Rosa's DopplerBench

Percentage Accurate: 72.3% → 96.2%

Time: 4.4s

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: 72.3% 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: 96.2% accurate, 0.8× speedup

Math

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

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)) * v) / Float64(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 69.3%

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

4. Applied rewrites 94.6%

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

5. Final simplification 94.6%

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

Alternative 2: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq -5.5 \cdot 10^{-204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{-172}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1} \cdot u, 2, -v\right)}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -1.8e+41)
     (/ (* -1.0 v) (+ (- u) t1))
     (if (<= t1 -5.5e-204)
       t_1
       (if (<= t1 1.75e-172)
         (* v (/ (/ (- t1) u) u))
         (if (<= t1 4.2e+76) t_1 (/ (fma (* (/ v t1) u) 2.0 (- v)) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -1.8e+41) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= -5.5e-204) {
		tmp = t_1;
	} else if (t1 <= 1.75e-172) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 4.2e+76) {
		tmp = t_1;
	} else {
		tmp = fma(((v / t1) * u), 2.0, -v) / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -1.8e+41)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= -5.5e-204)
		tmp = t_1;
	elseif (t1 <= 1.75e-172)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	elseif (t1 <= 4.2e+76)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(Float64(v / t1) * u), 2.0, Float64(-v)) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.8e+41], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -5.5e-204], t$95$1, If[LessEqual[t1, 1.75e-172], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.2e+76], t$95$1, N[(N[(N[(N[(v / t1), $MachinePrecision] * u), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.80000000000000013e41

   1. Initial program 60.7%

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 93.1%

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

   if -1.80000000000000013e41 < t1 < -5.4999999999999999e-204 or 1.75000000000000014e-172 < t1 < 4.20000000000000013e76

   1. Initial program 92.1%

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

   if -5.4999999999999999e-204 < t1 < 1.75000000000000014e-172

   1. Initial program 66.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 84.9%

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

   if 4.20000000000000013e76 < t1

   1. Initial program 43.9%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.8

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

   5. Applied rewrites 88.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq -5.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{-172}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1} \cdot u, 2, -v\right)}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq -5.5 \cdot 10^{-204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{-172}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\mathsf{fma}\left(\frac{u}{t1}, -2, 1\right)}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -1.8e+41)
     (/ (* -1.0 v) (+ (- u) t1))
     (if (<= t1 -5.5e-204)
       t_1
       (if (<= t1 1.75e-172)
         (* v (/ (/ (- t1) u) u))
         (if (<= t1 6.2e+76) t_1 (* (- v) (/ (fma (/ u t1) -2.0 1.0) t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -1.8e+41) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= -5.5e-204) {
		tmp = t_1;
	} else if (t1 <= 1.75e-172) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 6.2e+76) {
		tmp = t_1;
	} else {
		tmp = -v * (fma((u / t1), -2.0, 1.0) / t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -1.8e+41)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= -5.5e-204)
		tmp = t_1;
	elseif (t1 <= 1.75e-172)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	elseif (t1 <= 6.2e+76)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) * Float64(fma(Float64(u / t1), -2.0, 1.0) / t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.8e+41], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -5.5e-204], t$95$1, If[LessEqual[t1, 1.75e-172], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.2e+76], t$95$1, N[((-v) * N[(N[(N[(u / t1), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.80000000000000013e41

   1. Initial program 60.7%

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 93.1%

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

   if -1.80000000000000013e41 < t1 < -5.4999999999999999e-204 or 1.75000000000000014e-172 < t1 < 6.20000000000000023e76

