Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.1%

Time: 3.6s

Alternatives: 13

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: 72.6% 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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 98.1

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

3. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{-t1}{u + t1} \cdot \frac{v}{u + t1}} \]
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-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 98.0

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

5. Applied rewrites 98.0%

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

Alternative 2: 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 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 98.1

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

3. Applied rewrites 98.1%

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

Alternative 3: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u \cdot \frac{v}{t1}, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u + t1} \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -9e+95)
   (/ (fma (* u (/ v t1)) 2.0 (- v)) t1)
   (if (<= t1 1.25e+123)
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (* (/ (- t1) (+ u t1)) (/ v t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9e+95) {
		tmp = fma((u * (v / t1)), 2.0, -v) / t1;
	} else if (t1 <= 1.25e+123) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = (-t1 / (u + t1)) * (v / t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -9e+95)
		tmp = Float64(fma(Float64(u * Float64(v / t1)), 2.0, Float64(-v)) / t1);
	elseif (t1 <= 1.25e+123)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(u + t1)) * Float64(v / t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -9e+95], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, 1.25e+123], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.00000000000000033e95

   1. Initial program 49.7%

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

   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 86.8

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

   4. Applied rewrites 86.8%

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

   if -9.00000000000000033e95 < t1 < 1.24999999999999994e123

   1. Initial program 84.7%

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

   if 1.24999999999999994e123 < t1

   1. Initial program 46.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in u around 0

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

      1. lift-/.f64 91.4

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

   6. Applied rewrites 91.4%

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

Alternative 4: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u \cdot \frac{v}{t1}, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{+115}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.7e+95)
   (/ (fma (* u (/ v t1)) 2.0 (- v)) t1)
   (if (<= t1 1.45e+115)
     (* (- t1) (/ v (* (+ u t1) (+ u t1))))
     (* (- (/ u t1) 1.0) (/ v (+ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.7e+95) {
		tmp = fma((u * (v / t1)), 2.0, -v) / t1;
	} else if (t1 <= 1.45e+115) {
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	} else {
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.7e+95)
		tmp = Float64(fma(Float64(u * Float64(v / t1)), 2.0, Float64(-v)) / t1);
	elseif (t1 <= 1.45e+115)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))));
	else
		tmp = Float64(Float64(Float64(u / t1) - 1.0) * Float64(v / Float64(u + t1)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.7e+95], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, 1.45e+115], N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.69999999999999972e95

   1. Initial program 49.7%

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

   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 86.8

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

   4. Applied rewrites 86.8%

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

   if -4.69999999999999972e95 < t1 < 1.45000000000000002e115

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.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 84.7

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

   3. Applied rewrites 84.7%

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

   if 1.45000000000000002e115 < t1

   1. Initial program 47.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.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 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in u around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 90.5

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

   6. Applied rewrites 90.5%

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

Alternative 5: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{v \cdot \left(-t1\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u + t1} \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.55e-48)
   (* (- (/ u t1) 1.0) (/ v (+ u t1)))
   (if (<= t1 8.5e-66)
     (/ (/ (* v (- t1)) u) u)
     (* (/ (- t1) (+ u t1)) (/ v t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.55e-48) {
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	} else if (t1 <= 8.5e-66) {
		tmp = ((v * -t1) / u) / u;
	} else {
		tmp = (-t1 / (u + t1)) * (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 <= (-2.55d-48)) then
        tmp = ((u / t1) - 1.0d0) * (v / (u + t1))
    else if (t1 <= 8.5d-66) then
        tmp = ((v * -t1) / u) / u
    else
        tmp = (-t1 / (u + t1)) * (v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.55e-48) {
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	} else if (t1 <= 8.5e-66) {
		tmp = ((v * -t1) / u) / u;
	} else {
		tmp = (-t1 / (u + t1)) * (v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.55e-48:
		tmp = ((u / t1) - 1.0) * (v / (u + t1))
	elif t1 <= 8.5e-66:
		tmp = ((v * -t1) / u) / u
	else:
		tmp = (-t1 / (u + t1)) * (v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.55e-48)
		tmp = Float64(Float64(Float64(u / t1) - 1.0) * Float64(v / Float64(u + t1)));
	elseif (t1 <= 8.5e-66)
		tmp = Float64(Float64(Float64(v * Float64(-t1)) / u) / u);
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(u + t1)) * Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.55e-48)
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	elseif (t1 <= 8.5e-66)
		tmp = ((v * -t1) / u) / u;
	else
		tmp = (-t1 / (u + t1)) * (v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.55e-48], N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 8.5e-66], N[(N[(N[(v * (-t1)), $MachinePrecision] / u), $MachinePrecision] / u), $MachinePrecision], N[(N[((-t1) / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.55000000000000006e-48

