Rosa's DopplerBench

Percentage Accurate: 72.4% → 97.8%

Time: 6.2s

Alternatives: 9

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% 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: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.4%

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

3. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 75.9

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

5. Applied rewrites 75.9%

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

7. Applied rewrites 98.5%

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

8. Final simplification 98.5%

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

Alternative 2: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.6 \cdot 10^{+203}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{v}{u + t1}}{u + t1} \cdot \left(-t1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u, -v\right)}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.6e+203)
   (/ (- v) t1)
   (if (<= t1 7e+168)
     (* (/ (/ v (+ u t1)) (+ u t1)) (- t1))
     (/ (fma (/ v t1) u (- v)) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.6e+203) {
		tmp = -v / t1;
	} else if (t1 <= 7e+168) {
		tmp = ((v / (u + t1)) / (u + t1)) * -t1;
	} else {
		tmp = fma((v / t1), u, -v) / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.6e+203)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 7e+168)
		tmp = Float64(Float64(Float64(v / Float64(u + t1)) / Float64(u + t1)) * Float64(-t1));
	else
		tmp = Float64(fma(Float64(v / t1), u, Float64(-v)) / Float64(u + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.6e+203], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 7e+168], N[(N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision] * (-t1)), $MachinePrecision], N[(N[(N[(v / t1), $MachinePrecision] * u + (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.5999999999999998e203

   1. Initial program 34.5%

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

   3. Taylor expanded in u around 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 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.5999999999999998e203 < t1 < 7.0000000000000004e168

   1. Initial program 79.2%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 83.0

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

   5. Applied rewrites 83.0%

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

   7. Applied rewrites 93.2%

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

   if 7.0000000000000004e168 < t1

   1. Initial program 40.7%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 42.3

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

   5. Applied rewrites 42.3%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 95.3%

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

Alternative 3: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+108} \lor \neg \left(t1 \leq 4.7 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot t1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.7e+108) (not (<= t1 4.7e+141)))
   (/ (- v) (+ u t1))
   (* (/ (- v) (* (+ u t1) (+ u t1))) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.7e+108) || !(t1 <= 4.7e+141)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.7d+108)) .or. (.not. (t1 <= 4.7d+141))) then
        tmp = -v / (u + t1)
    else
        tmp = (-v / ((u + t1) * (u + t1))) * t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.7e+108) || !(t1 <= 4.7e+141)) {
		tmp = -v / (u + t1);
	} else {
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.7e+108) or not (t1 <= 4.7e+141):
		tmp = -v / (u + t1)
	else:
		tmp = (-v / ((u + t1) * (u + t1))) * t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.7e+108) || !(t1 <= 4.7e+141))
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(Float64(-v) / Float64(Float64(u + t1) * Float64(u + t1))) * t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.7e+108) || ~((t1 <= 4.7e+141)))
		tmp = -v / (u + t1);
	else
		tmp = (-v / ((u + t1) * (u + t1))) * t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.7e+108], N[Not[LessEqual[t1, 4.7e+141]], $MachinePrecision]], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-v) / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.7e108 or 4.69999999999999979e141 < t1

   1. Initial program 47.9%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 51.3

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

   5. Applied rewrites 51.3%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 94.0%

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

   if -2.7e108 < t1 < 4.69999999999999979e141

   1. Initial program 81.4%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+108} \lor \neg \left(t1 \leq 4.7 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot t1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5e-30)
   (/ (* -1.0 v) (+ (- u) t1))
   (if (<= t1 1.9e-62) (* (- v) (/ (/ t1 u) u)) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-30) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.9e-62) {
		tmp = -v * ((t1 / u) / u);
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5d-30)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= 1.9d-62) then
        tmp = -v * ((t1 / u) / u)
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-30) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.9e-62) {
		tmp = -v * ((t1 / u) / u);
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5e-30:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= 1.9e-62:
		tmp = -v * ((t1 / u) / u)
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5e-30)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= 1.9e-62)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5e-30)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= 1.9e-62)
		tmp = -v * ((t1 / u) / u);
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5e-30], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.9e-62], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.99999999999999972e-30

   1. Initial program 68.6%

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

   4. Applied rewrites 99.4%

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

   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 84.4%

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

   if -4.99999999999999972e-30 < t1 < 1.90000000000000003e-62

   1. Initial program 80.6%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-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 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 79.8%

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

   if 1.90000000000000003e-62 < t1

   1. Initial program 64.3%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 72.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5e-30)
   (/ (* -1.0 v) (+ (- u) t1))
   (if (<= t1 1.9e-62) (* v (/ (- t1) (* u u))) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-30) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.9e-62) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5d-30)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= 1.9d-62) then
        tmp = v * (-t1 / (u * u))
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-30) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.9e-62) {
		tmp = v * (-t1 / (u * u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5e-30:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= 1.9e-62:
		tmp = v * (-t1 / (u * u))
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5e-30)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= 1.9e-62)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5e-30)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= 1.9e-62)
		tmp = v * (-t1 / (u * u));
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5e-30], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.9e-62], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.99999999999999972e-30

   1. Initial program 68.6%

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

   4. Applied rewrites 99.4%

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

   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 84.4%

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

   if -4.99999999999999972e-30 < t1 < 1.90000000000000003e-62

   1. Initial program 80.6%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.9

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

   7. Applied rewrites 74.9%

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

   if 1.90000000000000003e-62 < t1

   1. Initial program 64.3%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 72.5%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5e-30)
   (/ (* -1.0 v) (+ (- u) t1))
   (if (<= t1 1.9e-62) (* (- t1) (/ v (* u u))) (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-30) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.9e-62) {
		tmp = -t1 * (v / (u * u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5d-30)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= 1.9d-62) then
        tmp = -t1 * (v / (u * u))
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-30) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= 1.9e-62) {
		tmp = -t1 * (v / (u * u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5e-30:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= 1.9e-62:
		tmp = -t1 * (v / (u * u))
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5e-30)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= 1.9e-62)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5e-30)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= 1.9e-62)
		tmp = -t1 * (v / (u * u));
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5e-30], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.9e-62], N[((-t1) * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.99999999999999972e-30

   1. Initial program 68.6%

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

   4. Applied rewrites 99.4%

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

   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 84.4%

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

   if -4.99999999999999972e-30 < t1 < 1.90000000000000003e-62

   1. Initial program 80.6%

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.6

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

   7. Applied rewrites 74.6%

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

   if 1.90000000000000003e-62 < t1

   1. Initial program 64.3%

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

   3. Taylor expanded in v around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 72.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = ((-1.0d0) * v) / (-u + t1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.4%

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 58.9%

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

8. Final simplification 58.9%

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

Alternative 8: 60.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.4%

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

3. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-neg.f64 75.9

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

5. Applied rewrites 75.9%

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 58.7%

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

Alternative 9: 53.8% 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 72.4%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 52.4

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

5. Applied rewrites 52.4%

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

Reproduce

herbie shell --seed 2024350 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))