   1. Initial program 92.1%

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

   if -5.4999999999999999e-204 < t1 < 1.75000000000000014e-172

   1. Initial program 66.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 84.9%

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

   if 6.20000000000000023e76 < t1

   1. Initial program 43.9%

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

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.0

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

   5. Applied rewrites 88.0%

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-neg-in N/A

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

      8. div-add N/A

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

      9. *-lft-identity N/A

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

      10. associate-*r/ N/A

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

      11. associate-*r* N/A

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

      12. associate-*l/ N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-in N/A

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

      15. associate-/l* N/A

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

      16. distribute-lft-neg-in N/A

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

      17. mul-1-neg N/A

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

   10. Applied rewrites 88.6%

       \[\leadsto \color{blue}{\left(-v\right) \cdot \frac{\mathsf{fma}\left(\frac{u}{t1}, -2, 1\right)}{t1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq -5.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{-172}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\mathsf{fma}\left(\frac{u}{t1}, -2, 1\right)}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -5.5 \cdot 10^{-204}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{-172}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 9.6 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
        (t_2 (/ (* -1.0 v) (+ (- u) t1))))
   (if (<= t1 -1.8e+41)
     t_2
     (if (<= t1 -5.5e-204)
       t_1
       (if (<= t1 1.75e-172)
         (* v (/ (/ (- t1) u) u))
         (if (<= t1 9.6e+69) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double t_2 = (-1.0 * v) / (-u + t1);
	double tmp;
	if (t1 <= -1.8e+41) {
		tmp = t_2;
	} else if (t1 <= -5.5e-204) {
		tmp = t_1;
	} else if (t1 <= 1.75e-172) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 9.6e+69) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
    t_2 = ((-1.0d0) * v) / (-u + t1)
    if (t1 <= (-1.8d+41)) then
        tmp = t_2
    else if (t1 <= (-5.5d-204)) then
        tmp = t_1
    else if (t1 <= 1.75d-172) then
        tmp = v * ((-t1 / u) / u)
    else if (t1 <= 9.6d+69) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double t_2 = (-1.0 * v) / (-u + t1);
	double tmp;
	if (t1 <= -1.8e+41) {
		tmp = t_2;
	} else if (t1 <= -5.5e-204) {
		tmp = t_1;
	} else if (t1 <= 1.75e-172) {
		tmp = v * ((-t1 / u) / u);
	} else if (t1 <= 9.6e+69) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
	t_2 = (-1.0 * v) / (-u + t1)
	tmp = 0
	if t1 <= -1.8e+41:
		tmp = t_2
	elif t1 <= -5.5e-204:
		tmp = t_1
	elif t1 <= 1.75e-172:
		tmp = v * ((-t1 / u) / u)
	elif t1 <= 9.6e+69:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	t_2 = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1))
	tmp = 0.0
	if (t1 <= -1.8e+41)
		tmp = t_2;
	elseif (t1 <= -5.5e-204)
		tmp = t_1;
	elseif (t1 <= 1.75e-172)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	elseif (t1 <= 9.6e+69)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	t_2 = (-1.0 * v) / (-u + t1);
	tmp = 0.0;
	if (t1 <= -1.8e+41)
		tmp = t_2;
	elseif (t1 <= -5.5e-204)
		tmp = t_1;
	elseif (t1 <= 1.75e-172)
		tmp = v * ((-t1 / u) / u);
	elseif (t1 <= 9.6e+69)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.8e+41], t$95$2, If[LessEqual[t1, -5.5e-204], t$95$1, If[LessEqual[t1, 1.75e-172], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 9.6e+69], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.80000000000000013e41 or 9.6000000000000007e69 < t1

   1. Initial program 52.0%

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 90.5%

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

   if -1.80000000000000013e41 < t1 < -5.4999999999999999e-204 or 1.75000000000000014e-172 < t1 < 9.6000000000000007e69

   1. Initial program 92.1%

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

   if -5.4999999999999999e-204 < t1 < 1.75000000000000014e-172

   1. Initial program 66.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 84.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq -5.5 \cdot 10^{-204}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{-172}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 9.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -4.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{-v}{u}}{u} \cdot t1\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* -1.0 v) (+ (- u) t1))))
   (if (<= t1 -2.8e+16)
     t_1
     (if (<= t1 -4.8e-132)
       (* (/ (/ (- v) u) u) t1)
       (if (<= t1 1.7e+54) (* v (/ (/ (- t1) u) u)) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = (-1.0 * v) / (-u + t1);
	double tmp;
	if (t1 <= -2.8e+16) {
		tmp = t_1;
	} else if (t1 <= -4.8e-132) {
		tmp = ((-v / u) / u) * t1;
	} else if (t1 <= 1.7e+54) {
		tmp = v * ((-t1 / u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-1.0d0) * v) / (-u + t1)
    if (t1 <= (-2.8d+16)) then
        tmp = t_1
    else if (t1 <= (-4.8d-132)) then
        tmp = ((-v / u) / u) * t1
    else if (t1 <= 1.7d+54) then
        tmp = v * ((-t1 / u) / u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-1.0 * v) / (-u + t1);
	double tmp;
	if (t1 <= -2.8e+16) {
		tmp = t_1;
	} else if (t1 <= -4.8e-132) {
		tmp = ((-v / u) / u) * t1;
	} else if (t1 <= 1.7e+54) {
		tmp = v * ((-t1 / u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-1.0 * v) / (-u + t1)
	tmp = 0
	if t1 <= -2.8e+16:
		tmp = t_1
	elif t1 <= -4.8e-132:
		tmp = ((-v / u) / u) * t1
	elif t1 <= 1.7e+54:
		tmp = v * ((-t1 / u) / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1))
	tmp = 0.0
	if (t1 <= -2.8e+16)
		tmp = t_1;
	elseif (t1 <= -4.8e-132)
		tmp = Float64(Float64(Float64(Float64(-v) / u) / u) * t1);
	elseif (t1 <= 1.7e+54)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-1.0 * v) / (-u + t1);
	tmp = 0.0;
	if (t1 <= -2.8e+16)
		tmp = t_1;
	elseif (t1 <= -4.8e-132)
		tmp = ((-v / u) / u) * t1;
	elseif (t1 <= 1.7e+54)
		tmp = v * ((-t1 / u) / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.8e+16], t$95$1, If[LessEqual[t1, -4.8e-132], N[(N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision] * t1), $MachinePrecision], If[LessEqual[t1, 1.7e+54], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.8e16 or 1.7e54 < t1

