Rosa's DopplerBench

Percentage Accurate: 72.9% → 98.1%

Time: 3.0s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

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

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

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) - u)) * Float64(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[(N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * N[(t1 / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. distribute-neg-in N/A

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

   11. lift-neg.f64 N/A

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

   12. sub-flip-reverse N/A

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

   13. lower--.f64 N/A

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

   14. lift-neg.f64 N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 98.1%

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 98.1%

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

3. Applied rewrites 98.1%

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

Alternative 2: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{-t1}{\left(u + t1\right) \cdot \left(u + t1\right)} \cdot v\\ \mathbf{if}\;t1 \leq -4.1 \cdot 10^{+117}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -5.8 \cdot 10^{-192}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.4 \cdot 10^{-175}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{u - t1}{t1} \cdot v}{u + t1}\\ \end{array} \]

FPCore

(FPCore (u v t1)
  :precision binary64
  (let* ((t_1 (* (/ (- t1) (* (+ u t1) (+ u t1))) v)))
  (if (<= t1 -4.1e+117)
    (/ (- v) t1)
    (if (<= t1 -5.8e-192)
      t_1
      (if (<= t1 1.4e-175)
        (* (/ v (- (- t1) u)) (/ t1 u))
        (if (<= t1 1.15e+165)
          t_1
          (/ (* (/ (- u t1) t1) v) (+ u t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 / ((u + t1) * (u + t1))) * v;
	double tmp;
	if (t1 <= -4.1e+117) {
		tmp = -v / t1;
	} else if (t1 <= -5.8e-192) {
		tmp = t_1;
	} else if (t1 <= 1.4e-175) {
		tmp = (v / (-t1 - u)) * (t1 / u);
	} else if (t1 <= 1.15e+165) {
		tmp = t_1;
	} else {
		tmp = (((u - t1) / t1) * v) / (u + t1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-t1 / ((u + t1) * (u + t1))) * v
    if (t1 <= (-4.1d+117)) then
        tmp = -v / t1
    else if (t1 <= (-5.8d-192)) then
        tmp = t_1
    else if (t1 <= 1.4d-175) then
        tmp = (v / (-t1 - u)) * (t1 / u)
    else if (t1 <= 1.15d+165) then
        tmp = t_1
    else
        tmp = (((u - t1) / t1) * v) / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-t1 / ((u + t1) * (u + t1))) * v;
	double tmp;
	if (t1 <= -4.1e+117) {
		tmp = -v / t1;
	} else if (t1 <= -5.8e-192) {
		tmp = t_1;
	} else if (t1 <= 1.4e-175) {
		tmp = (v / (-t1 - u)) * (t1 / u);
	} else if (t1 <= 1.15e+165) {
		tmp = t_1;
	} else {
		tmp = (((u - t1) / t1) * v) / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-t1 / ((u + t1) * (u + t1))) * v
	tmp = 0
	if t1 <= -4.1e+117:
		tmp = -v / t1
	elif t1 <= -5.8e-192:
		tmp = t_1
	elif t1 <= 1.4e-175:
		tmp = (v / (-t1 - u)) * (t1 / u)
	elif t1 <= 1.15e+165:
		tmp = t_1
	else:
		tmp = (((u - t1) / t1) * v) / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) / Float64(Float64(u + t1) * Float64(u + t1))) * v)
	tmp = 0.0
	if (t1 <= -4.1e+117)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= -5.8e-192)
		tmp = t_1;
	elseif (t1 <= 1.4e-175)
		tmp = Float64(Float64(v / Float64(Float64(-t1) - u)) * Float64(t1 / u));
	elseif (t1 <= 1.15e+165)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(u - t1) / t1) * v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-t1 / ((u + t1) * (u + t1))) * v;
	tmp = 0.0;
	if (t1 <= -4.1e+117)
		tmp = -v / t1;
	elseif (t1 <= -5.8e-192)
		tmp = t_1;
	elseif (t1 <= 1.4e-175)
		tmp = (v / (-t1 - u)) * (t1 / u);
	elseif (t1 <= 1.15e+165)
		tmp = t_1;
	else
		tmp = (((u - t1) / t1) * v) / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision]}, If[LessEqual[t1, -4.1e+117], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, -5.8e-192], t$95$1, If[LessEqual[t1, 1.4e-175], N[(N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.15e+165], t$95$1, N[(N[(N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision] * v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -4.0999999999999999e117

