Rosa's DopplerBench

Percentage Accurate: 72.6% → 96.3%

Time: 6.3s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

4. Applied rewrites 97.3%

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

5. Final simplification 97.3%

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

Alternative 2: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot v\\ \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1} \cdot u, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq -8.4 \cdot 10^{-168}:\\ \;\;\;\;\frac{t\_1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{-217}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+158}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(2, t1, u\right), u, t1 \cdot t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) v)))
   (if (<= t1 -9.2e+151)
     (/ (fma (* (/ v t1) u) 2.0 (- v)) t1)
     (if (<= t1 -8.4e-168)
       (/ t_1 (* (+ t1 u) (+ t1 u)))
       (if (<= t1 6.2e-217)
         (* (- v) (/ (/ t1 u) u))
         (if (<= t1 6e+158)
           (/ t_1 (fma (fma 2.0 t1 u) u (* t1 t1)))
           (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * v;
	double tmp;
	if (t1 <= -9.2e+151) {
		tmp = fma(((v / t1) * u), 2.0, -v) / t1;
	} else if (t1 <= -8.4e-168) {
		tmp = t_1 / ((t1 + u) * (t1 + u));
	} else if (t1 <= 6.2e-217) {
		tmp = -v * ((t1 / u) / u);
	} else if (t1 <= 6e+158) {
		tmp = t_1 / fma(fma(2.0, t1, u), u, (t1 * t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * v)
	tmp = 0.0
	if (t1 <= -9.2e+151)
		tmp = Float64(fma(Float64(Float64(v / t1) * u), 2.0, Float64(-v)) / t1);
	elseif (t1 <= -8.4e-168)
		tmp = Float64(t_1 / Float64(Float64(t1 + u) * Float64(t1 + u)));
	elseif (t1 <= 6.2e-217)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	elseif (t1 <= 6e+158)
		tmp = Float64(t_1 / fma(fma(2.0, t1, u), u, Float64(t1 * t1)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * v), $MachinePrecision]}, If[LessEqual[t1, -9.2e+151], N[(N[(N[(N[(v / t1), $MachinePrecision] * u), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, -8.4e-168], N[(t$95$1 / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.2e-217], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6e+158], N[(t$95$1 / N[(N[(2.0 * t1 + u), $MachinePrecision] * u + N[(t1 * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t1 < -9.2000000000000003e151

   1. Initial program 39.7%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if -9.2000000000000003e151 < t1 < -8.39999999999999976e-168

   1. Initial program 91.1%

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

   if -8.39999999999999976e-168 < t1 < 6.1999999999999997e-217

   1. Initial program 69.2%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 81.4

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 90.6%

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

   if 6.1999999999999997e-217 < t1 < 6e158

   1. Initial program 89.3%

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

   3. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if 6e158 < t1

   1. Initial program 37.1%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1} \cdot u, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq -8.4 \cdot 10^{-168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -9.2e+151)
     (/ (fma (* (/ v t1) u) 2.0 (- v)) t1)
     (if (<= t1 -8.4e-168)
       t_1
       (if (<= t1 5.8e-226)
         (* (- v) (/ (/ t1 u) u))
         (if (<= t1 6e+158) t_1 (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -9.2e+151) {
		tmp = fma(((v / t1) * u), 2.0, -v) / t1;
	} else if (t1 <= -8.4e-168) {
		tmp = t_1;
	} else if (t1 <= 5.8e-226) {
		tmp = -v * ((t1 / u) / u);
	} else if (t1 <= 6e+158) {
		tmp = t_1;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -9.2e+151)
		tmp = Float64(fma(Float64(Float64(v / t1) * u), 2.0, Float64(-v)) / t1);
	elseif (t1 <= -8.4e-168)
		tmp = t_1;
	elseif (t1 <= 5.8e-226)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	elseif (t1 <= 6e+158)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -9.2e+151], N[(N[(N[(N[(v / t1), $MachinePrecision] * u), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, -8.4e-168], t$95$1, If[LessEqual[t1, 5.8e-226], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6e+158], t$95$1, N[((-v) / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -9.2000000000000003e151

   1. Initial program 39.7%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. div-add N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if -9.2000000000000003e151 < t1 < -8.39999999999999976e-168 or 5.80000000000000003e-226 < t1 < 6e158

