Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.1%

Time: 13.7s

Alternatives: 14

Speedup: 1.1×

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: 72.7% 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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. associate-/l* 72.5%

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

   2. distribute-lft-neg-out 72.5%

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

   3. distribute-rgt-neg-in 72.5%

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

   4. associate-/r* 83.4%

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

   5. distribute-neg-frac2 83.4%

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

3. Simplified 83.4%

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

   1. distribute-frac-neg2 83.4%

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

   2. distribute-rgt-neg-out 83.4%

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

   3. associate-/r* 72.5%

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

   4. distribute-lft-neg-out 72.5%

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

   5. associate-/l* 69.1%

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

   6. times-frac 98.4%

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

   7. frac-2neg 98.4%

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

   8. associate-*r/ 99.1%

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

   9. add-sqr-sqrt 52.5%

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

   10. sqrt-unprod 44.3%

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

   11. sqr-neg 44.3%

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

   12. sqrt-unprod 14.1%

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

   13. add-sqr-sqrt 32.7%

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

   14. add-sqr-sqrt 16.4%

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

   15. sqrt-unprod 53.4%

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

   16. sqr-neg 53.4%

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

   17. sqrt-prod 48.6%

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

   18. add-sqr-sqrt 99.1%

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

6. Applied egg-rr 99.1%

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

Alternative 2: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 1.25 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-v\right) \cdot \frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.1e-88) (not (<= t1 1.25e+68)))
   (/ v (- (* u -2.0) t1))
   (/ (* (- v) (/ t1 u)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-88) || !(t1 <= 1.25e+68)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-v * (t1 / u)) / 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 <= (-3.1d-88)) .or. (.not. (t1 <= 1.25d+68))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (-v * (t1 / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-88) || !(t1 <= 1.25e+68)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-v * (t1 / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.1e-88) or not (t1 <= 1.25e+68):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-v * (t1 / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.1e-88) || !(t1 <= 1.25e+68))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-v) * Float64(t1 / u)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.1e-88) || ~((t1 <= 1.25e+68)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-v * (t1 / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.1e-88], N[Not[LessEqual[t1, 1.25e+68]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-v) * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.0999999999999998e-88 or 1.2500000000000001e68 < t1

   1. Initial program 58.0%

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

      1. associate-/l* 62.3%

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

      2. distribute-lft-neg-out 62.3%

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

      3. distribute-rgt-neg-in 62.3%

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

      4. associate-/r* 76.0%

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

      5. distribute-neg-frac2 76.0%

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

   3. Simplified 76.0%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.8%

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

      6. clear-num 99.8%

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

      7. frac-2neg 99.8%

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

      8. frac-times 98.5%

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

      9. *-un-lft-identity 98.5%

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

      10. frac-2neg 98.5%

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

      11. sub-neg 98.5%

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

      12. distribute-neg-in 98.5%

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

      13. +-commutative 98.5%

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

      14. remove-double-neg 98.5%

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

      15. add-sqr-sqrt 61.4%

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

      16. sqrt-unprod 43.0%

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

      17. sqr-neg 43.0%

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

      18. sqrt-unprod 9.7%

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

      19. add-sqr-sqrt 27.7%

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

      20. add-sqr-sqrt 17.1%

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

      21. sqrt-unprod 41.8%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in t1 around inf 98.5%

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

   8. Taylor expanded in v around 0 98.5%

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

      1. distribute-lft-in 97.8%

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

      2. associate-*r/ 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-lft-in 98.5%

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

      5. associate-*r/ 98.5%

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

      6. neg-mul-1 98.5%

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

      7. distribute-neg-frac2 98.5%

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

      8. *-commutative 98.5%

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

      9. distribute-rgt-neg-in 98.5%

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

      10. distribute-neg-in 98.5%

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

      11. metadata-eval 98.5%

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

      12. unsub-neg 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in u around 0 81.2%

