t1), $MachinePrecision] * u + (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.40000000000000002e116

   1. Initial program 72.7%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 54.9

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

   4. Applied rewrites 54.9%

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

   if -1.40000000000000002e116 < t1 < 5.7000000000000002e97

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 72.7

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

   3. Applied rewrites 72.7%

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

   if 5.7000000000000002e97 < t1

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 83.2

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

   3. Applied rewrites 83.2%

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

   4. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 52.1

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

   6. Applied rewrites 52.1%

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

Alternative 5: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -415000:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u + t1}\\ \mathbf{elif}\;u \leq 4.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v \cdot \left(-t1\right)}{u}}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -415000.0)
   (* (/ (- t1) u) (/ v (+ u t1)))
   (if (<= u 4.4e+41) (/ (- v) t1) (/ (/ (* v (- t1)) u) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -415000.0) {
		tmp = (-t1 / u) * (v / (u + t1));
	} else if (u <= 4.4e+41) {
		tmp = -v / t1;
	} else {
		tmp = ((v * -t1) / u) / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-415000.0d0)) then
        tmp = (-t1 / u) * (v / (u + t1))
    else if (u <= 4.4d+41) then
        tmp = -v / t1
    else
        tmp = ((v * -t1) / u) / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -415000.0) {
		tmp = (-t1 / u) * (v / (u + t1));
	} else if (u <= 4.4e+41) {
		tmp = -v / t1;
	} else {
		tmp = ((v * -t1) / u) / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -415000.0:
		tmp = (-t1 / u) * (v / (u + t1))
	elif u <= 4.4e+41:
		tmp = -v / t1
	else:
		tmp = ((v * -t1) / u) / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -415000.0)
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / Float64(u + t1)));
	elseif (u <= 4.4e+41)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(v * Float64(-t1)) / u) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -415000.0)
		tmp = (-t1 / u) * (v / (u + t1));
	elseif (u <= 4.4e+41)
		tmp = -v / t1;
	else
		tmp = ((v * -t1) / u) / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -415000.0], N[(N[((-t1) / u), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.4e+41], N[((-v) / t1), $MachinePrecision], N[(N[(N[(v * (-t1)), $MachinePrecision] / u), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -415000

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.0

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

   3. Applied rewrites 98.0%

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

   4. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 55.2

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

   6. Applied rewrites 55.2%

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

   if -415000 < u < 4.3999999999999998e41

   1. Initial program 72.7%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 54.9

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

   4. Applied rewrites 54.9%

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

   if 4.3999999999999998e41 < u

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. mul-1-neg N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 83.2

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

   3. Applied rewrites 83.2%

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

   4. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. lift-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 48.9

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

   6. Applied rewrites 48.9%

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

Alternative 6: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -415000:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u + t1}\\ \mathbf{elif}\;u \leq 4.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -415000.0)
   (* (/ (- t1) u) (/ v (+ u t1)))
   (if (<= u 4.4e+44) (/ (- v) t1) (* (- t1) (/ v (* u u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -415000.0) {
		tmp = (-t1 / u) * (v / (u + t1));
	} else if (u <= 4.4e+44) {
		tmp = -v / t1;
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-415000.0d0)) then
        tmp = (-t1 / u) * (v / (u + t1))
    else if (u <= 4.4d+44) then
        tmp = -v / t1
    else
        tmp = -t1 * (v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -415000.0) {
		tmp = (-t1 / u) * (v / (u + t1));
	} else if (u <= 4.4e+44) {
		tmp = -v / t1;
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -415000.0:
		tmp = (-t1 / u) * (v / (u + t1))
	elif u <= 4.4e+44:
		tmp = -v / t1
	else:
		tmp = -t1 * (v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -415000.0)
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / Float64(u + t1)));
	elseif (u <= 4.4e+44)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -415000.0)
		tmp = (-t1 / u) * (v / (u + t1));
	elseif (u <= 4.4e+44)
		tmp = -v / t1;
	else
		tmp = -t1 * (v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -415000.0], N[(N[((-t1) / u), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.4e+44], N[((-v) / t1), $MachinePrecision], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -415000

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.0

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

   3. Applied rewrites 98.0%

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

   4. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 55.2

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

   6. Applied rewrites 55.2%

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

   if -415000 < u < 4.39999999999999991e44

   1. Initial program 72.7%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 54.9

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

   4. Applied rewrites 54.9%

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

   if 4.39999999999999991e44 < u

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 72.7

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

   3. Applied rewrites 72.7%

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

   4. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 47.1

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

   6. Applied rewrites 47.1%

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

Alternative 7: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -52000000:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 4.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -52000000.0)
   (* (/ (- t1) u) (/ v u))
   (if (<= u 4.4e+44) (/ (- v) t1) (* (- t1) (/ v (* u u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -52000000.0) {
		tmp = (-t1 / u) * (v / u);
	} else if (u <= 4.4e+44) {
		tmp = -v / t1;
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-52000000.0d0)) then
        tmp = (-t1 / u) * (v / u)
    else if (u <= 4.4d+44) then
        tmp = -v / t1
    else
        tmp = -t1 * (v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -52000000.0) {
		tmp = (-t1 / u) * (v / u);
	} else if (u <= 4.4e+44) {
		tmp = -v / t1;
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -52000000.0:
		tmp = (-t1 / u) * (v / u)
	elif u <= 4.4e+44:
		tmp = -v / t1
	else:
		tmp = -t1 * (v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -52000000.0)
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	elseif (u <= 4.4e+44)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -52000000.0)
		tmp = (-t1 / u) * (v / u);
	elseif (u <= 4.4e+44)
		tmp = -v / t1;
	else
		tmp = -t1 * (v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -52000000.0], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.4e+44], N[((-v) / t1), $MachinePrecision], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.2e7

