Rosa's DopplerBench

Percentage Accurate: 71.8% → 98.0%

Time: 4.1s

Alternatives: 13

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.8× 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 71.8%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 98.1

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

3. Applied rewrites 98.1%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. +-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 98.0

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

5. Applied rewrites 98.0%

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

Alternative 2: 87.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -2.65 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u \cdot \frac{v}{t1}, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq -4.1 \cdot 10^{-166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{-169}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -2.65e+121)
     (/ (fma (* u (/ v t1)) 2.0 (- v)) t1)
     (if (<= t1 -4.1e-166)
       t_1
       (if (<= t1 2.3e-169)
         (* v (/ (/ (- t1) u) u))
         (if (<= t1 2.25e+52) t_1 (* (- (/ u t1) 1.0) (/ v (+ u t1)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -2.65e+121) {
		tmp = fma((u * (v / t1)), 2.0, -v) / t1;
	} else if (t1 <= -4.1e-166) {
		tmp = t_1;
	} else if (t1 <= 2.3e-169) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 2.25e+52) {
		tmp = t_1;
	} else {
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -2.65e+121)
		tmp = Float64(fma(Float64(u * Float64(v / t1)), 2.0, Float64(-v)) / t1);
	elseif (t1 <= -4.1e-166)
		tmp = t_1;
	elseif (t1 <= 2.3e-169)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	elseif (t1 <= 2.25e+52)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(u / t1) - 1.0) * Float64(v / Float64(u + t1)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.65e+121], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, -4.1e-166], t$95$1, If[LessEqual[t1, 2.3e-169], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.25e+52], t$95$1, N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.65000000000000005e121

   1. Initial program 47.9%

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

   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 89.9

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

   4. Applied rewrites 89.9%

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

   if -2.65000000000000005e121 < t1 < -4.0999999999999997e-166 or 2.3000000000000001e-169 < t1 < 2.25e52

   1. Initial program 87.8%

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

   if -4.0999999999999997e-166 < t1 < 2.3000000000000001e-169

   1. Initial program 77.2%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 85.8

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

   6. Applied rewrites 85.8%

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

      1. times-frac 85.8

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

      2. +-commutative 85.8

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

      3. +-commutative 85.8

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

      4. associate-*l/ 85.8

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. distribute-frac-neg N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if 2.25e52 < t1

   1. Initial program 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in u around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 86.9

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

   6. Applied rewrites 86.9%

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

Alternative 3: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1}\\ \mathbf{if}\;t1 \leq -6 \cdot 10^{+87}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -4.1 \cdot 10^{-166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{-169}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
        (t_2 (* (- (/ u t1) 1.0) (/ v (+ u t1)))))
   (if (<= t1 -6e+87)
     t_2
     (if (<= t1 -4.1e-166)
       t_1
       (if (<= t1 2.3e-169)
         (* v (/ (/ (- t1) u) u))
         (if (<= t1 2.25e+52) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double t_2 = ((u / t1) - 1.0) * (v / (u + t1));
	double tmp;
	if (t1 <= -6e+87) {
		tmp = t_2;
	} else if (t1 <= -4.1e-166) {
		tmp = t_1;
	} else if (t1 <= 2.3e-169) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 2.25e+52) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
    t_2 = ((u / t1) - 1.0d0) * (v / (u + t1))
    if (t1 <= (-6d+87)) then
        tmp = t_2
    else if (t1 <= (-4.1d-166)) then
        tmp = t_1
    else if (t1 <= 2.3d-169) then
        tmp = v * ((-t1 / u) / u)
    else if (t1 <= 2.25d+52) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double t_2 = ((u / t1) - 1.0) * (v / (u + t1));
	double tmp;
	if (t1 <= -6e+87) {
		tmp = t_2;
	} else if (t1 <= -4.1e-166) {
		tmp = t_1;
	} else if (t1 <= 2.3e-169) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 2.25e+52) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
	t_2 = ((u / t1) - 1.0) * (v / (u + t1))
	tmp = 0
	if t1 <= -6e+87:
		tmp = t_2
	elif t1 <= -4.1e-166:
		tmp = t_1
	elif t1 <= 2.3e-169:
		tmp = v * ((-t1 / u) / u)
	elif t1 <= 2.25e+52:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	t_2 = Float64(Float64(Float64(u / t1) - 1.0) * Float64(v / Float64(u + t1)))
	tmp = 0.0
	if (t1 <= -6e+87)
		tmp = t_2;
	elseif (t1 <= -4.1e-166)
		tmp = t_1;
	elseif (t1 <= 2.3e-169)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	elseif (t1 <= 2.25e+52)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	t_2 = ((u / t1) - 1.0) * (v / (u + t1));
	tmp = 0.0;
	if (t1 <= -6e+87)
		tmp = t_2;
	elseif (t1 <= -4.1e-166)
		tmp = t_1;
	elseif (t1 <= 2.3e-169)
		tmp = v * ((-t1 / u) / u);
	elseif (t1 <= 2.25e+52)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -6e+87], t$95$2, If[LessEqual[t1, -4.1e-166], t$95$1, If[LessEqual[t1, 2.3e-169], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.25e+52], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.9999999999999998e87 or 2.25e52 < t1

