Rosa's DopplerBench

Percentage Accurate: 72.5% → 98.3%

Time: 3.7s

Alternatives: 11

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: 72.5% 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: 98.3% 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 72.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.9

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

3. Applied rewrites 97.9%

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

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 98.3

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

5. Applied rewrites 98.3%

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

Alternative 2: 97.9% 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 72.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.9

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

3. Applied rewrites 97.9%

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

Alternative 3: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8.2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\left(\frac{u}{t1} - 1\right) \cdot v}{u + t1}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{+145}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u + t1}}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u \cdot \frac{v}{t1}, 2, -v\right)}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -8.2e+159)
   (/ (* (- (/ u t1) 1.0) v) (+ u t1))
   (if (<= t1 7.2e+145)
     (* (- t1) (/ (/ v (+ u t1)) (+ u t1)))
     (/ (fma (* u (/ v t1)) 2.0 (- v)) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -8.2e+159) {
		tmp = (((u / t1) - 1.0) * v) / (u + t1);
	} else if (t1 <= 7.2e+145) {
		tmp = -t1 * ((v / (u + t1)) / (u + t1));
	} else {
		tmp = fma((u * (v / t1)), 2.0, -v) / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -8.2e+159)
		tmp = Float64(Float64(Float64(Float64(u / t1) - 1.0) * v) / Float64(u + t1));
	elseif (t1 <= 7.2e+145)
		tmp = Float64(Float64(-t1) * Float64(Float64(v / Float64(u + t1)) / Float64(u + t1)));
	else
		tmp = Float64(fma(Float64(u * Float64(v / t1)), 2.0, Float64(-v)) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -8.2e+159], N[(N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 7.2e+145], N[((-t1) * N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.20000000000000027e159

   1. Initial program 40.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 100.0

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

   3. Applied rewrites 100.0%

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

   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 92.3

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

   6. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 91.6

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

   8. Applied rewrites 91.6%

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

   if -8.20000000000000027e159 < t1 < 7.19999999999999948e145

   1. Initial program 83.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 97.2

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

   3. Applied rewrites 97.2%

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

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 97.7

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

   5. Applied rewrites 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if 7.19999999999999948e145 < t1

   1. Initial program 41.4%

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

   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 91.9

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

   4. Applied rewrites 91.9%

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

Alternative 4: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u + t1}\\ \mathbf{elif}\;u \leq -8.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 3.55 \cdot 10^{+75}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.2e+154)
   (/ (* (- t1) (/ v u)) (+ u t1))
   (if (<= u -8.2e-134)
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (if (<= u 8.2e-69)
       (/ (- v) t1)
       (if (<= u 3.55e+75)
         (* (- t1) (/ v (* (+ u t1) (+ u t1))))
         (* (/ (- t1) u) (/ v (+ u t1))))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.2e+154) {
		tmp = (-t1 * (v / u)) / (u + t1);
	} else if (u <= -8.2e-134) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else if (u <= 8.2e-69) {
		tmp = -v / t1;
	} else if (u <= 3.55e+75) {
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	} else {
		tmp = (-t1 / u) * (v / (u + t1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.2d+154)) then
        tmp = (-t1 * (v / u)) / (u + t1)
    else if (u <= (-8.2d-134)) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else if (u <= 8.2d-69) then
        tmp = -v / t1
    else if (u <= 3.55d+75) then
        tmp = -t1 * (v / ((u + t1) * (u + t1)))
    else
        tmp = (-t1 / u) * (v / (u + t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.2e+154) {
		tmp = (-t1 * (v / u)) / (u + t1);
	} else if (u <= -8.2e-134) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else if (u <= 8.2e-69) {
		tmp = -v / t1;
	} else if (u <= 3.55e+75) {
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	} else {
		tmp = (-t1 / u) * (v / (u + t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.2e+154:
		tmp = (-t1 * (v / u)) / (u + t1)
	elif u <= -8.2e-134:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	elif u <= 8.2e-69:
		tmp = -v / t1
	elif u <= 3.55e+75:
		tmp = -t1 * (v / ((u + t1) * (u + t1)))
	else:
		tmp = (-t1 / u) * (v / (u + t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.2e+154)
		tmp = Float64(Float64(Float64(-t1) * Float64(v / u)) / Float64(u + t1));
	elseif (u <= -8.2e-134)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	elseif (u <= 8.2e-69)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 3.55e+75)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / Float64(u + t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.2e+154)
		tmp = (-t1 * (v / u)) / (u + t1);
	elseif (u <= -8.2e-134)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	elseif (u <= 8.2e-69)
		tmp = -v / t1;
	elseif (u <= 3.55e+75)
		tmp = -t1 * (v / ((u + t1) * (u + t1)));
	else
		tmp = (-t1 / u) * (v / (u + t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.2e+154], N[(N[((-t1) * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -8.2e-134], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.2e-69], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 3.55e+75], N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -1.20000000000000007e154