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.0

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

   3. Applied rewrites 98.0%

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

   4. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 55.2

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

   6. Applied rewrites 55.2%

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

   7. Taylor expanded in u around inf

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

      1. lower-/.f64 52.7

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

   9. Applied rewrites 52.7%

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

   if -5.2e7 < u < 4.39999999999999991e44

   1. Initial program 72.7%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 54.9

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

   4. Applied rewrites 54.9%

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

   if 4.39999999999999991e44 < u

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 72.7

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

   3. Applied rewrites 72.7%

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

   4. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 47.1

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

   6. Applied rewrites 47.1%

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

Alternative 8: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -4.6 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.35 \cdot 10^{-54}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -4.6e-77)
     t_1
     (if (<= t1 2.35e-54) (/ (* (- t1) v) (* u u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -4.6e-77) {
		tmp = t_1;
	} else if (t1 <= 2.35e-54) {
		tmp = (-t1 * v) / (u * u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-4.6d-77)) then
        tmp = t_1
    else if (t1 <= 2.35d-54) then
        tmp = (-t1 * v) / (u * u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -4.6e-77) {
		tmp = t_1;
	} else if (t1 <= 2.35e-54) {
		tmp = (-t1 * v) / (u * u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -4.6e-77:
		tmp = t_1
	elif t1 <= 2.35e-54:
		tmp = (-t1 * v) / (u * u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -4.6e-77)
		tmp = t_1;
	elseif (t1 <= 2.35e-54)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -4.6e-77)
		tmp = t_1;
	elseif (t1 <= 2.35e-54)
		tmp = (-t1 * v) / (u * u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -4.6e-77], t$95$1, If[LessEqual[t1, 2.35e-54], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.59999999999999997e-77 or 2.35e-54 < t1

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.0

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

   3. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if -4.59999999999999997e-77 < t1 < 2.35e-54

   1. Initial program 72.7%

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

   2. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 45.5

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

   4. Applied rewrites 45.5%

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

Alternative 9: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -4.6 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.35 \cdot 10^{-54}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -4.6e-77)
     t_1
     (if (<= t1 2.35e-54) (* (- t1) (/ v (* u u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -4.6e-77) {
		tmp = t_1;
	} else if (t1 <= 2.35e-54) {
		tmp = -t1 * (v / (u * u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-4.6d-77)) then
        tmp = t_1
    else if (t1 <= 2.35d-54) then
        tmp = -t1 * (v / (u * u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -4.6e-77) {
		tmp = t_1;
	} else if (t1 <= 2.35e-54) {
		tmp = -t1 * (v / (u * u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -4.6e-77:
		tmp = t_1
	elif t1 <= 2.35e-54:
		tmp = -t1 * (v / (u * u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -4.6e-77)
		tmp = t_1;
	elseif (t1 <= 2.35e-54)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -4.6e-77)
		tmp = t_1;
	elseif (t1 <= 2.35e-54)
		tmp = -t1 * (v / (u * u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -4.6e-77], t$95$1, If[LessEqual[t1, 2.35e-54], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.59999999999999997e-77 or 2.35e-54 < t1

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.0

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

   3. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if -4.59999999999999997e-77 < t1 < 2.35e-54

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 72.7

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

   3. Applied rewrites 72.7%

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

   4. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 47.1

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

   6. Applied rewrites 47.1%

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

Alternative 10: 62.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 98.0

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

3. Applied rewrites 98.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 62.2

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

8. Applied rewrites 62.2%

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

Alternative 11: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -1.2 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+177}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* -1.0 (/ v u))))
   (if (<= u -1.2e+82) t_1 (if (<= u 1.25e+177) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -1.0 * (v / u);
	double tmp;
	if (u <= -1.2e+82) {
		tmp = t_1;
	} else if (u <= 1.25e+177) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-1.0d0) * (v / u)
    if (u <= (-1.2d+82)) then
        tmp = t_1
    else if (u <= 1.25d+177) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -1.0 * (v / u);
	double tmp;
	if (u <= -1.2e+82) {
		tmp = t_1;
	} else if (u <= 1.25e+177) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -1.0 * (v / u)
	tmp = 0
	if u <= -1.2e+82:
		tmp = t_1
	elif u <= 1.25e+177:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-1.0 * Float64(v / u))
	tmp = 0.0
	if (u <= -1.2e+82)
		tmp = t_1;
	elseif (u <= 1.25e+177)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -1.0 * (v / u);
	tmp = 0.0;
	if (u <= -1.2e+82)
		tmp = t_1;
	elseif (u <= 1.25e+177)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(-1.0 * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.2e+82], t$95$1, If[LessEqual[u, 1.25e+177], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.19999999999999999e82 or 1.2500000000000001e177 < u

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.0

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

   3. Applied rewrites 98.0%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 62.2%

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

   7. Taylor expanded in u around inf

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

      1. lower-/.f64 17.3

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

   9. Applied rewrites 17.3%

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

   if -1.19999999999999999e82 < u < 1.2500000000000001e177

   1. Initial program 72.7%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 54.9

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

   4. Applied rewrites 54.9%

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

Alternative 12: 54.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

2. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 54.9

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

4. Applied rewrites 54.9%

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

Reproduce

herbie shell --seed 2025131 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))