   1. Initial program 72.9%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 54.3%

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

   4. Applied rewrites 54.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 54.3%

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

   6. Applied rewrites 54.3%

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

   if -4.0999999999999999e117 < t1 < -5.8000000000000003e-192 or 1.4e-175 < t1 < 1.1500000000000001e165

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 75.9%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 75.9%

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 75.9%

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

   3. Applied rewrites 75.9%

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

   if -5.8000000000000003e-192 < t1 < 1.4e-175

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. lift-neg.f64 N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 98.1%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in u around inf

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

      1. lower-/.f64 54.9%

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

   6. Applied rewrites 54.9%

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

   if 1.1500000000000001e165 < t1

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-neg-in N/A

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

      14. lift-neg.f64 N/A

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

      15. sub-flip-reverse N/A

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

      16. lower--.f64 98.3%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.3%

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in u around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 52.7%

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

   6. Applied rewrites 52.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-/.f64 N/A

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

      4. sub-to-fraction N/A

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

      5. distribute-neg-frac N/A

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

      6. *-lft-identity N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\left(t1 \cdot 1 - u \cdot {t1}^{0}\right)\right)}{t1} \cdot v}{u + t1} \]

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. inv-pow N/A

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

      13. associate-*l* N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. sub-negate-rev N/A

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

      17. lower-/.f64 N/A

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

   8. Applied rewrites 52.7%

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

Alternative 3: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -28500000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (u v t1)
  :precision binary64
  (let* ((t_1 (/ (- v) (+ u t1))))
  (if (<= t1 -28500000000000.0)
    t_1
    (if (<= t1 4.2e-28) (* (/ v (- (- t1) u)) (/ t1 u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -28500000000000.0) {
		tmp = t_1;
	} else if (t1 <= 4.2e-28) {
		tmp = (v / (-t1 - u)) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-28500000000000.0d0)) then
        tmp = t_1
    else if (t1 <= 4.2d-28) then
        tmp = (v / (-t1 - u)) * (t1 / u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -28500000000000.0) {
		tmp = t_1;
	} else if (t1 <= 4.2e-28) {
		tmp = (v / (-t1 - u)) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -28500000000000.0:
		tmp = t_1
	elif t1 <= 4.2e-28:
		tmp = (v / (-t1 - u)) * (t1 / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -28500000000000.0)
		tmp = t_1;
	elseif (t1 <= 4.2e-28)
		tmp = Float64(Float64(v / Float64(Float64(-t1) - u)) * Float64(t1 / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -28500000000000.0)
		tmp = t_1;
	elseif (t1 <= 4.2e-28)
		tmp = (v / (-t1 - u)) * (t1 / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -28500000000000.0], t$95$1, If[LessEqual[t1, 4.2e-28], N[(N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.85e13 or 4.2000000000000001e-28 < t1

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. sqr-neg-rev N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg-rev N/A

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

      19. lower-/.f64 98.1%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. remove-double-neg N/A

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

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

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

      8. lower-neg.f64 N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity 61.9%

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

   8. Applied rewrites 61.9%

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

   if -2.85e13 < t1 < 4.2000000000000001e-28

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. lift-neg.f64 N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 98.1%

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in u around inf

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

      1. lower-/.f64 54.9%

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

   6. Applied rewrites 54.9%

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

Alternative 4: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -23000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1}{u} \cdot \frac{t1 \cdot v}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (u v t1)
  :precision binary64
  (let* ((t_1 (/ (- v) (+ u t1))))
  (if (<= t1 -23000000000000.0)
    t_1
    (if (<= t1 4.2e-28) (* (/ -1.0 u) (/ (* t1 v) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -23000000000000.0) {
		tmp = t_1;
	} else if (t1 <= 4.2e-28) {
		tmp = (-1.0 / u) * ((t1 * v) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(u, v, t1)
use fmin_fmax_functions
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (u + t1)
    if (t1 <= (-23000000000000.0d0)) then
        tmp = t_1
    else if (t1 <= 4.2d-28) then
        tmp = ((-1.0d0) / u) * ((t1 * v) / u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (u + t1);
	double tmp;
	if (t1 <= -23000000000000.0) {
		tmp = t_1;
	} else if (t1 <= 4.2e-28) {
		tmp = (-1.0 / u) * ((t1 * v) / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (u + t1)
	tmp = 0
	if t1 <= -23000000000000.0:
		tmp = t_1
	elif t1 <= 4.2e-28:
		tmp = (-1.0 / u) * ((t1 * v) / u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(u + t1))
	tmp = 0.0
	if (t1 <= -23000000000000.0)
		tmp = t_1;
	elseif (t1 <= 4.2e-28)
		tmp = Float64(Float64(-1.0 / u) * Float64(Float64(t1 * v) / u));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (u + t1);
	tmp = 0.0;
	if (t1 <= -23000000000000.0)
		tmp = t_1;
	elseif (t1 <= 4.2e-28)
		tmp = (-1.0 / u) * ((t1 * v) / u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -23000000000000.0], t$95$1, If[LessEqual[t1, 4.2e-28], N[(N[(-1.0 / u), $MachinePrecision] * N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.3e13 or 4.2000000000000001e-28 < t1