   1. Initial program 72.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 98.4

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

   3. Applied rewrites 98.4%

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

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 90.8%

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

   if -1.20000000000000007e154 < u < -8.2000000000000004e-134

   1. Initial program 79.1%

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

   if -8.2000000000000004e-134 < u < 8.1999999999999998e-69

   1. Initial program 63.9%

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

   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 85.7

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

   4. Applied rewrites 85.7%

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

   if 8.1999999999999998e-69 < u < 3.54999999999999991e75

   1. Initial program 77.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 79.8

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

   3. Applied rewrites 79.8%

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

   if 3.54999999999999991e75 < u

   1. Initial program 77.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 97.6

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

   3. Applied rewrites 97.6%

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

   4. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 85.6

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

   6. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{u + t1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{if}\;u \leq -3.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u + t1}\\ \mathbf{elif}\;u \leq -1.6 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 3.55 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) (/ v (* (+ u t1) (+ u t1))))))
   (if (<= u -3.5e+153)
     (/ (* (- t1) (/ v u)) (+ u t1))
     (if (<= u -1.6e-145)
       t_1
       (if (<= u 8.2e-69)
         (/ (- v) t1)
         (if (<= u 3.55e+75) t_1 (* (/ (- t1) u) (/ v (+ u t1)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / ((u + t1) * (u + t1)));
	double tmp;
	if (u <= -3.5e+153) {
		tmp = (-t1 * (v / u)) / (u + t1);
	} else if (u <= -1.6e-145) {
		tmp = t_1;
	} else if (u <= 8.2e-69) {
		tmp = -v / t1;
	} else if (u <= 3.55e+75) {
		tmp = t_1;
	} else {
		tmp = (-t1 / u) * (v / (u + t1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -t1 * (v / ((u + t1) * (u + t1)))
    if (u <= (-3.5d+153)) then
        tmp = (-t1 * (v / u)) / (u + t1)
    else if (u <= (-1.6d-145)) then
        tmp = t_1
    else if (u <= 8.2d-69) then
        tmp = -v / t1
    else if (u <= 3.55d+75) then
        tmp = t_1
    else
        tmp = (-t1 / u) * (v / (u + t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 * (v / ((u + t1) * (u + t1)));
	double tmp;
	if (u <= -3.5e+153) {
		tmp = (-t1 * (v / u)) / (u + t1);
	} else if (u <= -1.6e-145) {
		tmp = t_1;
	} else if (u <= 8.2e-69) {
		tmp = -v / t1;
	} else if (u <= 3.55e+75) {
		tmp = t_1;
	} else {
		tmp = (-t1 / u) * (v / (u + t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 * (v / ((u + t1) * (u + t1)))
	tmp = 0
	if u <= -3.5e+153:
		tmp = (-t1 * (v / u)) / (u + t1)
	elif u <= -1.6e-145:
		tmp = t_1
	elif u <= 8.2e-69:
		tmp = -v / t1
	elif u <= 3.55e+75:
		tmp = t_1
	else:
		tmp = (-t1 / u) * (v / (u + t1))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * Float64(v / Float64(Float64(u + t1) * Float64(u + t1))))
	tmp = 0.0
	if (u <= -3.5e+153)
		tmp = Float64(Float64(Float64(-t1) * Float64(v / u)) / Float64(u + t1));
	elseif (u <= -1.6e-145)
		tmp = t_1;
	elseif (u <= 8.2e-69)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 3.55e+75)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / Float64(u + t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 * (v / ((u + t1) * (u + t1)));
	tmp = 0.0;
	if (u <= -3.5e+153)
		tmp = (-t1 * (v / u)) / (u + t1);
	elseif (u <= -1.6e-145)
		tmp = t_1;
	elseif (u <= 8.2e-69)
		tmp = -v / t1;
	elseif (u <= 3.55e+75)
		tmp = t_1;
	else
		tmp = (-t1 / u) * (v / (u + t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * N[(v / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.5e+153], N[(N[((-t1) * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.6e-145], t$95$1, If[LessEqual[u, 8.2e-69], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 3.55e+75], t$95$1, N[(N[((-t1) / u), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -3.4999999999999999e153