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

       1. neg-mul-1 81.2%

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

       2. unsub-neg 81.2%

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

       3. *-commutative 81.2%

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

   13. Simplified 81.2%

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

   if -3.0999999999999998e-88 < t1 < 1.2500000000000001e68

   1. Initial program 81.7%

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

      1. associate-*l/ 84.7%

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

      2. *-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in t1 around 0 71.1%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

      3. associate-/r* 75.0%

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

      4. frac-2neg 75.0%

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

      5. distribute-lft-neg-out 75.0%

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

      6. remove-double-neg 75.0%

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

      7. distribute-neg-in 75.0%

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

      8. add-sqr-sqrt 35.3%

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

      9. sqrt-unprod 76.4%

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

      10. sqr-neg 76.4%

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

      11. sqrt-unprod 40.2%

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

      12. add-sqr-sqrt 74.5%

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

      13. sub-neg 74.5%

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

   7. Applied egg-rr 74.5%

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

   8. Taylor expanded in t1 around 0 76.6%

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

      1. mul-1-neg 76.6%

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

      2. *-commutative 76.6%

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

      3. associate-*r/ 81.8%

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

      4. distribute-lft-neg-in 81.8%

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

   10. Simplified 81.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 1.25 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-v\right) \cdot \frac{t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 3.1 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.1e-88) (not (<= t1 3.1e+68)))
   (/ v (- (* u -2.0) t1))
   (* (/ t1 u) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-88) || !(t1 <= 3.1e+68)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / u) * (v / -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 <= (-3.1d-88)) .or. (.not. (t1 <= 3.1d+68))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (t1 / u) * (v / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-88) || !(t1 <= 3.1e+68)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / u) * (v / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.1e-88) or not (t1 <= 3.1e+68):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (t1 / u) * (v / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.1e-88) || !(t1 <= 3.1e+68))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(t1 / u) * Float64(v / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.1e-88) || ~((t1 <= 3.1e+68)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (t1 / u) * (v / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.1e-88], N[Not[LessEqual[t1, 3.1e+68]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[(v / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.0999999999999998e-88 or 3.0999999999999998e68 < t1

   1. Initial program 58.0%

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

      1. associate-/l* 62.3%

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

      2. distribute-lft-neg-out 62.3%

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

      3. distribute-rgt-neg-in 62.3%

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

      4. associate-/r* 76.0%

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

      5. distribute-neg-frac2 76.0%

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

   3. Simplified 76.0%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.8%

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

      6. clear-num 99.8%

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

      7. frac-2neg 99.8%

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

      8. frac-times 98.5%

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

      9. *-un-lft-identity 98.5%

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

      10. frac-2neg 98.5%

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

      11. sub-neg 98.5%

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

      12. distribute-neg-in 98.5%

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

      13. +-commutative 98.5%

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

      14. remove-double-neg 98.5%

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

      15. add-sqr-sqrt 61.4%

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

      16. sqrt-unprod 43.0%

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

      17. sqr-neg 43.0%

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

      18. sqrt-unprod 9.7%

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

      19. add-sqr-sqrt 27.7%

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

      20. add-sqr-sqrt 17.1%

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

      21. sqrt-unprod 41.8%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in t1 around inf 98.5%

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

   8. Taylor expanded in v around 0 98.5%

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

      1. distribute-lft-in 97.8%

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

      2. associate-*r/ 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-lft-in 98.5%

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

      5. associate-*r/ 98.5%

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

      6. neg-mul-1 98.5%

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

      7. distribute-neg-frac2 98.5%

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

      8. *-commutative 98.5%

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

      9. distribute-rgt-neg-in 98.5%

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

      10. distribute-neg-in 98.5%

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

      11. metadata-eval 98.5%

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

      12. unsub-neg 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in u around 0 81.2%

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

       1. neg-mul-1 81.2%

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

       2. unsub-neg 81.2%

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

       3. *-commutative 81.2%

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

   13. Simplified 81.2%

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

   if -3.0999999999999998e-88 < t1 < 3.0999999999999998e68

   1. Initial program 81.7%

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

      1. times-frac 96.7%

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

      2. distribute-frac-neg 96.7%

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

      3. distribute-neg-frac2 96.7%

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

      4. +-commutative 96.7%

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

      5. distribute-neg-in 96.7%

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

      6. unsub-neg 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in t1 around 0 78.7%

