Rosa's DopplerBench

Percentage Accurate: 73.3% → 98.0%

Time: 3.5s

Alternatives: 10

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: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 98.0

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

3. Applied rewrites 98.0%

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

Alternative 2: 77.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{if}\;u \leq -8.2 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq -2.05 \cdot 10^{-51}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{v}{\mathsf{fma}\left(2, t1, u\right) \cdot u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v u) (/ (- t1) u))))
   (if (<= u -8.2e+169)
     t_1
     (if (<= u -2.05e-51)
       (* (- t1) (/ v (* (fma 2.0 t1 u) u)))
       (if (<= u 5e+46) (/ (- v) t1) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / u) * (-t1 / u);
	double tmp;
	if (u <= -8.2e+169) {
		tmp = t_1;
	} else if (u <= -2.05e-51) {
		tmp = -t1 * (v / (fma(2.0, t1, u) * u));
	} else if (u <= 5e+46) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / u) * Float64(Float64(-t1) / u))
	tmp = 0.0
	if (u <= -8.2e+169)
		tmp = t_1;
	elseif (u <= -2.05e-51)
		tmp = Float64(Float64(-t1) * Float64(v / Float64(fma(2.0, t1, u) * u)));
	elseif (u <= 5e+46)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -8.2e+169], t$95$1, If[LessEqual[u, -2.05e-51], N[((-t1) * N[(v / N[(N[(2.0 * t1 + u), $MachinePrecision] * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 5e+46], N[((-v) / t1), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.2000000000000006e169 or 5.0000000000000002e46 < u

   1. Initial program 77.4%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 85.5

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

   4. Applied rewrites 85.5%

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

   if -8.2000000000000006e169 < u < -2.04999999999999987e-51

   1. Initial program 78.5%

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

   2. Taylor expanded in u around 0

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

   4. Applied rewrites 36.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 30.6

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{t1 \cdot \mathsf{Rewrite<=}\left(+-commutative, \left(u + t1\right)\right)} \]

      14. +-commutative N/A

          \[\leadsto \left(-t1\right) \cdot \frac{v}{t1 \cdot \mathsf{Rewrite<=}\left(lift-+.f64, \left(u + t1\right)\right)} \]

   6. Applied rewrites 30.6%

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

   7. Taylor expanded in t1 around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 66.5

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

   9. Applied rewrites 66.5%

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

   if -2.04999999999999987e-51 < u < 5.0000000000000002e46

   1. Initial program 69.0%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.9

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

   4. Applied rewrites 76.9%

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

Alternative 3: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+54}:\\ \;\;\;\;\left(\frac{u}{t1} - 1\right) \cdot \frac{v}{u + t1}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.4e+54)
   (* (- (/ u t1) 1.0) (/ v (+ u t1)))
   (if (<= t1 6.5e+140)
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (- v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.4e+54) {
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	} else if (t1 <= 6.5e+140) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.4d+54)) then
        tmp = ((u / t1) - 1.0d0) * (v / (u + t1))
    else if (t1 <= 6.5d+140) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = -v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.4e+54) {
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	} else if (t1 <= 6.5e+140) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.4e+54:
		tmp = ((u / t1) - 1.0) * (v / (u + t1))
	elif t1 <= 6.5e+140:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = -v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.4e+54)
		tmp = Float64(Float64(Float64(u / t1) - 1.0) * Float64(v / Float64(u + t1)));
	elseif (t1 <= 6.5e+140)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.4e+54)
		tmp = ((u / t1) - 1.0) * (v / (u + t1));
	elseif (t1 <= 6.5e+140)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = -v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.4e+54], N[(N[(N[(u / t1), $MachinePrecision] - 1.0), $MachinePrecision] * N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.5e+140], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.40000000000000008e54

   1. Initial program 54.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in u around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 86.9

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

   6. Applied rewrites 86.9%

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

   if -1.40000000000000008e54 < t1 < 6.4999999999999999e140

   1. Initial program 85.6%

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

   if 6.4999999999999999e140 < t1

   1. Initial program 42.8%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   3. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 4: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -7.7 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -7.7e+25)
     t_1
     (if (<= t1 6.5e+140) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -7.7e+25) {
		tmp = t_1;
	} else if (t1 <= 6.5e+140) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-7.7d+25)) then
        tmp = t_1
    else if (t1 <= 6.5d+140) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -7.7e+25) {
		tmp = t_1;
	} else if (t1 <= 6.5e+140) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -7.7e+25:
		tmp = t_1
	elif t1 <= 6.5e+140:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -7.7e+25)
		tmp = t_1;
	elseif (t1 <= 6.5e+140)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -7.7e+25)
		tmp = t_1;
	elseif (t1 <= 6.5e+140)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.7e+25], t$95$1, If[LessEqual[t1, 6.5e+140], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.70000000000000025e25 or 6.4999999999999999e140 < t1

   1. Initial program 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if -7.70000000000000025e25 < t1 < 6.4999999999999999e140