   1. Initial program 64.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.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 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in u around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 80.1

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

   6. Applied rewrites 80.1%

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

   if -2.55000000000000006e-48 < t1 < 8.49999999999999966e-66

   1. Initial program 82.6%

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

   2. Taylor expanded in u around inf

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in u around inf

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

   7. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 77.6

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

      15. +-commutative 77.6

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

      16. +-commutative N/A

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

   9. Applied rewrites 77.6%

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

   if 8.49999999999999966e-66 < t1

   1. Initial program 66.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in u around 0

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

      1. lift-/.f64 79.7

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

   6. Applied rewrites 79.7%

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

Alternative 6: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t1}{u + t1} \cdot \frac{v}{t1}\\ \mathbf{if}\;t1 \leq -2.5 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{v \cdot \left(-t1\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) (+ u t1)) (/ v t1))))
   (if (<= t1 -2.5e-84)
     t_1
     (if (<= t1 8.5e-66) (/ (/ (* v (- t1)) u) u) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / (u + t1)) * (v / t1);
	double tmp;
	if (t1 <= -2.5e-84) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		tmp = ((v * -t1) / 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 = (-t1 / (u + t1)) * (v / t1)
    if (t1 <= (-2.5d-84)) then
        tmp = t_1
    else if (t1 <= 8.5d-66) then
        tmp = ((v * -t1) / 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 = (-t1 / (u + t1)) * (v / t1);
	double tmp;
	if (t1 <= -2.5e-84) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		tmp = ((v * -t1) / u) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / (u + t1)) * (v / t1)
	tmp = 0
	if t1 <= -2.5e-84:
		tmp = t_1
	elif t1 <= 8.5e-66:
		tmp = ((v * -t1) / u) / u
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / Float64(u + t1)) * Float64(v / t1))
	tmp = 0.0
	if (t1 <= -2.5e-84)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		tmp = Float64(Float64(Float64(v * Float64(-t1)) / u) / u);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / (u + t1)) * (v / t1);
	tmp = 0.0;
	if (t1 <= -2.5e-84)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		tmp = ((v * -t1) / u) / u;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.5e-84], t$95$1, If[LessEqual[t1, 8.5e-66], N[(N[(N[(v * (-t1)), $MachinePrecision] / u), $MachinePrecision] / u), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.5000000000000001e-84 or 8.49999999999999966e-66 < t1

   1. Initial program 67.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in u around 0

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

      1. lift-/.f64 79.0

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

   6. Applied rewrites 79.0%

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

   if -2.5000000000000001e-84 < t1 < 8.49999999999999966e-66

   1. Initial program 81.8%

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

   2. Taylor expanded in u around inf

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in u around inf

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

   7. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 78.4

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

      15. +-commutative 78.4

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

      16. +-commutative N/A

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

   9. Applied rewrites 78.4%

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

Alternative 7: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{v \cdot \left(-t1\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2.55e-48)
     t_1
     (if (<= t1 8.5e-66) (/ (/ (* v (- t1)) u) u) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -2.55e-48) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		tmp = ((v * -t1) / 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 <= (-2.55d-48)) then
        tmp = t_1
    else if (t1 <= 8.5d-66) then
        tmp = ((v * -t1) / 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 <= -2.55e-48) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		tmp = ((v * -t1) / u) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -2.55e-48:
		tmp = t_1
	elif t1 <= 8.5e-66:
		tmp = ((v * -t1) / 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 <= -2.55e-48)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		tmp = Float64(Float64(Float64(v * Float64(-t1)) / 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 <= -2.55e-48)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		tmp = ((v * -t1) / 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, -2.55e-48], t$95$1, If[LessEqual[t1, 8.5e-66], N[(N[(N[(v * (-t1)), $MachinePrecision] / u), $MachinePrecision] / u), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.55000000000000006e-48 or 8.49999999999999966e-66 < t1

   1. Initial program 65.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      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 78.5