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

   6. Taylor expanded in t1 around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

   8. Simplified 80.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 3.1 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 8 \cdot 10^{+65}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.1e-88) (not (<= t1 8e+65)))
   (/ v (- (* u -2.0) t1))
   (* v (/ t1 (* u (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-88) || !(t1 <= 8e+65)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v * (t1 / (u * -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 <= (-3.1d-88)) .or. (.not. (t1 <= 8d+65))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = v * (t1 / (u * -u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-88) || !(t1 <= 8e+65)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v * (t1 / (u * -u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.1e-88) or not (t1 <= 8e+65):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = v * (t1 / (u * -u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.1e-88) || !(t1 <= 8e+65))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(-u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.1e-88) || ~((t1 <= 8e+65)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = v * (t1 / (u * -u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.1e-88], N[Not[LessEqual[t1, 8e+65]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(t1 / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.0999999999999998e-88 or 7.9999999999999999e65 < t1

   1. Initial program 57.6%

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

      1. associate-/l* 61.8%

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

      2. distribute-lft-neg-out 61.8%

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

      3. distribute-rgt-neg-in 61.8%

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

      4. associate-/r* 76.1%

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

      5. distribute-neg-frac2 76.1%

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

   3. Simplified 76.1%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.8%

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

      6. clear-num 99.8%

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

      7. frac-2neg 99.8%

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

      8. frac-times 98.5%

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

      9. *-un-lft-identity 98.5%

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

      10. frac-2neg 98.5%

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

      11. sub-neg 98.5%

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

      12. distribute-neg-in 98.5%

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

      13. +-commutative 98.5%

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

      14. remove-double-neg 98.5%

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

      15. add-sqr-sqrt 60.9%

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

      16. sqrt-unprod 42.7%

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

      17. sqr-neg 42.7%

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

      18. sqrt-unprod 9.6%

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

      19. add-sqr-sqrt 27.5%

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

      20. add-sqr-sqrt 17.0%

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

      21. sqrt-unprod 41.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in t1 around inf 98.5%

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

   8. Taylor expanded in v around 0 98.5%

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

      1. distribute-lft-in 97.8%

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

      2. associate-*r/ 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-lft-in 98.5%

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

      5. associate-*r/ 98.5%

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

      6. neg-mul-1 98.5%

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

      7. distribute-neg-frac2 98.5%

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

      8. *-commutative 98.5%

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

      9. distribute-rgt-neg-in 98.5%

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

      10. distribute-neg-in 98.5%

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

      11. metadata-eval 98.5%

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

      12. unsub-neg 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in u around 0 80.7%

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

       1. neg-mul-1 80.7%

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

       2. unsub-neg 80.7%

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

       3. *-commutative 80.7%

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

   13. Simplified 80.7%

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

   if -3.0999999999999998e-88 < t1 < 7.9999999999999999e65

   1. Initial program 82.4%

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

      1. associate-*l/ 85.4%

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

      2. *-commutative 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in t1 around 0 71.7%

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

   6. Taylor expanded in t1 around 0 73.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 8 \cdot 10^{+65}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 1.3 \cdot 10^{-216}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.1e-88) (not (<= t1 1.3e-216)))
   (/ (- v) (+ t1 u))
   (* v (/ (/ t1 u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-88) || !(t1 <= 1.3e-216)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * ((t1 / u) / 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 <= (-3.1d-88)) .or. (.not. (t1 <= 1.3d-216))) then
        tmp = -v / (t1 + u)
    else
        tmp = v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e-88) || !(t1 <= 1.3e-216)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.1e-88) or not (t1 <= 1.3e-216):
		tmp = -v / (t1 + u)
	else:
		tmp = v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.1e-88) || !(t1 <= 1.3e-216))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.1e-88) || ~((t1 <= 1.3e-216)))
		tmp = -v / (t1 + u);
	else
		tmp = v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.1e-88], N[Not[LessEqual[t1, 1.3e-216]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.0999999999999998e-88 or 1.2999999999999999e-216 < t1

