Rosa's DopplerBench

Percentage Accurate: 73.5% → 98.1%

Time: 4.5s

Alternatives: 11

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: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.8× speedup

Math

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

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[(N[(t1 / N[(u + t1), $MachinePrecision]), $MachinePrecision] * (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing
3. 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 98.4

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

4. Applied rewrites 98.4%

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

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 98.7

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 2: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{-v}{\mathsf{fma}\left(\mathsf{fma}\left(2, u, t1\right), t1, u \cdot u\right)}\\ \mathbf{if}\;t1 \leq -3.75 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}{u + t1}\\ \mathbf{elif}\;t1 \leq -3.2 \cdot 10^{-98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3.7 \cdot 10^{-176}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (- v) (fma (fma 2.0 u t1) t1 (* u u))))))
   (if (<= t1 -3.75e+124)
     (/ (fma u (/ v t1) (- v)) (+ u t1))
     (if (<= t1 -3.2e-98)
       t_1
       (if (<= t1 3.7e-176)
         (/ (* v (/ (- t1) u)) u)
         (if (<= t1 1.55e+110) t_1 (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (-v / fma(fma(2.0, u, t1), t1, (u * u)));
	double tmp;
	if (t1 <= -3.75e+124) {
		tmp = fma(u, (v / t1), -v) / (u + t1);
	} else if (t1 <= -3.2e-98) {
		tmp = t_1;
	} else if (t1 <= 3.7e-176) {
		tmp = (v * (-t1 / u)) / u;
	} else if (t1 <= 1.55e+110) {
		tmp = t_1;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(Float64(-v) / fma(fma(2.0, u, t1), t1, Float64(u * u))))
	tmp = 0.0
	if (t1 <= -3.75e+124)
		tmp = Float64(fma(u, Float64(v / t1), Float64(-v)) / Float64(u + t1));
	elseif (t1 <= -3.2e-98)
		tmp = t_1;
	elseif (t1 <= 3.7e-176)
		tmp = Float64(Float64(v * Float64(Float64(-t1) / u)) / u);
	elseif (t1 <= 1.55e+110)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[((-v) / N[(N[(2.0 * u + t1), $MachinePrecision] * t1 + N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.75e+124], N[(N[(u * N[(v / t1), $MachinePrecision] + (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -3.2e-98], t$95$1, If[LessEqual[t1, 3.7e-176], N[(N[(v * N[((-t1) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[t1, 1.55e+110], t$95$1, N[((-v) / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -3.75000000000000019e124

   1. Initial program 45.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 68.9

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

   4. Applied rewrites 68.9%

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

   5. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 96.8

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

   7. Applied rewrites 96.8%

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

   if -3.75000000000000019e124 < t1 < -3.2000000000000001e-98 or 3.69999999999999984e-176 < t1 < 1.55000000000000009e110

   1. Initial program 86.9%

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

   3. Taylor expanded in u around 0

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

   5. Applied rewrites 58.0%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \color{blue}{t1}} \]
   6. 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. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 63.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 63.9

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

      11. +-commutative 63.9

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

   7. Applied rewrites 63.9%

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

   8. Taylor expanded in t1 around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 91.4

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

   10. Applied rewrites 91.4%

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

   if -3.2000000000000001e-98 < t1 < 3.69999999999999984e-176

   1. Initial program 81.9%

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

   3. Taylor expanded in u around inf

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

      1. 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 89.8

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

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\frac{v}{u} \cdot \frac{-t1}{u}} \]
   6. 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 91.0