   1. Initial program 89.8%

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

   if -8.39999999999999976e-168 < t1 < 5.80000000000000003e-226

   1. Initial program 69.1%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 91.8%

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

   if 6e158 < t1

   1. Initial program 37.1%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\mathsf{fma}\left(\frac{u}{t1}, -2, 1\right)}{t1}\\ \mathbf{elif}\;t1 \leq -8.4 \cdot 10^{-168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -1.3e+154)
     (* (- v) (/ (fma (/ u t1) -2.0 1.0) t1))
     (if (<= t1 -8.4e-168)
       t_1
       (if (<= t1 5.8e-226)
         (* (- v) (/ (/ t1 u) u))
         (if (<= t1 6e+158) t_1 (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -1.3e+154) {
		tmp = -v * (fma((u / t1), -2.0, 1.0) / t1);
	} else if (t1 <= -8.4e-168) {
		tmp = t_1;
	} else if (t1 <= 5.8e-226) {
		tmp = -v * ((t1 / u) / u);
	} else if (t1 <= 6e+158) {
		tmp = t_1;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -1.3e+154)
		tmp = Float64(Float64(-v) * Float64(fma(Float64(u / t1), -2.0, 1.0) / t1));
	elseif (t1 <= -8.4e-168)
		tmp = t_1;
	elseif (t1 <= 5.8e-226)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	elseif (t1 <= 6e+158)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.3e+154], N[((-v) * N[(N[(N[(u / t1), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -8.4e-168], t$95$1, If[LessEqual[t1, 5.8e-226], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6e+158], t$95$1, N[((-v) / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.29999999999999994e154

   1. Initial program 39.7%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in t1 around -inf

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

      1. *-lft-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 91.7

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

   8. Applied rewrites 91.7%

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

   if -1.29999999999999994e154 < t1 < -8.39999999999999976e-168 or 5.80000000000000003e-226 < t1 < 6e158

   1. Initial program 89.8%

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

   if -8.39999999999999976e-168 < t1 < 5.80000000000000003e-226

   1. Initial program 69.1%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 91.8%

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

   if 6e158 < t1

   1. Initial program 37.1%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq -8.4 \cdot 10^{-168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -9.2e+151)
     (/ (* -1.0 v) (+ (- u) t1))
     (if (<= t1 -8.4e-168)
       t_1
       (if (<= t1 5.8e-226)
         (* (- v) (/ (/ t1 u) u))
         (if (<= t1 6e+158) t_1 (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -9.2e+151) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= -8.4e-168) {
		tmp = t_1;
	} else if (t1 <= 5.8e-226) {
		tmp = -v * ((t1 / u) / u);
	} else if (t1 <= 6e+158) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
    if (t1 <= (-9.2d+151)) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else if (t1 <= (-8.4d-168)) then
        tmp = t_1
    else if (t1 <= 5.8d-226) then
        tmp = -v * ((t1 / u) / u)
    else if (t1 <= 6d+158) then
        tmp = t_1
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -9.2e+151) {
		tmp = (-1.0 * v) / (-u + t1);
	} else if (t1 <= -8.4e-168) {
		tmp = t_1;
	} else if (t1 <= 5.8e-226) {
		tmp = -v * ((t1 / u) / u);
	} else if (t1 <= 6e+158) {
		tmp = t_1;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u))
	tmp = 0
	if t1 <= -9.2e+151:
		tmp = (-1.0 * v) / (-u + t1)
	elif t1 <= -8.4e-168:
		tmp = t_1
	elif t1 <= 5.8e-226:
		tmp = -v * ((t1 / u) / u)
	elif t1 <= 6e+158:
		tmp = t_1
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -9.2e+151)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	elseif (t1 <= -8.4e-168)
		tmp = t_1;
	elseif (t1 <= 5.8e-226)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	elseif (t1 <= 6e+158)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	tmp = 0.0;
	if (t1 <= -9.2e+151)
		tmp = (-1.0 * v) / (-u + t1);
	elseif (t1 <= -8.4e-168)
		tmp = t_1;
	elseif (t1 <= 5.8e-226)
		tmp = -v * ((t1 / u) / u);
	elseif (t1 <= 6e+158)
		tmp = t_1;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -9.2e+151], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -8.4e-168], t$95$1, If[LessEqual[t1, 5.8e-226], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6e+158], t$95$1, N[((-v) / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -9.2000000000000003e151

   1. Initial program 39.7%

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 91.4%

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

   if -9.2000000000000003e151 < t1 < -8.39999999999999976e-168 or 5.80000000000000003e-226 < t1 < 6e158

   1. Initial program 89.8%

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

   if -8.39999999999999976e-168 < t1 < 5.80000000000000003e-226

   1. Initial program 69.1%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 91.8%

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

   if 6e158 < t1

   1. Initial program 37.1%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{elif}\;t1 \leq -8.4 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-226}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-82} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.1e-82) (not (<= t1 1.4e+20)))
   (/ (* -1.0 v) (+ (- u) t1))
   (/ (* (/ t1 u) v) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-82) || !(t1 <= 1.4e+20)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((t1 / u) * v) / -u;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.1d-82)) .or. (.not. (t1 <= 1.4d+20))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = ((t1 / u) * v) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-82) || !(t1 <= 1.4e+20)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((t1 / u) * v) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.1e-82) or not (t1 <= 1.4e+20):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = ((t1 / u) * v) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.1e-82) || !(t1 <= 1.4e+20))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(t1 / u) * v) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.1e-82) || ~((t1 <= 1.4e+20)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = ((t1 / u) * v) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.1e-82], N[Not[LessEqual[t1, 1.4e+20]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * v), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.1e-82 or 1.4e20 < t1

