Rosa's DopplerBench

Percentage Accurate: 73.3% → 96.8%

Time: 5.4s

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: 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: 96.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

4. Applied rewrites 96.9%

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

5. Final simplification 96.9%

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

Alternative 2: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1} \cdot u, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;v \cdot \frac{-t1}{\mathsf{fma}\left(\mathsf{fma}\left(t1, 2, u\right), u, t1 \cdot t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.25e+154)
   (/ (fma (* (/ v t1) u) 2.0 (- v)) t1)
   (if (<= t1 2e+151)
     (* v (/ (- t1) (fma (fma t1 2.0 u) u (* t1 t1))))
     (/ (* -1.0 v) (+ (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.25e+154) {
		tmp = fma(((v / t1) * u), 2.0, -v) / t1;
	} else if (t1 <= 2e+151) {
		tmp = v * (-t1 / fma(fma(t1, 2.0, u), u, (t1 * t1)));
	} else {
		tmp = (-1.0 * v) / (-u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.25e+154)
		tmp = Float64(fma(Float64(Float64(v / t1) * u), 2.0, Float64(-v)) / t1);
	elseif (t1 <= 2e+151)
		tmp = Float64(v * Float64(Float64(-t1) / fma(fma(t1, 2.0, u), u, Float64(t1 * t1))));
	else
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.25e+154], N[(N[(N[(N[(v / t1), $MachinePrecision] * u), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, 2e+151], N[(v * N[((-t1) / N[(N[(t1 * 2.0 + u), $MachinePrecision] * u + N[(t1 * t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.25000000000000001e154

   1. Initial program 28.4%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.25000000000000001e154 < t1 < 2.00000000000000003e151

   1. Initial program 86.8%

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

   3. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.9

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

   7. Applied rewrites 91.9%

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

   if 2.00000000000000003e151 < t1

   1. Initial program 36.6%

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

   4. Applied rewrites 98.7%

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

   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 92.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1} \cdot u, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;v \cdot \frac{-t1}{\mathsf{fma}\left(\mathsf{fma}\left(t1, 2, u\right), u, t1 \cdot t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+82} \lor \neg \left(t1 \leq 7 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7e+82) (not (<= t1 7e+132)))
   (/ (* -1.0 v) (+ (- u) t1))
   (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7e+82) || !(t1 <= 7e+132)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-t1 * v) / ((t1 + 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 <= (-7d+82)) .or. (.not. (t1 <= 7d+132))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7e+82) || !(t1 <= 7e+132)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7e+82) or not (t1 <= 7e+132):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7e+82) || !(t1 <= 7e+132))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -7e+82) || ~((t1 <= 7e+132)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7e+82], N[Not[LessEqual[t1, 7e+132]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.0000000000000001e82 or 7.00000000000000041e132 < t1

   1. Initial program 39.9%

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 94.1%

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

   if -7.0000000000000001e82 < t1 < 7.00000000000000041e132

   1. Initial program 90.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+82} \lor \neg \left(t1 \leq 7 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-118} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u} \cdot v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.3e-118) (not (<= t1 1.8e-56)))
   (/ (* -1.0 v) (+ (- u) t1))
   (/ (* (/ (- t1) u) v) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.3e-118) || !(t1 <= 1.8e-56)) {
		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 <= (-1.3d-118)) .or. (.not. (t1 <= 1.8d-56))) 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 <= -1.3e-118) || !(t1 <= 1.8e-56)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = ((-t1 / u) * v) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.3e-118) or not (t1 <= 1.8e-56):
		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 <= -1.3e-118) || !(t1 <= 1.8e-56))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(Float64(-t1) / u) * v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.3e-118) || ~((t1 <= 1.8e-56)))
		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, -1.3e-118], N[Not[LessEqual[t1, 1.8e-56]], $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 < -1.3e-118 or 1.79999999999999989e-56 < t1

   1. Initial program 66.5%

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 82.1%

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

   if -1.3e-118 < t1 < 1.79999999999999989e-56

   1. Initial program 86.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 83.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-118} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u} \cdot v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-108} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.2e-108) (not (<= t1 1.8e-56)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (- t1) (/ (/ v u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.2e-108) || !(t1 <= 1.8e-56)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = -t1 * ((v / u) / u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.2d-108)) .or. (.not. (t1 <= 1.8d-56))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = -t1 * ((v / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.2e-108) || !(t1 <= 1.8e-56)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = -t1 * ((v / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.2e-108) or not (t1 <= 1.8e-56):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = -t1 * ((v / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.2e-108) || !(t1 <= 1.8e-56))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(-t1) * Float64(Float64(v / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.2e-108) || ~((t1 <= 1.8e-56)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = -t1 * ((v / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.2e-108], N[Not[LessEqual[t1, 1.8e-56]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[((-t1) * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.20000000000000009e-108 or 1.79999999999999989e-56 < t1