   1. Initial program 72.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 98.4

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

   3. Applied rewrites 98.4%

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

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 90.8%

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

   if -3.4999999999999999e153 < u < -1.60000000000000004e-145 or 8.1999999999999998e-69 < u < 3.54999999999999991e75

   1. Initial program 78.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 78.9

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

   3. Applied rewrites 78.9%

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

   if -1.60000000000000004e-145 < u < 8.1999999999999998e-69

   1. Initial program 63.8%

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

   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 86.0

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

   4. Applied rewrites 86.0%

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

   if 3.54999999999999991e75 < u

   1. Initial program 77.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 97.6

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

   3. Applied rewrites 97.6%

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

   4. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 85.6

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

   6. Applied rewrites 85.6%

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

Alternative 6: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v}{u + t1}\\ \mathbf{if}\;u \leq -6.8 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) u) (/ v (+ u t1)))))
   (if (<= u -6.8e-99) t_1 (if (<= u 9.5e-29) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / (u + t1));
	double tmp;
	if (u <= -6.8e-99) {
		tmp = t_1;
	} else if (u <= 9.5e-29) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-t1 / u) * (v / (u + t1))
    if (u <= (-6.8d-99)) then
        tmp = t_1
    else if (u <= 9.5d-29) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / (u + t1));
	double tmp;
	if (u <= -6.8e-99) {
		tmp = t_1;
	} else if (u <= 9.5e-29) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / u) * (v / (u + t1))
	tmp = 0
	if u <= -6.8e-99:
		tmp = t_1
	elif u <= 9.5e-29:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / u) * Float64(v / Float64(u + t1)))
	tmp = 0.0
	if (u <= -6.8e-99)
		tmp = t_1;
	elseif (u <= 9.5e-29)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / u) * (v / (u + t1));
	tmp = 0.0;
	if (u <= -6.8e-99)
		tmp = t_1;
	elseif (u <= 9.5e-29)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / u), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -6.8e-99], t$95$1, If[LessEqual[u, 9.5e-29], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -6.80000000000000014e-99 or 9.50000000000000023e-29 < u

   1. Initial program 77.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 97.7

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 75.1

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

   6. Applied rewrites 75.1%

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

   if -6.80000000000000014e-99 < u < 9.50000000000000023e-29

   1. Initial program 65.3%

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

   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 82.7

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

   4. Applied rewrites 82.7%

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

Alternative 7: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -6.8 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 0.00038:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- t1) u) (/ v u))))
   (if (<= u -6.8e-99) t_1 (if (<= u 0.00038) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -6.8e-99) {
		tmp = t_1;
	} else if (u <= 0.00038) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-t1 / u) * (v / u)
    if (u <= (-6.8d-99)) then
        tmp = t_1
    else if (u <= 0.00038d0) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 / u) * (v / u);
	double tmp;
	if (u <= -6.8e-99) {
		tmp = t_1;
	} else if (u <= 0.00038) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / u) * (v / u)
	tmp = 0
	if u <= -6.8e-99:
		tmp = t_1
	elif u <= 0.00038:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / u) * Float64(v / u))
	tmp = 0.0
	if (u <= -6.8e-99)
		tmp = t_1;
	elseif (u <= 0.00038)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / u) * (v / u);
	tmp = 0.0;
	if (u <= -6.8e-99)
		tmp = t_1;
	elseif (u <= 0.00038)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -6.8e-99], t$95$1, If[LessEqual[u, 0.00038], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -6.80000000000000014e-99 or 3.8000000000000002e-4 < u