   1. Initial program 66.3%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.8%

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

   if -3.1e-82 < t1 < 1.4e20

   1. Initial program 82.8%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 79.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-82} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-82} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.8e-82) (not (<= t1 1.4e+20)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e-82) || !(t1 <= 1.4e+20)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.8d-82)) .or. (.not. (t1 <= 1.4d+20))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = (v / u) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e-82) || !(t1 <= 1.4e+20)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.8e-82) or not (t1 <= 1.4e+20):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.8e-82) || !(t1 <= 1.4e+20))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.8e-82) || ~((t1 <= 1.4e+20)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.8e-82], N[Not[LessEqual[t1, 1.4e+20]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.80000000000000024e-82 or 1.4e20 < t1

   1. Initial program 66.3%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 85.8%

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

   if -2.80000000000000024e-82 < t1 < 1.4e20

   1. Initial program 82.8%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-82} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.75 \cdot 10^{-114} \lor \neg \left(t1 \leq 1.15 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.75e-114) (not (<= t1 1.15e+20)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (- v) (/ (/ t1 u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.75e-114) || !(t1 <= 1.15e+20)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.75d-114)) .or. (.not. (t1 <= 1.15d+20))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = -v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.75e-114) || !(t1 <= 1.15e+20)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.75e-114) or not (t1 <= 1.15e+20):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = -v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.75e-114) || !(t1 <= 1.15e+20))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.75e-114) || ~((t1 <= 1.15e+20)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = -v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.75e-114], N[Not[LessEqual[t1, 1.15e+20]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.75000000000000005e-114 or 1.15e20 < t1

   1. Initial program 68.6%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 83.6%

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

   if -2.75000000000000005e-114 < t1 < 1.15e20

   1. Initial program 81.2%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 78.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.75 \cdot 10^{-114} \lor \neg \left(t1 \leq 1.15 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.75 \cdot 10^{-114} \lor \neg \left(t1 \leq 1.15 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.75e-114) (not (<= t1 1.15e+20)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (- v) (/ t1 (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.75e-114) || !(t1 <= 1.15e+20)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = -v * (t1 / (u * u));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.75d-114)) .or. (.not. (t1 <= 1.15d+20))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = -v * (t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.75e-114) || !(t1 <= 1.15e+20)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = -v * (t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.75e-114) or not (t1 <= 1.15e+20):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = -v * (t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.75e-114) || !(t1 <= 1.15e+20))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.75e-114) || ~((t1 <= 1.15e+20)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = -v * (t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.75e-114], N[Not[LessEqual[t1, 1.15e+20]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.75000000000000005e-114 or 1.15e20 < t1

   1. Initial program 68.6%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 83.6%

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

   if -2.75000000000000005e-114 < t1 < 1.15e20

   1. Initial program 81.2%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 74.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.75 \cdot 10^{-114} \lor \neg \left(t1 \leq 1.15 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 28000000:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 28000000.0) (/ (* -1.0 v) (+ (- u) t1)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 28000000.0) {
		tmp = (-1.0 * v) / (-u + t1);
	} 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 (v <= 28000000.0d0) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 28000000.0) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 28000000.0:
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 28000000.0)
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 28000000.0)
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 28000000.0], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 2.8e7

   1. Initial program 78.8%

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 66.0%

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

   if 2.8e7 < v

   1. Initial program 58.6%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 53.2

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

   5. Applied rewrites 53.2%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 28000000:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 26500000:\\ \;\;\;\;\left(-v\right) \cdot \frac{-1}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 26500000.0) (* (- v) (/ -1.0 (- u t1))) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 26500000.0) {
		tmp = -v * (-1.0 / (u - t1));
	} 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 (v <= 26500000.0d0) then
        tmp = -v * ((-1.0d0) / (u - t1))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 26500000.0) {
		tmp = -v * (-1.0 / (u - t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 26500000.0:
		tmp = -v * (-1.0 / (u - t1))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 26500000.0)
		tmp = Float64(Float64(-v) * Float64(-1.0 / Float64(u - t1)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 26500000.0)
		tmp = -v * (-1.0 / (u - t1));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 26500000.0], N[((-v) * N[(-1.0 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 2.65e7

   1. Initial program 78.8%

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 65.8

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

   9. Applied rewrites 65.8%

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

   if 2.65e7 < v

   1. Initial program 58.6%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 53.2

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

   5. Applied rewrites 53.2%

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

Alternative 12: 54.5% 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 73.9%

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

3. Taylor expanded in u around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 55.9

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

5. Applied rewrites 55.9%

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

Reproduce

herbie shell --seed 2025006 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))