   1. Initial program 64.2%

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

      1. associate-/l* 67.6%

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

      2. distribute-lft-neg-out 67.6%

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

      3. distribute-rgt-neg-in 67.6%

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

      4. associate-/r* 80.8%

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

      5. distribute-neg-frac2 80.8%

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

   3. Simplified 80.8%

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

      1. distribute-frac-neg2 80.8%

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

      2. distribute-rgt-neg-out 80.8%

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

      3. associate-/r* 67.6%

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

      4. distribute-lft-neg-out 67.6%

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

      5. associate-/l* 64.2%

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

      6. times-frac 99.3%

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

      7. frac-2neg 99.3%

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

      8. associate-*r/ 99.8%

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

      9. add-sqr-sqrt 45.6%

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

      10. sqrt-unprod 39.7%

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

      11. sqr-neg 39.7%

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

      12. sqrt-unprod 14.6%

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

      13. add-sqr-sqrt 27.8%

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

      14. add-sqr-sqrt 15.9%

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

      15. sqrt-unprod 50.4%

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

      16. sqr-neg 50.4%

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

      17. sqrt-prod 50.5%

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

      18. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t1 around inf 71.2%

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

      1. mul-1-neg 71.2%

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

   9. Simplified 71.2%

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

   if -3.0999999999999998e-88 < t1 < 1.2999999999999999e-216

   1. Initial program 82.1%

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

      1. associate-*l/ 84.9%

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

      2. *-commutative 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in t1 around 0 79.5%

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

   6. Taylor expanded in t1 around 0 79.6%

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

      1. associate-/r* 86.1%

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

      2. div-inv 86.2%

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

      3. add-sqr-sqrt 59.6%

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

      4. sqrt-unprod 50.5%

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

      5. sqr-neg 50.5%

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

      6. sqrt-unprod 12.8%

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

      7. add-sqr-sqrt 45.4%

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

   8. Applied egg-rr 45.4%

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

      1. associate-*r/ 45.4%

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

      2. *-rgt-identity 45.4%

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

   10. Simplified 45.4%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-88} \lor \neg \left(t1 \leq 1.3 \cdot 10^{-216}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+108}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+246}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3e+108)
   (/ v (- u))
   (if (<= u 5.2e+246) (/ v (- t1)) (/ 1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3e+108) {
		tmp = v / -u;
	} else if (u <= 5.2e+246) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	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 (u <= (-3d+108)) then
        tmp = v / -u
    else if (u <= 5.2d+246) then
        tmp = v / -t1
    else
        tmp = 1.0d0 / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3e+108) {
		tmp = v / -u;
	} else if (u <= 5.2e+246) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3e+108:
		tmp = v / -u
	elif u <= 5.2e+246:
		tmp = v / -t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3e+108)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 5.2e+246)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3e+108)
		tmp = v / -u;
	elseif (u <= 5.2e+246)
		tmp = v / -t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3e+108], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 5.2e+246], N[(v / (-t1)), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.99999999999999984e108

   1. Initial program 73.4%

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

      1. associate-*l/ 76.4%

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

      2. *-commutative 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in t1 around 0 76.4%

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

   6. Taylor expanded in t1 around inf 41.2%

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

      1. associate-*r/ 41.2%

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

      2. mul-1-neg 41.2%

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

   8. Simplified 41.2%

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

   if -2.99999999999999984e108 < u < 5.20000000000000028e246

   1. Initial program 66.6%

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

      1. associate-/l* 70.7%

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

      2. distribute-lft-neg-out 70.7%

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

      3. distribute-rgt-neg-in 70.7%

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

      4. associate-/r* 81.8%

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

      5. distribute-neg-frac2 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in t1 around inf 61.2%

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

      1. associate-*r/ 61.2%

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

      2. neg-mul-1 61.2%

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

   7. Simplified 61.2%

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

   if 5.20000000000000028e246 < u

   1. Initial program 92.9%

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

      1. associate-*l/ 93.2%

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

      2. *-commutative 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in t1 around 0 93.2%