   1. Initial program 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in u around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 88.1

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

   6. Applied rewrites 88.1%

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

   if -5.9999999999999998e87 < t1 < -4.0999999999999997e-166 or 2.3000000000000001e-169 < t1 < 2.25e52

   1. Initial program 89.2%

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

   if -4.0999999999999997e-166 < t1 < 2.3000000000000001e-169

   1. Initial program 77.2%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 85.6

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

   4. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 85.8

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

   6. Applied rewrites 85.8%

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

      1. times-frac 85.8

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

      2. +-commutative 85.8

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

      3. +-commutative 85.8

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

      4. associate-*l/ 85.8

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. distribute-frac-neg N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 86.8

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

   8. Applied rewrites 86.8%

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

Alternative 4: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ t_2 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -6 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -5.8 \cdot 10^{-164}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{-169}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))) (t_2 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -6e+87)
     t_1
     (if (<= t1 -5.8e-164)
       t_2
       (if (<= t1 2.3e-169)
         (* v (/ (/ (- t1) u) u))
         (if (<= t1 2.25e+52) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double t_2 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -6e+87) {
		tmp = t_1;
	} else if (t1 <= -5.8e-164) {
		tmp = t_2;
	} else if (t1 <= 2.3e-169) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 2.25e+52) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = -v / (u + t1)
    t_2 = (-t1 * v) / ((t1 + u) * (t1 + u))
    if (t1 <= (-6d+87)) then
        tmp = t_1
    else if (t1 <= (-5.8d-164)) then
        tmp = t_2
    else if (t1 <= 2.3d-169) then
        tmp = v * ((-t1 / u) / u)
    else if (t1 <= 2.25d+52) then
        tmp = t_2
    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 t_2 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -6e+87) {
		tmp = t_1;
	} else if (t1 <= -5.8e-164) {
		tmp = t_2;
	} else if (t1 <= 2.3e-169) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 2.25e+52) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	t_2 = (-t1 * v) / ((t1 + u) * (t1 + u))
	tmp = 0
	if t1 <= -6e+87:
		tmp = t_1
	elif t1 <= -5.8e-164:
		tmp = t_2
	elif t1 <= 2.3e-169:
		tmp = v * ((-t1 / u) / u)
	elif t1 <= 2.25e+52:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	t_2 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -6e+87)
		tmp = t_1;
	elseif (t1 <= -5.8e-164)
		tmp = t_2;
	elseif (t1 <= 2.3e-169)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	elseif (t1 <= 2.25e+52)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	t_2 = (-t1 * v) / ((t1 + u) * (t1 + u));
	tmp = 0.0;
	if (t1 <= -6e+87)
		tmp = t_1;
	elseif (t1 <= -5.8e-164)
		tmp = t_2;
	elseif (t1 <= 2.3e-169)
		tmp = v * ((-t1 / u) / u);
	elseif (t1 <= 2.25e+52)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -6e+87], t$95$1, If[LessEqual[t1, -5.8e-164], t$95$2, If[LessEqual[t1, 2.3e-169], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.25e+52], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.9999999999999998e87 or 2.25e52 < t1

   1. Initial program 50.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 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-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. 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 88.1

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

   8. Applied rewrites 88.1%

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

   if -5.9999999999999998e87 < t1 < -5.8e-164 or 2.3000000000000001e-169 < t1 < 2.25e52

   1. Initial program 89.2%

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

   if -5.8e-164 < t1 < 2.3000000000000001e-169

   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

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 85.7

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

   4. Applied rewrites 85.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 85.8

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

   6. Applied rewrites 85.8%

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

      1. times-frac 85.8

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

      2. +-commutative 85.8

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

      3. +-commutative 85.8

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

      4. associate-*l/ 85.8

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. distribute-frac-neg N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 86.8