   1. Initial program 77.3%

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

   2. Taylor expanded in u around inf

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

   4. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. times-frac N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 71.6

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

      13. +-commutative 71.6

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -6.80000000000000014e-99 < u < 3.8000000000000002e-4

   1. Initial program 65.9%

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

   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 81.6

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

   4. Applied rewrites 81.6%

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

Alternative 8: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-62}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -1.15e-100)
     t_1
     (if (<= t1 4.4e-62) (* (- 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 <= -1.15e-100) {
		tmp = t_1;
	} else if (t1 <= 4.4e-62) {
		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 <= (-1.15d-100)) then
        tmp = t_1
    else if (t1 <= 4.4d-62) 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 <= -1.15e-100) {
		tmp = t_1;
	} else if (t1 <= 4.4e-62) {
		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 <= -1.15e-100:
		tmp = t_1
	elif t1 <= 4.4e-62:
		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 <= -1.15e-100)
		tmp = t_1;
	elseif (t1 <= 4.4e-62)
		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 <= -1.15e-100)
		tmp = t_1;
	elseif (t1 <= 4.4e-62)
		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, -1.15e-100], t$95$1, If[LessEqual[t1, 4.4e-62], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.14999999999999997e-100 or 4.40000000000000035e-62 < t1

   1. Initial program 67.3%

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

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.0

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

   8. Applied rewrites 78.0%

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

   if -1.14999999999999997e-100 < t1 < 4.40000000000000035e-62

   1. Initial program 81.3%

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

   2. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.7

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

   4. Applied rewrites 74.7%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]
   5. 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 74.1

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

   6. Applied rewrites 74.1%

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

Alternative 9: 61.9% 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 72.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 97.9

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

3. Applied rewrites 97.9%

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

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 61.9

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

8. Applied rewrites 61.9%

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

Alternative 10: 58.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -1.8 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* -1.0 (/ v u))))
   (if (<= u -1.8e+173) t_1 (if (<= u 1.12e+156) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -1.0 * (v / u);
	double tmp;
	if (u <= -1.8e+173) {
		tmp = t_1;
	} else if (u <= 1.12e+156) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-1.0d0) * (v / u)
    if (u <= (-1.8d+173)) then
        tmp = t_1
    else if (u <= 1.12d+156) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -1.0 * (v / u);
	double tmp;
	if (u <= -1.8e+173) {
		tmp = t_1;
	} else if (u <= 1.12e+156) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -1.0 * (v / u)
	tmp = 0
	if u <= -1.8e+173:
		tmp = t_1
	elif u <= 1.12e+156:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-1.0 * Float64(v / u))
	tmp = 0.0
	if (u <= -1.8e+173)
		tmp = t_1;
	elseif (u <= 1.12e+156)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -1.0 * (v / u);
	tmp = 0.0;
	if (u <= -1.8e+173)
		tmp = t_1;
	elseif (u <= 1.12e+156)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(-1.0 * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.8e+173], t$95$1, If[LessEqual[u, 1.12e+156], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.8000000000000001e173 or 1.12000000000000007e156 < u

   1. Initial program 74.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 98.8

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in u around inf

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

   9. Applied rewrites 40.1%

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

   if -1.8000000000000001e173 < u < 1.12000000000000007e156

   1. Initial program 72.0%

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

   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 64.4

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

   4. Applied rewrites 64.4%

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

Alternative 11: 54.3% 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 72.5%

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

2. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 54.3

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

4. Applied rewrites 54.3%

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

Reproduce

herbie shell --seed 2025095 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))