Rosa's DopplerBench

Percentage Accurate: 72.9% → 95.9%

Time: 4.4s

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: 72.9% 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: 95.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

3. Applied rewrites 95.1%

   \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1} \cdot \left(-v\right)}{u - t1}} \]
4. 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-neg.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lift-neg.f64 95.7

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

5. Applied rewrites 95.7%

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

Alternative 2: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{-1 \cdot \left(-v\right)}{u - t1}\\ \mathbf{elif}\;t1 \leq -2.9 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{-253}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u + u, -v\right)}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -2.6e+92)
     (/ (* -1.0 (- v)) (- u t1))
     (if (<= t1 -2.9e-84)
       t_1
       (if (<= t1 2.25e-253)
         (* (- v) (/ (/ t1 u) (- u t1)))
         (if (<= t1 1.2e+138) t_1 (/ (fma (/ v t1) (+ u u) (- v)) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t1 <= -2.6e+92) {
		tmp = (-1.0 * -v) / (u - t1);
	} else if (t1 <= -2.9e-84) {
		tmp = t_1;
	} else if (t1 <= 2.25e-253) {
		tmp = -v * ((t1 / u) / (u - t1));
	} else if (t1 <= 1.2e+138) {
		tmp = t_1;
	} else {
		tmp = fma((v / t1), (u + u), -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 <= -2.6e+92)
		tmp = Float64(Float64(-1.0 * Float64(-v)) / Float64(u - t1));
	elseif (t1 <= -2.9e-84)
		tmp = t_1;
	elseif (t1 <= 2.25e-253)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / Float64(u - t1)));
	elseif (t1 <= 1.2e+138)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(v / t1), Float64(u + u), 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, -2.6e+92], N[(N[(-1.0 * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -2.9e-84], t$95$1, If[LessEqual[t1, 2.25e-253], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.2e+138], t$95$1, N[(N[(N[(v / t1), $MachinePrecision] * N[(u + u), $MachinePrecision] + (-v)), $MachinePrecision] / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.5999999999999999e92

   1. Initial program 47.9%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 95.1%

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

   if -2.5999999999999999e92 < t1 < -2.90000000000000019e-84 or 2.25000000000000014e-253 < t1 < 1.2e138

   1. Initial program 88.8%

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

   if -2.90000000000000019e-84 < t1 < 2.25000000000000014e-253

   1. Initial program 68.2%

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

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1} \cdot \left(-v\right)}{u - t1}} \]
   4. 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-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      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 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in u around inf

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

      1. lower-/.f64 82.4

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

   8. Applied rewrites 82.4%

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

   if 1.2e138 < t1

   1. Initial program 45.4%

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

   2. Taylor expanded in t1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, v, 2 \cdot \frac{u \cdot v}{t1}\right)}{t1}} \]
   5. Step-by-step derivation

   6. Applied rewrites 93.8%

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

Alternative 3: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{-1 \cdot \left(-v\right)}{u - t1}\\ \mathbf{elif}\;t1 \leq -2.9 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{-253}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))
   (if (<= t1 -2.6e+92)
     (/ (* -1.0 (- v)) (- u t1))
     (if (<= t1 -2.9e-84)
       t_1
       (if (<= t1 2.25e-253)
         (* (- v) (/ (/ t1 u) (- u t1)))
         (if (<= t1 1.2e+138) 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 <= -2.6e+92) {
		tmp = (-1.0 * -v) / (u - t1);
	} else if (t1 <= -2.9e-84) {
		tmp = t_1;
	} else if (t1 <= 2.25e-253) {
		tmp = -v * ((t1 / u) / (u - t1));
	} else if (t1 <= 1.2e+138) {
		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 <= (-2.6d+92)) then
        tmp = ((-1.0d0) * -v) / (u - t1)
    else if (t1 <= (-2.9d-84)) then
        tmp = t_1
    else if (t1 <= 2.25d-253) then
        tmp = -v * ((t1 / u) / (u - t1))
    else if (t1 <= 1.2d+138) 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 <= -2.6e+92) {
		tmp = (-1.0 * -v) / (u - t1);
	} else if (t1 <= -2.9e-84) {
		tmp = t_1;
	} else if (t1 <= 2.25e-253) {
		tmp = -v * ((t1 / u) / (u - t1));
	} else if (t1 <= 1.2e+138) {
		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 <= -2.6e+92:
		tmp = (-1.0 * -v) / (u - t1)
	elif t1 <= -2.9e-84:
		tmp = t_1
	elif t1 <= 2.25e-253:
		tmp = -v * ((t1 / u) / (u - t1))
	elif t1 <= 1.2e+138:
		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 <= -2.6e+92)
		tmp = Float64(Float64(-1.0 * Float64(-v)) / Float64(u - t1));
	elseif (t1 <= -2.9e-84)
		tmp = t_1;
	elseif (t1 <= 2.25e-253)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / Float64(u - t1)));
	elseif (t1 <= 1.2e+138)
		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 <= -2.6e+92)
		tmp = (-1.0 * -v) / (u - t1);
	elseif (t1 <= -2.9e-84)
		tmp = t_1;
	elseif (t1 <= 2.25e-253)
		tmp = -v * ((t1 / u) / (u - t1));
	elseif (t1 <= 1.2e+138)
		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, -2.6e+92], N[(N[(-1.0 * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -2.9e-84], t$95$1, If[LessEqual[t1, 2.25e-253], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.2e+138], t$95$1, N[((-v) / t1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.5999999999999999e92

