Rosa's DopplerBench

Percentage Accurate: 72.0% → 98.0%

Time: 12.0s

Alternatives: 11

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

real(8) function code(u, v, t1)
    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: 72.0% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 98.4

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

4. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

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

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

   4. distribute-rgt-neg-in N/A

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

   5. lift-neg.f64 N/A

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

   6. remove-double-neg N/A

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

   7. lower-*.f64 98.4

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

6. Applied rewrites 98.4%

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

7. Final simplification 98.4%

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

Alternative 2: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-1 + \frac{u}{t1}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{+108}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -1}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -9.2e+156)
   (* v (/ (+ -1.0 (/ u t1)) (+ t1 u)))
   (if (<= t1 1.5e+108)
     (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))
     (/ (* v -1.0) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.2e+156) {
		tmp = v * ((-1.0 + (u / t1)) / (t1 + u));
	} else if (t1 <= 1.5e+108) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = (v * -1.0) / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-9.2d+156)) then
        tmp = v * (((-1.0d0) + (u / t1)) / (t1 + u))
    else if (t1 <= 1.5d+108) then
        tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
    else
        tmp = (v * (-1.0d0)) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.2e+156) {
		tmp = v * ((-1.0 + (u / t1)) / (t1 + u));
	} else if (t1 <= 1.5e+108) {
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = (v * -1.0) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -9.2e+156:
		tmp = v * ((-1.0 + (u / t1)) / (t1 + u))
	elif t1 <= 1.5e+108:
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = (v * -1.0) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -9.2e+156)
		tmp = Float64(v * Float64(Float64(-1.0 + Float64(u / t1)) / Float64(t1 + u)));
	elseif (t1 <= 1.5e+108)
		tmp = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(Float64(v * -1.0) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -9.2e+156)
		tmp = v * ((-1.0 + (u / t1)) / (t1 + u));
	elseif (t1 <= 1.5e+108)
		tmp = v * (-t1 / ((t1 + u) * (t1 + u)));
	else
		tmp = (v * -1.0) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -9.2e+156], N[(v * N[(N[(-1.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.5e+108], N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v * -1.0), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.1999999999999995e156

   1. Initial program 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 40.1

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

   4. Applied rewrites 40.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-/.f64 96.9

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-neg-frac2 N/A

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

      12. lift-neg.f64 N/A

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

      13. lower-/.f64 96.9

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

      14. lift-neg.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-neg-in N/A

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

      17. lift-neg.f64 N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 96.9

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

   6. Applied rewrites 96.9%

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

   7. Taylor expanded in t1 around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 90.3

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

   9. Applied rewrites 90.3%

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

   if -9.1999999999999995e156 < t1 < 1.49999999999999992e108

   1. Initial program 83.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.3

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

   4. Applied rewrites 87.3%

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

   if 1.49999999999999992e108 < t1

   1. Initial program 47.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t1 around inf

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

   7. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. distribute-frac-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. remove-double-neg N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 91.4

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

   9. Applied rewrites 91.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+156}:\\ \;\;\;\;v \cdot \frac{-1 + \frac{u}{t1}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{+108}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot -1}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024225 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))