Rosa's DopplerBench

Percentage Accurate: 73.5% → 96.8%

Time: 6.7s

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: 96.8% 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 73.2%

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

4. Applied rewrites 96.2%

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

5. Final simplification 96.2%

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

Alternative 2: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\mathsf{fma}\left(\mathsf{fma}\left(2, t1, u\right), u, t1 \cdot t1\right)}\\ t_2 := \frac{-1 \cdot v}{t1 - u}\\ \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -6.5 \cdot 10^{-259}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 3.9 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;t1 \leq 1.95 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (fma (fma 2.0 t1 u) u (* t1 t1)))))
        (t_2 (/ (* -1.0 v) (- t1 u))))
   (if (<= t1 -1.5e+81)
     t_2
     (if (<= t1 -6.5e-259)
       t_1
       (if (<= t1 3.9e-176)
         (/ (* (/ (- v) u) t1) u)
         (if (<= t1 1.95e+111) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / fma(fma(2.0, t1, u), u, (t1 * t1)));
	double t_2 = (-1.0 * v) / (t1 - u);
	double tmp;
	if (t1 <= -1.5e+81) {
		tmp = t_2;
	} else if (t1 <= -6.5e-259) {
		tmp = t_1;
	} else if (t1 <= 3.9e-176) {
		tmp = ((-v / u) * t1) / u;
	} else if (t1 <= 1.95e+111) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / fma(fma(2.0, t1, u), u, Float64(t1 * t1))))
	t_2 = Float64(Float64(-1.0 * v) / Float64(t1 - u))
	tmp = 0.0
	if (t1 <= -1.5e+81)
		tmp = t_2;
	elseif (t1 <= -6.5e-259)
		tmp = t_1;
	elseif (t1 <= 3.9e-176)
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	elseif (t1 <= 1.95e+111)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(2.0 * t1 + u), $MachinePrecision] * u + N[(t1 * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 * v), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.5e+81], t$95$2, If[LessEqual[t1, -6.5e-259], t$95$1, If[LessEqual[t1, 3.9e-176], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[t1, 1.95e+111], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.49999999999999999e81 or 1.9499999999999999e111 < t1

   1. Initial program 58.6%

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

   4. Applied rewrites 99.9%

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

   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 89.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. +-commutative N/A

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

   9. Applied rewrites 89.7%

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

   if -1.49999999999999999e81 < t1 < -6.50000000000000045e-259 or 3.8999999999999997e-176 < t1 < 1.9499999999999999e111

   1. Initial program 85.8%

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

   3. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.7

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

   7. Applied rewrites 90.7%

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

   if -6.50000000000000045e-259 < t1 < 3.8999999999999997e-176

   1. Initial program 65.7%

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

   3. Taylor expanded in u around 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-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 87.8%

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

Alternative 3: 88.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \frac{-1 \cdot v}{t1 - u}\\ \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -1.12 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{-220}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
        (t_2 (/ (* -1.0 v) (- t1 u))))
   (if (<= t1 -1.5e+81)
     t_2
     (if (<= t1 -1.12e-111)
       t_1
       (if (<= t1 1.25e-220)
         (* (- v) (/ (/ t1 u) u))
         (if (<= t1 2.1e+110) 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) / (t1 - u);
	double tmp;
	if (t1 <= -1.5e+81) {
		tmp = t_2;
	} else if (t1 <= -1.12e-111) {
		tmp = t_1;
	} else if (t1 <= 1.25e-220) {
		tmp = -v * ((t1 / u) / u);
	} else if (t1 <= 2.1e+110) {
		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) / (t1 - u)
    if (t1 <= (-1.5d+81)) then
        tmp = t_2
    else if (t1 <= (-1.12d-111)) then
        tmp = t_1
    else if (t1 <= 1.25d-220) then
        tmp = -v * ((t1 / u) / u)
    else if (t1 <= 2.1d+110) 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) / (t1 - u);
	double tmp;
	if (t1 <= -1.5e+81) {
		tmp = t_2;
	} else if (t1 <= -1.12e-111) {
		tmp = t_1;
	} else if (t1 <= 1.25e-220) {
		tmp = -v * ((t1 / u) / u);
	} else if (t1 <= 2.1e+110) {
		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) / (t1 - u)
	tmp = 0
	if t1 <= -1.5e+81:
		tmp = t_2
	elif t1 <= -1.12e-111:
		tmp = t_1
	elif t1 <= 1.25e-220:
		tmp = -v * ((t1 / u) / u)
	elif t1 <= 2.1e+110:
		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(t1 - u))
	tmp = 0.0
	if (t1 <= -1.5e+81)
		tmp = t_2;
	elseif (t1 <= -1.12e-111)
		tmp = t_1;
	elseif (t1 <= 1.25e-220)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	elseif (t1 <= 2.1e+110)
		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) / (t1 - u);
	tmp = 0.0;
	if (t1 <= -1.5e+81)
		tmp = t_2;
	elseif (t1 <= -1.12e-111)
		tmp = t_1;
	elseif (t1 <= 1.25e-220)
		tmp = -v * ((t1 / u) / u);
	elseif (t1 <= 2.1e+110)
		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[(t1 - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.5e+81], t$95$2, If[LessEqual[t1, -1.12e-111], t$95$1, If[LessEqual[t1, 1.25e-220], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.1e+110], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.49999999999999999e81 or 2.10000000000000015e110 < t1