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

   7. Applied rewrites 91.0%

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

   if 1.55000000000000009e110 < t1

   1. Initial program 48.9%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      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 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.75 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}{u + t1}\\ \mathbf{elif}\;t1 \leq -3.2 \cdot 10^{-98}:\\ \;\;\;\;t1 \cdot \frac{-v}{\mathsf{fma}\left(\mathsf{fma}\left(2, u, t1\right), t1, u \cdot u\right)}\\ \mathbf{elif}\;t1 \leq 3.7 \cdot 10^{-176}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;t1 \cdot \frac{-v}{\mathsf{fma}\left(\mathsf{fma}\left(2, u, t1\right), t1, u \cdot u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.3 \cdot 10^{+129} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.3e+129) (not (<= t1 3.6e+85)))
   (/ (fma u (/ v t1) (- v)) (+ u t1))
   (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.3e+129) || !(t1 <= 3.6e+85)) {
		tmp = fma(u, (v / t1), -v) / (u + t1);
	} else {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.3e+129) || !(t1 <= 3.6e+85))
		tmp = Float64(fma(u, Float64(v / t1), Float64(-v)) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.3e+129], N[Not[LessEqual[t1, 3.6e+85]], $MachinePrecision]], N[(N[(u * N[(v / t1), $MachinePrecision] + (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.2999999999999999e129 or 3.5999999999999998e85 < t1

   1. Initial program 47.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 92.9

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

   7. Applied rewrites 92.9%

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

   if -2.2999999999999999e129 < t1 < 3.5999999999999998e85

   1. Initial program 85.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.3 \cdot 10^{+129} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(u, \frac{v}{t1}, -v\right)}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8.2 \cdot 10^{+119} \lor \neg \left(t1 \leq 1.45 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -8.2e+119) (not (<= t1 1.45e+41)))
   (/ (- v) (+ u t1))
   (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8.2e+119) || !(t1 <= 1.45e+41)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-8.2d+119)) .or. (.not. (t1 <= 1.45d+41))) then
        tmp = -v / (u + t1)
    else
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8.2e+119) || !(t1 <= 1.45e+41)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -8.2e+119) or not (t1 <= 1.45e+41):
		tmp = -v / (u + t1)
	else:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -8.2e+119) || !(t1 <= 1.45e+41))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -8.2e+119) || ~((t1 <= 1.45e+41)))
		tmp = -v / (u + t1);
	else
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -8.2e+119], N[Not[LessEqual[t1, 1.45e+41]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.1999999999999994e119 or 1.44999999999999994e41 < t1

   1. Initial program 54.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 71.2

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 91.5

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

   7. Applied rewrites 91.5%

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

   if -8.1999999999999994e119 < t1 < 1.44999999999999994e41

   1. Initial program 85.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.2 \cdot 10^{+119} \lor \neg \left(t1 \leq 1.45 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-47} \lor \neg \left(t1 \leq 9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.2e-47) (not (<= t1 9e-107)))
   (/ (- v) (+ u t1))
   (/ (* v (/ (- t1) u)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (v * (-t1 / u)) / u;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.2d-47)) .or. (.not. (t1 <= 9d-107))) then
        tmp = -v / (u + t1)
    else
        tmp = (v * (-t1 / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (v * (-t1 / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.2e-47) or not (t1 <= 9e-107):
		tmp = -v / (u + t1)
	else:
		tmp = (v * (-t1 / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(v * Float64(Float64(-t1) / u)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.2e-47) || ~((t1 <= 9e-107)))
		tmp = -v / (u + t1);
	else
		tmp = (v * (-t1 / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.2e-47], N[Not[LessEqual[t1, 9e-107]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[((-t1) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.20000000000000019e-47 or 9.00000000000000032e-107 < t1

   1. Initial program 70.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if -2.20000000000000019e-47 < t1 < 9.00000000000000032e-107

   1. Initial program 84.8%

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

   3. Taylor expanded in u around inf

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

      1. 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 83.1

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

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{v}{u} \cdot \frac{-t1}{u}} \]
   6. 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 84.0