   1. Initial program 65.9%

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 82.4%

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

   if -1.20000000000000009e-108 < t1 < 1.79999999999999989e-56

   1. Initial program 86.7%

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

   3. Taylor expanded in u around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-108} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1} \cdot u, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -7e+82)
   (/ (fma (* (/ v t1) u) 2.0 (- v)) t1)
   (if (<= t1 7e+132)
     (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
     (/ (* -1.0 v) (+ (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -7e+82) {
		tmp = fma(((v / t1) * u), 2.0, -v) / t1;
	} else if (t1 <= 7e+132) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = (-1.0 * v) / (-u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -7e+82)
		tmp = Float64(fma(Float64(Float64(v / t1) * u), 2.0, Float64(-v)) / t1);
	elseif (t1 <= 7e+132)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -7e+82], N[(N[(N[(N[(v / t1), $MachinePrecision] * u), $MachinePrecision] * 2.0 + (-v)), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, 7e+132], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -7.0000000000000001e82

   1. Initial program 42.5%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1} + 2 \cdot \frac{u \cdot v}{{t1}^{2}}} \]
   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 96.2

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

   5. Applied rewrites 96.2%

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

   if -7.0000000000000001e82 < t1 < 7.00000000000000041e132

   1. Initial program 90.2%

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

   if 7.00000000000000041e132 < t1

   1. Initial program 35.6%

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

   4. Applied rewrites 98.1%

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

   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 90.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1} \cdot u, 2, -v\right)}{t1}\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-118} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.3e-118) (not (<= t1 1.8e-56)))
   (/ (* -1.0 v) (+ (- u) t1))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.3e-118) || !(t1 <= 1.8e-56)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.3d-118)) .or. (.not. (t1 <= 1.8d-56))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = (-t1 * v) / (u * u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.3e-118) || !(t1 <= 1.8e-56)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.3e-118) or not (t1 <= 1.8e-56):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (-t1 * v) / (u * u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.3e-118) || !(t1 <= 1.8e-56))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.3e-118) || ~((t1 <= 1.8e-56)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (-t1 * v) / (u * u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.3e-118], N[Not[LessEqual[t1, 1.8e-56]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.3e-118 or 1.79999999999999989e-56 < t1

   1. Initial program 66.5%

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 82.1%

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

   if -1.3e-118 < t1 < 1.79999999999999989e-56

   1. Initial program 86.2%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-118} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-118} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.3e-118) (not (<= t1 1.8e-56)))
   (/ (* -1.0 v) (+ (- u) t1))
   (* (/ (- v) (* u u)) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.3e-118) || !(t1 <= 1.8e-56)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-v / (u * u)) * t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.3d-118)) .or. (.not. (t1 <= 1.8d-56))) then
        tmp = ((-1.0d0) * v) / (-u + t1)
    else
        tmp = (-v / (u * u)) * t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.3e-118) || !(t1 <= 1.8e-56)) {
		tmp = (-1.0 * v) / (-u + t1);
	} else {
		tmp = (-v / (u * u)) * t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.3e-118) or not (t1 <= 1.8e-56):
		tmp = (-1.0 * v) / (-u + t1)
	else:
		tmp = (-v / (u * u)) * t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.3e-118) || !(t1 <= 1.8e-56))
		tmp = Float64(Float64(-1.0 * v) / Float64(Float64(-u) + t1));
	else
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.3e-118) || ~((t1 <= 1.8e-56)))
		tmp = (-1.0 * v) / (-u + t1);
	else
		tmp = (-v / (u * u)) * t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.3e-118], N[Not[LessEqual[t1, 1.8e-56]], $MachinePrecision]], N[(N[(-1.0 * v), $MachinePrecision] / N[((-u) + t1), $MachinePrecision]), $MachinePrecision], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.3e-118 or 1.79999999999999989e-56 < t1

   1. Initial program 66.5%

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in u around 0

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

   7. Applied rewrites 82.1%

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

   if -1.3e-118 < t1 < 1.79999999999999989e-56

   1. Initial program 86.2%

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

   3. Taylor expanded in t1 around 0

      \[\leadsto \color{blue}{t1 \cdot \left(-1 \cdot \frac{v}{{u}^{2}} + 2 \cdot \frac{t1 \cdot v}{{u}^{3}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}} + 2 \cdot \frac{t1 \cdot v}{{u}^{3}}\right) \cdot t1} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}} + 2 \cdot \frac{t1 \cdot v}{{u}^{3}}\right) \cdot t1} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{-1 \cdot v}{{u}^{2}}} + 2 \cdot \frac{t1 \cdot v}{{u}^{3}}\right) \cdot t1 \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{-1 \cdot v}{{u}^{2}} + \color{blue}{\frac{2 \cdot \left(t1 \cdot v\right)}{{u}^{3}}}\right) \cdot t1 \]

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 76.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.3 \cdot 10^{-118} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-1 \cdot v}{\left(-u\right) + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

4. Applied rewrites 96.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lift-neg.f64 N/A

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

   10. lower-/.f64 95.3

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

6. Applied rewrites 95.3%

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

Alternative 10: 62.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 65.5%

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

8. Final simplification 65.5%

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

Alternative 11: 62.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

4. Applied rewrites 96.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lift-neg.f64 N/A

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

   10. lower-/.f64 95.3

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

6. Applied rewrites 95.3%

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

7. Taylor expanded in u around 0

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

9. Applied rewrites 65.3%

   \[\leadsto \left(-v\right) \cdot \frac{\color{blue}{-1}}{u - t1} \]
10. 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 72.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 58.6

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

5. Applied rewrites 58.6%

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

Reproduce

herbie shell --seed 2025007 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))