   1. Initial program 58.6%

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

   4. Applied rewrites 99.9%

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

   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 89.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. +-commutative N/A

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

   9. Applied rewrites 89.7%

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

   if -1.49999999999999999e81 < t1 < -1.12000000000000009e-111 or 1.25e-220 < t1 < 2.10000000000000015e110

   1. Initial program 88.8%

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

   if -1.12000000000000009e-111 < t1 < 1.25e-220

   1. Initial program 66.0%

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 87.2%

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

Alternative 4: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-83} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{-1 \cdot v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9e-83) (not (<= t1 3.6e+34)))
   (/ (* -1.0 v) (- t1 u))
   (/ (* (/ t1 u) v) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9e-83) || !(t1 <= 3.6e+34)) {
		tmp = (-1.0 * v) / (t1 - u);
	} else {
		tmp = ((t1 / u) * v) / -u;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-9d-83)) .or. (.not. (t1 <= 3.6d+34))) then
        tmp = ((-1.0d0) * v) / (t1 - u)
    else
        tmp = ((t1 / u) * v) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9e-83) || !(t1 <= 3.6e+34)) {
		tmp = (-1.0 * v) / (t1 - u);
	} else {
		tmp = ((t1 / u) * v) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9e-83) or not (t1 <= 3.6e+34):
		tmp = (-1.0 * v) / (t1 - u)
	else:
		tmp = ((t1 / u) * v) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9e-83) || !(t1 <= 3.6e+34))
		tmp = Float64(Float64(-1.0 * v) / Float64(t1 - u));
	else
		tmp = Float64(Float64(Float64(t1 / u) * v) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9e-83) || ~((t1 <= 3.6e+34)))
		tmp = (-1.0 * v) / (t1 - u);
	else
		tmp = ((t1 / u) * v) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9e-83], N[Not[LessEqual[t1, 3.6e+34]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * v), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.99999999999999995e-83 or 3.6e34 < t1

   1. Initial program 70.4%

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

   4. Applied rewrites 99.9%

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

   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 85.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. +-commutative N/A

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

   9. Applied rewrites 85.2%

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

   if -8.99999999999999995e-83 < t1 < 3.6e34

   1. Initial program 76.0%

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 81.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-83} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{-1 \cdot v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-83} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{-1 \cdot v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9e-83) (not (<= t1 3.6e+34)))
   (/ (* -1.0 v) (- t1 u))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9e-83) || !(t1 <= 3.6e+34)) {
		tmp = (-1.0 * v) / (t1 - u);
	} 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 <= (-9d-83)) .or. (.not. (t1 <= 3.6d+34))) then
        tmp = ((-1.0d0) * v) / (t1 - u)
    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 <= -9e-83) || !(t1 <= 3.6e+34)) {
		tmp = (-1.0 * v) / (t1 - u);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9e-83) or not (t1 <= 3.6e+34):
		tmp = (-1.0 * v) / (t1 - u)
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9e-83) || !(t1 <= 3.6e+34))
		tmp = Float64(Float64(-1.0 * v) / Float64(t1 - u));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9e-83) || ~((t1 <= 3.6e+34)))
		tmp = (-1.0 * v) / (t1 - u);
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9e-83], N[Not[LessEqual[t1, 3.6e+34]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.99999999999999995e-83 or 3.6e34 < t1

   1. Initial program 70.4%

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

   4. Applied rewrites 99.9%

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

   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 85.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. +-commutative N/A

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

   9. Applied rewrites 85.2%

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

   if -8.99999999999999995e-83 < t1 < 3.6e34

   1. Initial program 76.0%

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-83} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{-1 \cdot v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-83} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-1 \cdot v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9e-83) (not (<= t1 1.35e-16)))
   (/ (* -1.0 v) (- t1 u))
   (* (- v) (/ (/ t1 u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9e-83) || !(t1 <= 1.35e-16)) {
		tmp = (-1.0 * v) / (t1 - u);
	} 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 <= (-9d-83)) .or. (.not. (t1 <= 1.35d-16))) then
        tmp = ((-1.0d0) * v) / (t1 - u)
    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 <= -9e-83) || !(t1 <= 1.35e-16)) {
		tmp = (-1.0 * v) / (t1 - u);
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9e-83) or not (t1 <= 1.35e-16):
		tmp = (-1.0 * v) / (t1 - u)
	else:
		tmp = -v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9e-83) || !(t1 <= 1.35e-16))
		tmp = Float64(Float64(-1.0 * v) / Float64(t1 - u));
	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 <= -9e-83) || ~((t1 <= 1.35e-16)))
		tmp = (-1.0 * v) / (t1 - u);
	else
		tmp = -v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9e-83], N[Not[LessEqual[t1, 1.35e-16]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.99999999999999995e-83 or 1.35e-16 < t1