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

   6. Taylor expanded in t1 around inf 50.8%

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

      1. associate-*r/ 50.8%

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

      2. mul-1-neg 50.8%

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

   8. Simplified 50.8%

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

      1. clear-num 59.3%

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

      2. inv-pow 59.3%

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

      3. add-sqr-sqrt 29.0%

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

      4. sqrt-unprod 59.1%

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

      5. sqr-neg 59.1%

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

      6. sqrt-unprod 30.8%

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

      7. add-sqr-sqrt 59.8%

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

   10. Applied egg-rr 59.8%

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

       1. unpow-1 59.8%

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

   12. Simplified 59.8%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+108}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+246}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+114} \lor \neg \left(u \leq 1.35 \cdot 10^{+246}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.35e+114) (not (<= u 1.35e+246))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.35e+114) || !(u <= 1.35e+246)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	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 ((u <= (-1.35d+114)) .or. (.not. (u <= 1.35d+246))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.35e+114) || !(u <= 1.35e+246)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.35e+114) or not (u <= 1.35e+246):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.35e+114) || !(u <= 1.35e+246))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.35e+114) || ~((u <= 1.35e+246)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.35e+114], N[Not[LessEqual[u, 1.35e+246]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.35e114 or 1.35e246 < u

   1. Initial program 78.1%

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

      1. associate-*l/ 80.4%

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

      2. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in t1 around 0 80.4%

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

      1. associate-*r/ 78.1%

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

      2. *-commutative 78.1%

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

      3. associate-/r* 86.1%

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

      4. frac-2neg 86.1%

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

      5. distribute-lft-neg-out 86.1%

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

      6. remove-double-neg 86.1%

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

      7. distribute-neg-in 86.1%

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

      8. add-sqr-sqrt 48.4%

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

      9. sqrt-unprod 84.9%

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

      10. sqr-neg 84.9%

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

      11. sqrt-unprod 37.8%

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

      12. add-sqr-sqrt 86.1%

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

      13. sub-neg 86.1%

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

   7. Applied egg-rr 86.1%

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

   8. Taylor expanded in t1 around inf 44.3%

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

   if -1.35e114 < u < 1.35e246

   1. Initial program 66.8%

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

      1. associate-/l* 70.8%

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

      2. distribute-lft-neg-out 70.8%

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

      3. distribute-rgt-neg-in 70.8%

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

      4. associate-/r* 81.9%

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

      5. distribute-neg-frac2 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in t1 around inf 60.9%

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

      1. associate-*r/ 60.9%

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

      2. neg-mul-1 60.9%

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

   7. Simplified 60.9%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+114} \lor \neg \left(u \leq 1.35 \cdot 10^{+246}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+108}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+246}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6e+108) (/ v (- u)) (if (<= u 1.35e+246) (/ v (- t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e+108) {
		tmp = v / -u;
	} else if (u <= 1.35e+246) {
		tmp = v / -t1;
	} else {
		tmp = v / 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 (u <= (-6d+108)) then
        tmp = v / -u
    else if (u <= 1.35d+246) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e+108) {
		tmp = v / -u;
	} else if (u <= 1.35e+246) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6e+108:
		tmp = v / -u
	elif u <= 1.35e+246:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6e+108)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 1.35e+246)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6e+108)
		tmp = v / -u;
	elseif (u <= 1.35e+246)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6e+108], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 1.35e+246], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.99999999999999968e108

   1. Initial program 73.4%

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

      1. associate-*l/ 76.4%

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

      2. *-commutative 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in t1 around 0 76.4%

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

   6. Taylor expanded in t1 around inf 41.2%

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

      1. associate-*r/ 41.2%

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

      2. mul-1-neg 41.2%

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

   8. Simplified 41.2%

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

   if -5.99999999999999968e108 < u < 1.35e246

   1. Initial program 66.6%

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

      1. associate-/l* 70.7%

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

      2. distribute-lft-neg-out 70.7%

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

      3. distribute-rgt-neg-in 70.7%

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

      4. associate-/r* 81.8%

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

      5. distribute-neg-frac2 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in t1 around inf 61.2%