   1. Initial program 47.9%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 95.1%

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

   if -2.5999999999999999e92 < t1 < -2.90000000000000019e-84 or 2.25000000000000014e-253 < t1 < 1.2e138

   1. Initial program 88.8%

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

   if -2.90000000000000019e-84 < t1 < 2.25000000000000014e-253

   1. Initial program 68.2%

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

   3. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1} \cdot \left(-v\right)}{u - t1}} \]
   4. 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-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      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 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in u around inf

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

      1. lower-/.f64 82.4

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

   8. Applied rewrites 82.4%

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

   if 1.2e138 < t1

   1. Initial program 45.4%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 93.1

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

   4. Applied rewrites 93.1%

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

   6. Applied rewrites 93.1%

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

Alternative 4: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.75 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1 \cdot \left(-v\right)}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{+185}:\\ \;\;\;\;\left(-v\right) \cdot \frac{-t1}{\left(-\left(u - t1\right)\right) \cdot \left(u - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{v}{t1}, u + u, -v\right)}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.75e+103)
   (/ (* -1.0 (- v)) (- u t1))
   (if (<= t1 1.5e+185)
     (* (- v) (/ (- t1) (* (- (- u t1)) (- u t1))))
     (/ (fma (/ v t1) (+ u u) (- v)) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.75e+103) {
		tmp = (-1.0 * -v) / (u - t1);
	} else if (t1 <= 1.5e+185) {
		tmp = -v * (-t1 / (-(u - t1) * (u - t1)));
	} else {
		tmp = fma((v / t1), (u + u), -v) / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.75e+103)
		tmp = Float64(Float64(-1.0 * Float64(-v)) / Float64(u - t1));
	elseif (t1 <= 1.5e+185)
		tmp = Float64(Float64(-v) * Float64(Float64(-t1) / Float64(Float64(-Float64(u - t1)) * Float64(u - t1))));
	else
		tmp = Float64(fma(Float64(v / t1), Float64(u + u), Float64(-v)) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.75e+103], N[(N[(-1.0 * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.5e+185], N[((-v) * N[((-t1) / N[((-N[(u - t1), $MachinePrecision]) * N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v / t1), $MachinePrecision] * N[(u + u), $MachinePrecision] + (-v)), $MachinePrecision] / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.75e103

   1. Initial program 46.7%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 95.0%

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

   if -1.75e103 < t1 < 1.49999999999999997e185

   1. Initial program 81.5%

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

   3. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1} \cdot \left(-v\right)}{u - t1}} \]
   4. 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-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      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 91.7

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

   5. Applied rewrites 91.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if 1.49999999999999997e185 < t1

   1. Initial program 40.5%

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

   2. Taylor expanded in t1 around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 96.1

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

   4. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, v, 2 \cdot \frac{u \cdot v}{t1}\right)}{t1}} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.9%

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

Alternative 5: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{-1 \cdot \left(-v\right)}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.6e+92)
   (/ (* -1.0 (- v)) (- u t1))
   (if (<= t1 1.2e+138) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.6e+92) {
		tmp = (-1.0 * -v) / (u - t1);
	} else if (t1 <= 1.2e+138) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.6d+92)) then
        tmp = ((-1.0d0) * -v) / (u - t1)
    else if (t1 <= 1.2d+138) then
        tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.6e+92) {
		tmp = (-1.0 * -v) / (u - t1);
	} else if (t1 <= 1.2e+138) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.6e+92:
		tmp = (-1.0 * -v) / (u - t1)
	elif t1 <= 1.2e+138:
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.6e+92)
		tmp = Float64(Float64(-1.0 * Float64(-v)) / Float64(u - t1));
	elseif (t1 <= 1.2e+138)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.6e+92)
		tmp = (-1.0 * -v) / (u - t1);
	elseif (t1 <= 1.2e+138)
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.6e+92], N[(N[(-1.0 * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.2e+138], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.5999999999999999e92