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

   8. Applied rewrites 86.8%

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

Alternative 5: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{-t1}{u + t1} \cdot \frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{-169}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.8e-130)
   (* (/ (- t1) (+ u t1)) (/ v t1))
   (if (<= t1 2.3e-169)
     (* v (/ (/ (- t1) u) u))
     (if (<= t1 2.25e+52)
       (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
       (/ (- v) (+ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.8e-130) {
		tmp = (-t1 / (u + t1)) * (v / t1);
	} else if (t1 <= 2.3e-169) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 2.25e+52) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5.8d-130)) then
        tmp = (-t1 / (u + t1)) * (v / t1)
    else if (t1 <= 2.3d-169) then
        tmp = v * ((-t1 / u) / u)
    else if (t1 <= 2.25d+52) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.8e-130) {
		tmp = (-t1 / (u + t1)) * (v / t1);
	} else if (t1 <= 2.3e-169) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 2.25e+52) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.8e-130:
		tmp = (-t1 / (u + t1)) * (v / t1)
	elif t1 <= 2.3e-169:
		tmp = v * ((-t1 / u) / u)
	elif t1 <= 2.25e+52:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.8e-130)
		tmp = Float64(Float64(Float64(-t1) / Float64(u + t1)) * Float64(v / t1));
	elseif (t1 <= 2.3e-169)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	elseif (t1 <= 2.25e+52)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5.8e-130)
		tmp = (-t1 / (u + t1)) * (v / t1);
	elseif (t1 <= 2.3e-169)
		tmp = v * ((-t1 / u) / u);
	elseif (t1 <= 2.25e+52)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.8e-130], N[(N[((-t1) / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.3e-169], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.25e+52], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -5.8e-130

   1. Initial program 70.8%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 53.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 74.7

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

      13. +-commutative 74.7

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

   6. Applied rewrites 74.7%

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

   if -5.8e-130 < t1 < 2.3000000000000001e-169

   1. Initial program 78.3%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 84.9

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

   4. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 85.1

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

   6. Applied rewrites 85.1%

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

      1. times-frac 85.1

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

      2. +-commutative 85.1

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

      3. +-commutative 85.1

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

      4. associate-*l/ 85.1

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. mul-1-neg N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. distribute-frac-neg N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 85.6

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

   8. Applied rewrites 85.6%

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

   if 2.3000000000000001e-169 < t1 < 2.25e52

   1. Initial program 88.8%

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

   if 2.25e52 < t1

   1. Initial program 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 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-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. 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 87.0

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

   8. Applied rewrites 87.0%

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

Alternative 6: 90.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u + t1}\\ \mathbf{if}\;t1 \leq -1.02 \cdot 10^{+131}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u \cdot \frac{v}{t1}, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+76}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{t\_1}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{u}{t1} - 1\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ u t1))))
   (if (<= t1 -1.02e+131)
     (/ (fma (* u (/ v t1)) 2.0 (- v)) t1)
     (if (<= t1 4.1e+76)
       (* (- t1) (/ t_1 (+ u t1)))
       (* (- (/ u t1) 1.0) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (u + t1);
	double tmp;
	if (t1 <= -1.02e+131) {
		tmp = fma((u * (v / t1)), 2.0, -v) / t1;
	} else if (t1 <= 4.1e+76) {
		tmp = -t1 * (t_1 / (u + t1));
	} else {
		tmp = ((u / t1) - 1.0) * t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -1.02e+131)
		tmp = Float64(fma(Float64(u * Float64(v / t1)), 2.0, Float64(-v)) / t1);
	elseif (t1 <= 4.1e+76)
		tmp = Float64(Float64(-t1) * Float64(t_1 / Float64(u + t1)));
	else
		tmp = Float64(Float64(Float64(u / t1) - 1.0) * t_1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.02e+131], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, 4.1e+76], N[((-t1) * N[(t$95$1 / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.0199999999999999e131

   1. Initial program 46.0%

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

   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 90.6

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

   4. Applied rewrites 90.6%

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

   if -1.0199999999999999e131 < t1 < 4.0999999999999998e76

   1. Initial program 83.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 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-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 97.1

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

   5. Applied rewrites 97.1%

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

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 90.2

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

   7. Applied rewrites 90.2%

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

   if 4.0999999999999998e76 < t1

   1. Initial program 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in u around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 88.8

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

   6. Applied rewrites 88.8%

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

Alternative 7: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -5.8e-130)
     t_1
     (if (<= t1 5.1e-42) (/ (* v (/ (- t1) u)) u) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -5.8e-130) {
		tmp = t_1;
	} else if (t1 <= 5.1e-42) {
		tmp = (v * (-t1 / u)) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-5.8d-130)) then
        tmp = t_1
    else if (t1 <= 5.1d-42) then
        tmp = (v * (-t1 / u)) / u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -5.8e-130) {
		tmp = t_1;
	} else if (t1 <= 5.1e-42) {
		tmp = (v * (-t1 / u)) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -5.8e-130:
		tmp = t_1
	elif t1 <= 5.1e-42:
		tmp = (v * (-t1 / u)) / u
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -5.8e-130)
		tmp = t_1;
	elseif (t1 <= 5.1e-42)
		tmp = Float64(Float64(v * Float64(Float64(-t1) / u)) / u);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -5.8e-130)
		tmp = t_1;
	elseif (t1 <= 5.1e-42)
		tmp = (v * (-t1 / u)) / u;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.8e-130], t$95$1, If[LessEqual[t1, 5.1e-42], N[(N[(v * N[((-t1) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.8e-130 or 5.1e-42 < t1