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

      1. associate-*r/ 61.2%

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

      2. neg-mul-1 61.2%

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

   7. Simplified 61.2%

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

   if 1.35e246 < u

   1. Initial program 92.9%

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

      1. associate-*l/ 93.2%

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

      2. *-commutative 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in t1 around 0 93.2%

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

      1. associate-*r/ 92.9%

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

      2. *-commutative 92.9%

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

      3. associate-/r* 93.0%

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

      4. frac-2neg 93.0%

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

      5. distribute-lft-neg-out 93.0%

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

      6. remove-double-neg 93.0%

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

      7. distribute-neg-in 93.0%

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

      8. add-sqr-sqrt 57.3%

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

      9. sqrt-unprod 92.9%

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

      10. sqr-neg 92.9%

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

      11. sqrt-unprod 35.7%

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

      12. add-sqr-sqrt 93.0%

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

      13. sub-neg 93.0%

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

   7. Applied egg-rr 93.0%

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

   8. Taylor expanded in t1 around inf 51.3%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+108}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+246}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{+64} \lor \neg \left(t1 \leq 5 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.45e+64) (not (<= t1 5e+126))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.45e+64) || !(t1 <= 5e+126)) {
		tmp = v / t1;
	} else {
		tmp = v / 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 <= (-1.45d+64)) .or. (.not. (t1 <= 5d+126))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.45e+64) || !(t1 <= 5e+126)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.45e+64) or not (t1 <= 5e+126):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.45e+64) || !(t1 <= 5e+126))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.45e+64) || ~((t1 <= 5e+126)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.45e+64], N[Not[LessEqual[t1, 5e+126]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.44999999999999997e64 or 4.99999999999999977e126 < t1

   1. Initial program 43.5%

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

      1. associate-/l* 48.8%

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

      2. distribute-lft-neg-out 48.8%

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

      3. distribute-rgt-neg-in 48.8%

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

      4. associate-/r* 67.4%

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

      5. distribute-neg-frac2 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in t1 around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   7. Simplified 86.2%

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

      1. neg-sub0 86.2%

         \[\leadsto \frac{\color{blue}{0 - v}}{t1} \]

      2. sub-neg 86.2%

         \[\leadsto \frac{\color{blue}{0 + \left(-v\right)}}{t1} \]

      3. add-sqr-sqrt 39.1%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

      4. sqrt-unprod 41.3%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1} \]

      5. sqr-neg 41.3%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{v \cdot v}}}{t1} \]

      6. sqrt-unprod 12.7%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

      7. add-sqr-sqrt 26.7%

         \[\leadsto \frac{0 + \color{blue}{v}}{t1} \]

   9. Applied egg-rr 26.7%

      \[\leadsto \frac{\color{blue}{0 + v}}{t1} \]
   10. Step-by-step derivation

       1. +-lft-identity 26.7%

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

   11. Simplified 26.7%

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

   if -1.44999999999999997e64 < t1 < 4.99999999999999977e126

   1. Initial program 83.0%

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

      1. associate-*l/ 87.1%

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

      2. *-commutative 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in t1 around 0 63.2%

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

      1. associate-*r/ 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-/r* 67.1%

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

      4. frac-2neg 67.1%

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

      5. distribute-lft-neg-out 67.1%

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

      6. remove-double-neg 67.1%

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

      7. distribute-neg-in 67.1%

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

      8. add-sqr-sqrt 35.9%

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

      9. sqrt-unprod 68.1%

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

      10. sqr-neg 68.1%

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

      11. sqrt-unprod 31.6%

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

      12. add-sqr-sqrt 67.2%

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

      13. sub-neg 67.2%

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

   7. Applied egg-rr 67.2%

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

   8. Taylor expanded in t1 around inf 18.4%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{+64} \lor \neg \left(t1 \leq 5 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 4.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 4.2e+117) (/ v (- (* u -2.0) t1)) (* v (/ (/ t1 u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 4.2e+117) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v * ((t1 / u) / 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 (u <= 4.2d+117) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 4.2e+117) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 4.2e+117:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 4.2e+117)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 4.2e+117)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 4.2e+117], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 4.2000000000000002e117