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. sqr-neg-rev N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg-rev N/A

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

      19. lower-/.f64 98.1%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 61.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. remove-double-neg N/A

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

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

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

      8. lower-neg.f64 N/A

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

      9. metadata-eval N/A

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

      10. *-rgt-identity 61.9%

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

   8. Applied rewrites 61.9%

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

   if -2.3e13 < t1 < 4.2000000000000001e-28

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. sqr-neg-rev N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg-rev N/A

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

      19. lower-/.f64 98.1%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 61.7%

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

   7. Taylor expanded in u around inf

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

      1. lower-/.f64 16.9%

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

   9. Applied rewrites 16.9%

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

   10. Taylor expanded in u around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 49.6%

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

   12. Applied rewrites 49.6%

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

Alternative 5: 61.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. sqr-neg-rev N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. frac-2neg-rev N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l/ N/A

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

   16. lower-*.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. frac-2neg-rev N/A

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

   19. lower-/.f64 98.1%

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

   20. lift-+.f64 N/A

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

   21. +-commutative N/A

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

   22. lower-+.f64 98.1%

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

3. Applied rewrites 98.1%

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

4. Taylor expanded in u around 0

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

6. Applied rewrites 61.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. remove-double-neg N/A

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

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

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

   8. lower-neg.f64 N/A

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

   9. metadata-eval N/A

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

   10. *-rgt-identity 61.9%

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

8. Applied rewrites 61.9%

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

Alternative 6: 59.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \frac{-1}{u} \cdot v\\ \mathbf{if}\;u \leq -3.6 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (u v t1)
  :precision binary64
  (let* ((t_1 (* (/ -1.0 u) v)))
  (if (<= u -3.6e+145) t_1 (if (<= u 1.6e+116) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-1.0 / u) * v;
	double tmp;
	if (u <= -3.6e+145) {
		tmp = t_1;
	} else if (u <= 1.6e+116) {
		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) / u) * v
    if (u <= (-3.6d+145)) then
        tmp = t_1
    else if (u <= 1.6d+116) 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 / u) * v;
	double tmp;
	if (u <= -3.6e+145) {
		tmp = t_1;
	} else if (u <= 1.6e+116) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-1.0 / u) * v
	tmp = 0
	if u <= -3.6e+145:
		tmp = t_1
	elif u <= 1.6e+116:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-1.0 / u) * v)
	tmp = 0.0
	if (u <= -3.6e+145)
		tmp = t_1;
	elseif (u <= 1.6e+116)
		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 / u) * v;
	tmp = 0.0;
	if (u <= -3.6e+145)
		tmp = t_1;
	elseif (u <= 1.6e+116)
		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 / u), $MachinePrecision] * v), $MachinePrecision]}, If[LessEqual[u, -3.6e+145], t$95$1, If[LessEqual[u, 1.6e+116], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.5999999999999997e145 or 1.6e116 < u

   1. Initial program 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. sqr-neg-rev N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lower-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. frac-2neg-rev N/A

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

      19. lower-/.f64 98.1%

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in u around 0

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

   6. Applied rewrites 61.7%

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

   7. Taylor expanded in u around inf

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

      1. lower-/.f64 16.9%

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

   9. Applied rewrites 16.9%

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

   if -3.5999999999999997e145 < u < 1.6e116

   1. Initial program 72.9%

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

   2. Taylor expanded in u around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 54.3%

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

   4. Applied rewrites 54.3%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 54.3%

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

   6. Applied rewrites 54.3%

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

Alternative 7: 54.3% accurate, 2.9× speedup

Math

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

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 72.9%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 54.3%

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

4. Applied rewrites 54.3%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 54.3%

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

6. Applied rewrites 54.3%

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

Reproduce

herbie shell --seed 2025212 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))