   1. Initial program 66.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

   3. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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 76.8

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

   8. Applied rewrites 76.8%

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

   if -5.8e-130 < t1 < 5.1e-42

   1. Initial program 81.0%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 81.0

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

   4. Applied rewrites 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 81.0

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

   6. Applied rewrites 81.0%

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

Alternative 8: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -5.8e-130)
     t_1
     (if (<= t1 5.1e-42) (* (/ v u) (/ (- t1) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -5.8e-130) {
		tmp = t_1;
	} else if (t1 <= 5.1e-42) {
		tmp = (v / u) * (-t1 / 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 <= (-5.8d-130)) then
        tmp = t_1
    else if (t1 <= 5.1d-42) then
        tmp = (v / u) * (-t1 / 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 <= -5.8e-130) {
		tmp = t_1;
	} else if (t1 <= 5.1e-42) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -5.8e-130:
		tmp = t_1
	elif t1 <= 5.1e-42:
		tmp = (v / u) * (-t1 / 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 <= -5.8e-130)
		tmp = t_1;
	elseif (t1 <= 5.1e-42)
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / 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 <= -5.8e-130)
		tmp = t_1;
	elseif (t1 <= 5.1e-42)
		tmp = (v / u) * (-t1 / 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, -5.8e-130], t$95$1, If[LessEqual[t1, 5.1e-42], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.8e-130 or 5.1e-42 < t1

   1. Initial program 66.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

   3. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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 76.8

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

   8. Applied rewrites 76.8%

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

   if -5.8e-130 < t1 < 5.1e-42

   1. Initial program 81.0%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 81.0

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

   4. Applied rewrites 81.0%

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

Alternative 9: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-50}:\\ \;\;\;\;-t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -5.8e-130) t_1 (if (<= t1 7e-50) (- (* 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 <= -5.8e-130) {
		tmp = t_1;
	} else if (t1 <= 7e-50) {
		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 <= (-5.8d-130)) then
        tmp = t_1
    else if (t1 <= 7d-50) 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 <= -5.8e-130) {
		tmp = t_1;
	} else if (t1 <= 7e-50) {
		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 <= -5.8e-130:
		tmp = t_1
	elif t1 <= 7e-50:
		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 <= -5.8e-130)
		tmp = t_1;
	elseif (t1 <= 7e-50)
		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 <= -5.8e-130)
		tmp = t_1;
	elseif (t1 <= 7e-50)
		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, -5.8e-130], t$95$1, If[LessEqual[t1, 7e-50], (-N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.8e-130 or 6.99999999999999993e-50 < t1

   1. Initial program 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.8

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

   3. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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 76.6

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

   8. Applied rewrites 76.6%

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

   if -5.8e-130 < t1 < 6.99999999999999993e-50

   1. Initial program 80.8%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 81.4

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

   4. Applied rewrites 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 73.6

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

   6. Applied rewrites 73.6%

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

Alternative 10: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 98.1

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

3. Applied rewrites 98.1%

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

Alternative 11: 55.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 3.1e+101) (/ (- v) t1) (/ (- v) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.1e+101) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 3.1d+101) then
        tmp = -v / t1
    else
        tmp = -v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.1e+101) {
		tmp = -v / t1;
	} else {
		tmp = -v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3.1e+101:
		tmp = -v / t1
	else:
		tmp = -v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3.1e+101)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 3.1e+101)
		tmp = -v / t1;
	else
		tmp = -v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3.1e+101], N[((-v) / t1), $MachinePrecision], N[((-v) / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.09999999999999999e101

   1. Initial program 71.1%

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

   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 59.8

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

   4. Applied rewrites 59.8%

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

   if 3.09999999999999999e101 < u

   1. Initial program 75.1%

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

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. 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 46.0

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

   8. Applied rewrites 46.0%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 35.3%

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

Alternative 12: 61.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 98.1

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

3. Applied rewrites 98.1%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. +-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 98.0

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

5. Applied rewrites 98.0%

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

6. 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 61.1

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

8. Applied rewrites 61.1%

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

Alternative 13: 53.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

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 53.7

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

4. Applied rewrites 53.7%

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

Reproduce

herbie shell --seed 2025098 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))