   1. Initial program 52.5%

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 90.6%

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

   if -2.8e16 < t1 < -4.80000000000000031e-132

   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 71.3%

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

   if -4.80000000000000031e-132 < t1 < 1.7e54

   1. Initial program 80.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 81.4

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 83.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq -4.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{-v}{u}}{u} \cdot t1\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+16} \lor \neg \left(t1 \leq 3.05 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.8e+16) (not (<= t1 3.05e+56)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (/ (- v) u) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e+16) || !(t1 <= 3.05e+56)) {
		tmp = (-1.0 * 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 <= (-2.8d+16)) .or. (.not. (t1 <= 3.05d+56))) then
        tmp = ((-1.0d0) * 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 <= -2.8e+16) || !(t1 <= 3.05e+56)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-v / u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.8e+16) or not (t1 <= 3.05e+56):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (-v / u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.8e+16) || !(t1 <= 3.05e+56))
		tmp = Float64(Float64(-1.0 * v) / Float64(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 <= -2.8e+16) || ~((t1 <= 3.05e+56)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (-v / u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.8e+16], N[Not[LessEqual[t1, 3.05e+56]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / 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 < -2.8e16 or 3.0500000000000001e56 < t1

   1. Initial program 52.5%

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 90.6%

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

   if -2.8e16 < t1 < 3.0500000000000001e56

   1. Initial program 81.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 79.2

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

   5. Applied rewrites 79.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+16} \lor \neg \left(t1 \leq 3.05 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+16} \lor \neg \left(t1 \leq 1.7 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u}}{u} \cdot t1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.8e+16) (not (<= t1 1.7e+54)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (/ (/ (- v) u) u) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e+16) || !(t1 <= 1.7e+54)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((-v / u) / u) * t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.8d+16)) .or. (.not. (t1 <= 1.7d+54))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = ((-v / u) / u) * t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e+16) || !(t1 <= 1.7e+54)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((-v / u) / u) * t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.8e+16) or not (t1 <= 1.7e+54):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = ((-v / u) / u) * t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.8e+16) || !(t1 <= 1.7e+54))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(Float64(-v) / u) / u) * t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.8e+16) || ~((t1 <= 1.7e+54)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = ((-v / u) / u) * t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.8e+16], N[Not[LessEqual[t1, 1.7e+54]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision] * t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.8e16 or 1.7e54 < t1

   1. Initial program 52.5%

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 90.6%

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

   if -2.8e16 < t1 < 1.7e54

   1. Initial program 81.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 77.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+16} \lor \neg \left(t1 \leq 1.7 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u}}{u} \cdot t1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+14} \lor \neg \left(t1 \leq 5.6 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.7e+14) (not (<= t1 5.6e+14)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (- v) (/ t1 (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.7e+14) || !(t1 <= 5.6e+14)) {
		tmp = (-1.0 * 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 <= (-4.7d+14)) .or. (.not. (t1 <= 5.6d+14))) then
        tmp = ((-1.0d0) * 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 <= -4.7e+14) || !(t1 <= 5.6e+14)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = -v * (t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.7e+14) or not (t1 <= 5.6e+14):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = -v * (t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.7e+14) || !(t1 <= 5.6e+14))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.7e+14) || ~((t1 <= 5.6e+14)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = -v * (t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.7e+14], N[Not[LessEqual[t1, 5.6e+14]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.7e14 or 5.6e14 < t1

   1. Initial program 53.7%

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 87.8%

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

   if -4.7e14 < t1 < 5.6e14

   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 80.2

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 71.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+14} \lor \neg \left(t1 \leq 5.6 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{-1 \cdot v}{\left(-u\right) + t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* -1.0 v) (+ (- u) t1)))

C

double code(double u, double v, double t1) {
	return (-1.0 * 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 = ((-1.0d0) * v) / (-u + t1)
end function

Java

public static double code(double u, double v, double t1) {
	return (-1.0 * v) / (-u + t1);
}

Python

def code(u, v, t1):
	return (-1.0 * v) / (-u + t1)

Julia

function code(u, v, t1)
	return Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-1.0 * v) / (-u + t1);
end

Wolfram

code[u_, v_, t1_] := N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

4. Applied rewrites 94.6%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 59.5%

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

8. Final simplification 59.5%

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

Alternative 10: 60.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(-v\right) \cdot \frac{-1}{u - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (* (- v) (/ -1.0 (- u t1))))

C

double code(double u, double v, double t1) {
	return -v * (-1.0 / (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 * ((-1.0d0) / (u - t1))
end function

Java

public static double code(double u, double v, double t1) {
	return -v * (-1.0 / (u - t1));
}

Python

def code(u, v, t1):
	return -v * (-1.0 / (u - t1))

Julia

function code(u, v, t1)
	return Float64(Float64(-v) * Float64(-1.0 / Float64(u - t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = -v * (-1.0 / (u - t1));
end

Wolfram

code[u_, v_, t1_] := N[((-v) * N[(-1.0 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

4. Applied rewrites 94.6%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 59.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 59.4

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

9. Applied rewrites 59.4%

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

Alternative 11: 52.9% 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 69.3%

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

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 52.5

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

5. Applied rewrites 52.5%

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

Reproduce

herbie shell --seed 2025017 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))