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 80.2

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

   6. Applied rewrites 80.2%

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

   if -2.55000000000000006e-48 < t1 < 8.49999999999999966e-66

   1. Initial program 82.6%

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

   2. Taylor expanded in u around inf

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in u around inf

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

   7. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 77.6

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

      15. +-commutative 77.6

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

      16. +-commutative N/A

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

   9. Applied rewrites 77.6%

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

Alternative 8: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2e-48) t_1 (if (<= t1 8.5e-66) (/ (* (- 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 <= -2e-48) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		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 <= (-2d-48)) then
        tmp = t_1
    else if (t1 <= 8.5d-66) 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 <= -2e-48) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		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 <= -2e-48:
		tmp = t_1
	elif t1 <= 8.5e-66:
		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 <= -2e-48)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		tmp = Float64(Float64(Float64(-t1) * Float64(v / 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 <= -2e-48)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		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, -2e-48], t$95$1, If[LessEqual[t1, 8.5e-66], N[(N[((-t1) * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.9999999999999999e-48 or 8.49999999999999966e-66 < t1

   1. Initial program 65.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      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 78.5

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 80.2

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

   6. Applied rewrites 80.2%

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

   if -1.9999999999999999e-48 < t1 < 8.49999999999999966e-66

   1. Initial program 82.6%

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

   2. Taylor expanded in u around inf

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in u around inf

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

   7. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

         \[\leadsto \frac{-t1}{u} \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{v}{u}\right)\right) \]

      9. +-commutative N/A

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

   9. Applied rewrites 80.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 79.8

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

   11. Applied rewrites 79.8%

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

Alternative 9: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2e-48) t_1 (if (<= t1 8.5e-66) (/ (* (- 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 <= -2e-48) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		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 <= (-2d-48)) then
        tmp = t_1
    else if (t1 <= 8.5d-66) 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 <= -2e-48) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		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 <= -2e-48:
		tmp = t_1
	elif t1 <= 8.5e-66:
		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 <= -2e-48)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		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 <= -2e-48)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		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, -2e-48], t$95$1, If[LessEqual[t1, 8.5e-66], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.9999999999999999e-48 or 8.49999999999999966e-66 < t1

   1. Initial program 65.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      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 78.5

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 80.2

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

   6. Applied rewrites 80.2%

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

   if -1.9999999999999999e-48 < t1 < 8.49999999999999966e-66

   1. Initial program 82.6%

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

   2. Taylor expanded in u around inf

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in u around inf

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

   7. Applied rewrites 74.4%

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

Alternative 10: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-66}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -2e-48) t_1 (if (<= t1 8.5e-66) (* (- 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 <= -2e-48) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		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 <= (-2d-48)) then
        tmp = t_1
    else if (t1 <= 8.5d-66) 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 <= -2e-48) {
		tmp = t_1;
	} else if (t1 <= 8.5e-66) {
		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 <= -2e-48:
		tmp = t_1
	elif t1 <= 8.5e-66:
		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 <= -2e-48)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		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 <= -2e-48)
		tmp = t_1;
	elseif (t1 <= 8.5e-66)
		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, -2e-48], t$95$1, If[LessEqual[t1, 8.5e-66], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.9999999999999999e-48 or 8.49999999999999966e-66 < t1

   1. Initial program 65.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      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 78.5

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

   3. Applied rewrites 78.5%

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

   4. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 80.2

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

   6. Applied rewrites 80.2%

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

   if -1.9999999999999999e-48 < t1 < 8.49999999999999966e-66

   1. Initial program 82.6%

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

   2. Taylor expanded in u around inf

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in u around inf

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

   7. Applied rewrites 74.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 74.2

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

      6. +-commutative 74.2

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

      7. +-commutative N/A

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

   9. Applied rewrites 74.2%

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

Alternative 11: 62.6% 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 72.6%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   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 82.5

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

3. Applied rewrites 82.5%

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

4. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 62.6

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

6. Applied rewrites 62.6%

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

Alternative 12: 55.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 4.7 \cdot 10^{-273}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 4.7e-273) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 4.7e-273) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (v <= 4.7d-273) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 4.7e-273) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 4.7e-273:
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 4.7e-273)
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 4.7e-273)
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 4.7e-273], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 4.69999999999999962e-273

   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-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      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 81.9

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

   3. Applied rewrites 81.9%

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

   4. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 64.8

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 56.7

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

   9. Applied rewrites 56.7%

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

   10. Taylor expanded in u around inf

       \[\leadsto \frac{-v}{u} \]
   11. Step-by-step derivation

   12. Applied rewrites 19.3%

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

   if 4.69999999999999962e-273 < v

   1. Initial program 72.4%

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

   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 53.8

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

   4. Applied rewrites 53.8%

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

Alternative 13: 35.6% 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 72.6%

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

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 55.0

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

4. Applied rewrites 55.0%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))