   1. Initial program 47.9%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 95.1%

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

   if -2.5999999999999999e92 < t1 < 1.2e138

   1. Initial program 81.9%

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

   if 1.2e138 < t1

   1. Initial program 45.4%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 93.1

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

   4. Applied rewrites 93.1%

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

   6. Applied rewrites 93.1%

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

Alternative 6: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1} \cdot \frac{t1}{u}\\ \mathbf{if}\;u \leq -2.25 \cdot 10^{+226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+189}:\\ \;\;\;\;\frac{-1 \cdot \left(-v\right)}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v t1) (/ t1 u))))
   (if (<= u -2.25e+226)
     t_1
     (if (<= u 4.8e+189) (/ (* -1.0 (- v)) (- u t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / t1) * (t1 / u);
	double tmp;
	if (u <= -2.25e+226) {
		tmp = t_1;
	} else if (u <= 4.8e+189) {
		tmp = (-1.0 * -v) / (u - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / t1) * (t1 / u)
    if (u <= (-2.25d+226)) then
        tmp = t_1
    else if (u <= 4.8d+189) then
        tmp = ((-1.0d0) * -v) / (u - t1)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / t1) * (t1 / u);
	double tmp;
	if (u <= -2.25e+226) {
		tmp = t_1;
	} else if (u <= 4.8e+189) {
		tmp = (-1.0 * -v) / (u - t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / t1) * (t1 / u)
	tmp = 0
	if u <= -2.25e+226:
		tmp = t_1
	elif u <= 4.8e+189:
		tmp = (-1.0 * -v) / (u - t1)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / t1) * Float64(t1 / u))
	tmp = 0.0
	if (u <= -2.25e+226)
		tmp = t_1;
	elseif (u <= 4.8e+189)
		tmp = Float64(Float64(-1.0 * Float64(-v)) / Float64(u - t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / t1) * (t1 / u);
	tmp = 0.0;
	if (u <= -2.25e+226)
		tmp = t_1;
	elseif (u <= 4.8e+189)
		tmp = (-1.0 * -v) / (u - t1);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / t1), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.25e+226], t$95$1, If[LessEqual[u, 4.8e+189], N[(N[(-1.0 * (-v)), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.24999999999999995e226 or 4.8000000000000001e189 < u

   1. Initial program 88.4%

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

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1} \cdot \left(-v\right)}{u - t1}} \]
   4. 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-neg.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in u around 0

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

      1. lower-/.f64 71.1

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in u around inf

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

       1. lower-/.f64 69.7

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

   11. Applied rewrites 69.7%

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

   if -2.24999999999999995e226 < u < 4.8000000000000001e189

   1. Initial program 68.3%

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

   3. Applied rewrites 94.4%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 61.4%

      \[\leadsto \frac{\color{blue}{-1} \cdot \left(-v\right)}{u - t1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.0% 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 71.5%

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

3. Applied rewrites 95.1%

   \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1} \cdot \left(-v\right)}{u - t1}} \]
4. 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-neg.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-*.f64 N/A

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

   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 93.6

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

5. Applied rewrites 93.6%

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

Alternative 8: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1 \cdot v}{u - t1}\\ \mathbf{if}\;u \leq -2.1 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+162}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* -1.0 v) (- u t1))))
   (if (<= u -2.1e+83) t_1 (if (<= u 1.15e+162) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-1.0 * v) / (u - t1);
	double tmp;
	if (u <= -2.1e+83) {
		tmp = t_1;
	} else if (u <= 1.15e+162) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-1.0d0) * v) / (u - t1)
    if (u <= (-2.1d+83)) then
        tmp = t_1
    else if (u <= 1.15d+162) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-1.0 * v) / (u - t1);
	double tmp;
	if (u <= -2.1e+83) {
		tmp = t_1;
	} else if (u <= 1.15e+162) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-1.0 * v) / (u - t1)
	tmp = 0
	if u <= -2.1e+83:
		tmp = t_1
	elif u <= 1.15e+162:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-1.0 * v) / Float64(u - t1))
	tmp = 0.0
	if (u <= -2.1e+83)
		tmp = t_1;
	elseif (u <= 1.15e+162)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-1.0 * v) / (u - t1);
	tmp = 0.0;
	if (u <= -2.1e+83)
		tmp = t_1;
	elseif (u <= 1.15e+162)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(-1.0 * v), $MachinePrecision] / N[(u - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.1e+83], t$95$1, If[LessEqual[u, 1.15e+162], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.10000000000000002e83 or 1.14999999999999997e162 < u

   1. Initial program 79.9%

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

   3. Applied rewrites 76.1%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 41.8%

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

   if -2.10000000000000002e83 < u < 1.14999999999999997e162

   1. Initial program 68.0%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.0

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

   4. Applied rewrites 64.0%

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

   6. Applied rewrites 64.0%

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

Alternative 9: 61.4% accurate, 1.4× speedup

Math

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

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 * Float64(-v)) / 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 71.5%

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

3. Applied rewrites 95.1%

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

4. Taylor expanded in u around 0

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

6. Applied rewrites 59.1%

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

Alternative 10: 61.3% 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 71.5%

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

3. Applied rewrites 95.1%

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

4. Taylor expanded in u around 0

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

6. Applied rewrites 59.1%

   \[\leadsto \frac{\color{blue}{-1} \cdot \left(-v\right)}{u - t1} \]
7. 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 58.9

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

8. Applied rewrites 58.9%

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

Alternative 11: 53.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 50.3

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

4. Applied rewrites 50.3%

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

6. Applied rewrites 50.3%

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

Alternative 12: 13.9% accurate, 2.5× 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(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 71.5%

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

3. Applied rewrites 37.5%

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

4. Taylor expanded in u around 0

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

   1. lower-/.f64 13.3

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

6. Applied rewrites 13.3%

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

Reproduce

herbie shell --seed 2024363 -o localize:costs -o setup:simplify -o generate:simplify
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))