   1. Initial program 71.5%

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

   4. Applied rewrites 99.3%

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

   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 82.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. +-commutative N/A

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

   9. Applied rewrites 82.1%

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

   if -8.99999999999999995e-83 < t1 < 1.35e-16

   1. Initial program 75.3%

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

   3. Taylor expanded in u around 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-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 83.6

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

   5. Applied rewrites 83.6%

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

   7. Applied rewrites 81.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-83} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-1 \cdot v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-83} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-1 \cdot v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9e-83) (not (<= t1 1.35e-16)))
   (/ (* -1.0 v) (- t1 u))
   (* (- t1) (/ v (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9e-83) || !(t1 <= 1.35e-16)) {
		tmp = (-1.0 * v) / (t1 - u);
	} 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 <= (-9d-83)) .or. (.not. (t1 <= 1.35d-16))) then
        tmp = ((-1.0d0) * v) / (t1 - u)
    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 <= -9e-83) || !(t1 <= 1.35e-16)) {
		tmp = (-1.0 * v) / (t1 - u);
	} else {
		tmp = -t1 * (v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9e-83) or not (t1 <= 1.35e-16):
		tmp = (-1.0 * v) / (t1 - u)
	else:
		tmp = -t1 * (v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9e-83) || !(t1 <= 1.35e-16))
		tmp = Float64(Float64(-1.0 * v) / Float64(t1 - u));
	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 <= -9e-83) || ~((t1 <= 1.35e-16)))
		tmp = (-1.0 * v) / (t1 - u);
	else
		tmp = -t1 * (v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9e-83], N[Not[LessEqual[t1, 1.35e-16]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.99999999999999995e-83 or 1.35e-16 < t1

   1. Initial program 71.5%

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

   4. Applied rewrites 99.3%

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

   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 82.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-frac N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. distribute-neg-in N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-neg.f64 N/A

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

      13. sqr-neg-rev N/A

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

      14. +-commutative N/A

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

   9. Applied rewrites 82.1%

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

   if -8.99999999999999995e-83 < t1 < 1.35e-16

   1. Initial program 75.3%

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

   3. Taylor expanded in u around 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-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 75.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-83} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{-1 \cdot v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+116} \lor \neg \left(u \leq 6 \cdot 10^{+138}\right):\\ \;\;\;\;v \cdot \frac{-1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.2e+116) (not (<= u 6e+138)))
   (* v (/ -1.0 (- u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+116) || !(u <= 6e+138)) {
		tmp = v * (-1.0 / -u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.2d+116)) .or. (.not. (u <= 6d+138))) then
        tmp = v * ((-1.0d0) / -u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+116) || !(u <= 6e+138)) {
		tmp = v * (-1.0 / -u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.2e+116) or not (u <= 6e+138):
		tmp = v * (-1.0 / -u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.2e+116) || !(u <= 6e+138))
		tmp = Float64(v * Float64(-1.0 / Float64(-u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.2e+116) || ~((u <= 6e+138)))
		tmp = v * (-1.0 / -u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.2e+116], N[Not[LessEqual[u, 6e+138]], $MachinePrecision]], N[(v * N[(-1.0 / (-u)), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.2e116 or 6.0000000000000002e138 < u

   1. Initial program 77.2%

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

   4. Applied rewrites 98.4%

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

   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 51.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-frac N/A

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

   9. Applied rewrites 51.7%

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

   10. Taylor expanded in u around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 45.0

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

   12. Applied rewrites 45.0%

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

   if -1.2e116 < u < 6.0000000000000002e138

   1. Initial program 71.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 61.3

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

   5. Applied rewrites 61.3%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+116} \lor \neg \left(u \leq 6 \cdot 10^{+138}\right):\\ \;\;\;\;v \cdot \frac{-1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

4. Applied rewrites 96.2%

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

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 59.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. +-commutative N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-*.f64 N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. distribute-neg-in N/A

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

   10. lift-neg.f64 N/A

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

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

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

   12. lift-neg.f64 N/A

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

   13. sqr-neg-rev N/A

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

   14. +-commutative N/A

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

9. Applied rewrites 59.0%

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

Alternative 10: 61.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

4. Applied rewrites 96.2%

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

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 59.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

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

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

   6. remove-double-neg N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lift--.f64 N/A

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

   10. flip-- N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-frac N/A

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

9. Applied rewrites 59.0%

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

Alternative 11: 54.1% 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 73.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 48.4

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

5. Applied rewrites 48.4%

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

Reproduce

herbie shell --seed 2024354 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))