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

   7. Applied rewrites 84.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-47} \lor \neg \left(t1 \leq 9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-47} \lor \neg \left(t1 \leq 9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.2e-47) (not (<= t1 9e-107)))
   (/ (- v) (+ u t1))
   (* (/ (- v) u) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-v / u) * (t1 / u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.2d-47)) .or. (.not. (t1 <= 9d-107))) then
        tmp = -v / (u + t1)
    else
        tmp = (-v / u) * (t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-v / u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.2e-47) or not (t1 <= 9e-107):
		tmp = -v / (u + t1)
	else:
		tmp = (-v / u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.2e-47) || ~((t1 <= 9e-107)))
		tmp = -v / (u + t1);
	else
		tmp = (-v / u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.2e-47], N[Not[LessEqual[t1, 9e-107]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.20000000000000019e-47 or 9.00000000000000032e-107 < t1

   1. Initial program 70.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if -2.20000000000000019e-47 < t1 < 9.00000000000000032e-107

   1. Initial program 84.8%

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

   3. Taylor expanded in u around inf

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

      1. 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 83.1

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

   5. Applied rewrites 83.1%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-47} \lor \neg \left(t1 \leq 9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-47} \lor \neg \left(t1 \leq 9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.2e-47) (not (<= t1 9e-107)))
   (/ (- v) (+ u t1))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.2d-47)) .or. (.not. (t1 <= 9d-107))) then
        tmp = -v / (u + t1)
    else
        tmp = (-t1 * v) / (u * u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.2e-47) or not (t1 <= 9e-107):
		tmp = -v / (u + t1)
	else:
		tmp = (-t1 * v) / (u * u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.2e-47) || !(t1 <= 9e-107))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.2e-47) || ~((t1 <= 9e-107)))
		tmp = -v / (u + t1);
	else
		tmp = (-t1 * v) / (u * u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.2e-47], N[Not[LessEqual[t1, 9e-107]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.20000000000000019e-47 or 9.00000000000000032e-107 < t1

   1. Initial program 70.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if -2.20000000000000019e-47 < t1 < 9.00000000000000032e-107

   1. Initial program 84.8%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-47} \lor \neg \left(t1 \leq 9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-48} \lor \neg \left(t1 \leq 9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.2e-48) (not (<= t1 9e-107)))
   (/ (- v) (+ u t1))
   (* (- t1) (/ v (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.2e-48) || !(t1 <= 9e-107)) {
		tmp = -v / (u + t1);
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-9.2d-48)) .or. (.not. (t1 <= 9d-107))) then
        tmp = -v / (u + t1)
    else
        tmp = -t1 * (v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.2e-48) || !(t1 <= 9e-107)) {
		tmp = -v / (u + t1);
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.2e-48) or not (t1 <= 9e-107):
		tmp = -v / (u + t1)
	else:
		tmp = -t1 * (v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.2e-48) || !(t1 <= 9e-107))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9.2e-48) || ~((t1 <= 9e-107)))
		tmp = -v / (u + t1);
	else
		tmp = -t1 * (v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.2e-48], N[Not[LessEqual[t1, 9e-107]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.2000000000000003e-48 or 9.00000000000000032e-107 < t1

   1. Initial program 70.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing
   3. 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. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if -9.2000000000000003e-48 < t1 < 9.00000000000000032e-107

   1. Initial program 84.8%

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

   3. Taylor expanded in u around inf

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

      1. 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 83.1

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

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{v}{u} \cdot \frac{-t1}{u}} \]
   6. 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 75.6

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

   7. Applied rewrites 75.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-48} \lor \neg \left(t1 \leq 9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% 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 75.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing
3. 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 98.4

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

4. Applied rewrites 98.4%

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

Alternative 10: 61.8% 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 75.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing
3. 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. distribute-lft-neg-out N/A

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

   5. mul-1-neg N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. mul-1-neg N/A

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

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

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

   14. lift-*.f64 N/A

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

   15. lift-neg.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 83.4

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

4. Applied rewrites 83.4%

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

5. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 62.7

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

7. Applied rewrites 62.7%

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

Alternative 11: 53.6% 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 75.7%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 55.4

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

5. Applied rewrites 55.4%

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

Reproduce

herbie shell --seed 2025060 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))