   1. Initial program 85.5%

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

Alternative 5: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{if}\;u \leq -2.05 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v u) (/ (- t1) u))))
   (if (<= u -2.05e-51) t_1 (if (<= u 5e+46) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / u) * (-t1 / u);
	double tmp;
	if (u <= -2.05e-51) {
		tmp = t_1;
	} else if (u <= 5e+46) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / u) * (-t1 / u)
    if (u <= (-2.05d-51)) then
        tmp = t_1
    else if (u <= 5d+46) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / u) * (-t1 / u);
	double tmp;
	if (u <= -2.05e-51) {
		tmp = t_1;
	} else if (u <= 5e+46) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / u) * (-t1 / u)
	tmp = 0
	if u <= -2.05e-51:
		tmp = t_1
	elif u <= 5e+46:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / u) * Float64(Float64(-t1) / u))
	tmp = 0.0
	if (u <= -2.05e-51)
		tmp = t_1;
	elseif (u <= 5e+46)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / u) * (-t1 / u);
	tmp = 0.0;
	if (u <= -2.05e-51)
		tmp = t_1;
	elseif (u <= 5e+46)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.05e-51], t$95$1, If[LessEqual[u, 5e+46], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.04999999999999987e-51 or 5.0000000000000002e46 < u

   1. Initial program 77.8%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 76.7

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

   4. Applied rewrites 76.7%

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

   if -2.04999999999999987e-51 < u < 5.0000000000000002e46

   1. Initial program 69.0%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.9

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

   4. Applied rewrites 76.9%

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

Alternative 6: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -13000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.755 \cdot 10^{-51}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -13000000000.0)
     t_1
     (if (<= t1 5.755e-51) (* v (/ (/ (- t1) u) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -13000000000.0) {
		tmp = t_1;
	} else if (t1 <= 5.755e-51) {
		tmp = v * ((-t1 / u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-13000000000.0d0)) then
        tmp = t_1
    else if (t1 <= 5.755d-51) then
        tmp = v * ((-t1 / u) / u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -13000000000.0) {
		tmp = t_1;
	} else if (t1 <= 5.755e-51) {
		tmp = v * ((-t1 / u) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -13000000000.0:
		tmp = t_1
	elif t1 <= 5.755e-51:
		tmp = v * ((-t1 / u) / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -13000000000.0)
		tmp = t_1;
	elseif (t1 <= 5.755e-51)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -13000000000.0)
		tmp = t_1;
	elseif (t1 <= 5.755e-51)
		tmp = v * ((-t1 / u) / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -13000000000.0], t$95$1, If[LessEqual[t1, 5.755e-51], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.3e10 or 5.75500000000000015e-51 < t1

   1. Initial program 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.4

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

   8. Applied rewrites 82.4%

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

   if -1.3e10 < t1 < 5.75500000000000015e-51

   1. Initial program 84.7%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 76.8

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

   4. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. distribute-frac-neg N/A

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

      12. lift-/.f64 N/A

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

      13. lift-neg.f64 74.5

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

   8. Applied rewrites 74.5%

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

Alternative 7: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -13000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.755 \cdot 10^{-51}:\\ \;\;\;\;-t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ u t1))))
   (if (<= t1 -13000000000.0)
     t_1
     (if (<= t1 5.755e-51) (- (* t1 (/ v (* u u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -13000000000.0) {
		tmp = t_1;
	} else if (t1 <= 5.755e-51) {
		tmp = -(t1 * (v / (u * u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-13000000000.0d0)) then
        tmp = t_1
    else if (t1 <= 5.755d-51) then
        tmp = -(t1 * (v / (u * u)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -13000000000.0) {
		tmp = t_1;
	} else if (t1 <= 5.755e-51) {
		tmp = -(t1 * (v / (u * u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -13000000000.0:
		tmp = t_1
	elif t1 <= 5.755e-51:
		tmp = -(t1 * (v / (u * u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -13000000000.0)
		tmp = t_1;
	elseif (t1 <= 5.755e-51)
		tmp = Float64(-Float64(t1 * Float64(v / Float64(u * u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -13000000000.0)
		tmp = t_1;
	elseif (t1 <= 5.755e-51)
		tmp = -(t1 * (v / (u * u)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -13000000000.0], t$95$1, If[LessEqual[t1, 5.755e-51], (-N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.3e10 or 5.75500000000000015e-51 < t1

   1. Initial program 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.4

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

   8. Applied rewrites 82.4%

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

   if -1.3e10 < t1 < 5.75500000000000015e-51

   1. Initial program 84.7%

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

   2. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-neg.f64 76.8

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

   4. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

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

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

      8. mul-1-neg N/A

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

      9. pow2 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 70.6

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

   6. Applied rewrites 70.6%

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

Alternative 8: 55.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+79}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.45e+79) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.45e+79) {
		tmp = -v / 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 <= (-1.45d+79)) then
        tmp = -v / 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 <= -1.45e+79) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.45e+79:
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.45e+79)
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.45e+79)
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.45e+79], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.44999999999999996e79

   1. Initial program 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 97.7

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

   3. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.6

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 34.1%

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

   if -1.44999999999999996e79 < u

   1. Initial program 72.9%

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

   2. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 60.4

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

   4. Applied rewrites 60.4%

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

Alternative 9: 61.7% 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 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 98.0

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

3. Applied rewrites 98.0%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. +-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 61.7

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

8. Applied rewrites 61.7%

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

Alternative 10: 54.4% 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 73.3%

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

2. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 54.4

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

4. Applied rewrites 54.4%

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

Reproduce

herbie shell --seed 2025105 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))