   1. Initial program 69.3%

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

      1. associate-/l* 73.1%

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

      2. distribute-lft-neg-out 73.1%

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

      3. distribute-rgt-neg-in 73.1%

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

      4. associate-/r* 82.6%

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

      5. distribute-neg-frac2 82.6%

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

   3. Simplified 82.6%

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

      1. associate-*r/ 98.1%

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

      2. +-commutative 98.1%

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

      3. distribute-neg-in 98.1%

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

      4. sub-neg 98.1%

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

      5. associate-*l/ 98.1%

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

      6. clear-num 98.0%

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

      7. frac-2neg 98.0%

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

      8. frac-times 96.8%

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

      9. *-un-lft-identity 96.8%

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

      10. frac-2neg 96.8%

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

      11. sub-neg 96.8%

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

      12. distribute-neg-in 96.8%

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

      13. +-commutative 96.8%

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

      14. remove-double-neg 96.8%

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

      15. add-sqr-sqrt 51.6%

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

      16. sqrt-unprod 39.3%

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

      17. sqr-neg 39.3%

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

      18. sqrt-unprod 12.0%

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

      19. add-sqr-sqrt 27.4%

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

      20. add-sqr-sqrt 19.0%

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

      21. sqrt-unprod 50.5%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in t1 around inf 96.8%

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

   8. Taylor expanded in v around 0 96.8%

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

      1. distribute-lft-in 95.4%

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

      2. associate-*r/ 95.4%

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

      3. +-commutative 95.4%

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

      4. distribute-lft-in 96.8%

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

      5. associate-*r/ 96.8%

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

      6. neg-mul-1 96.8%

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

      7. distribute-neg-frac2 96.8%

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

      8. *-commutative 96.8%

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

      9. distribute-rgt-neg-in 96.8%

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

      10. distribute-neg-in 96.8%

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

      11. metadata-eval 96.8%

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

      12. unsub-neg 96.8%

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

   10. Simplified 96.8%

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

   11. Taylor expanded in u around 0 63.9%

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

       1. neg-mul-1 63.9%

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

       2. unsub-neg 63.9%

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

       3. *-commutative 63.9%

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

   13. Simplified 63.9%

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

   if 4.2000000000000002e117 < u

   1. Initial program 68.5%

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

      1. associate-*l/ 69.0%

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

      2. *-commutative 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in t1 around 0 69.0%

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

   6. Taylor expanded in t1 around 0 69.1%

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

      1. associate-/r* 82.9%

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

      2. div-inv 82.8%

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

      3. add-sqr-sqrt 49.9%

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

      4. sqrt-unprod 68.2%

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

      5. sqr-neg 68.2%

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

      6. sqrt-unprod 25.8%

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

      7. add-sqr-sqrt 61.4%

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

   8. Applied egg-rr 61.4%

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

      1. associate-*r/ 61.4%

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

      2. *-rgt-identity 61.4%

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

   10. Simplified 61.4%

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

Alternative 11: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. times-frac 98.4%

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

   2. distribute-frac-neg 98.4%

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

   3. distribute-neg-frac2 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-neg-in 98.4%

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

   6. unsub-neg 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 12: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return v / ((t1 + u) * (-1.0 - (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 = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. associate-/l* 72.5%

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

   2. distribute-lft-neg-out 72.5%

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

   3. distribute-rgt-neg-in 72.5%

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

   4. associate-/r* 83.4%

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

   5. distribute-neg-frac2 83.4%

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

3. Simplified 83.4%

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

   1. associate-*r/ 98.3%

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

   2. +-commutative 98.3%

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

   3. distribute-neg-in 98.3%

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

   4. sub-neg 98.3%

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

   5. associate-*l/ 98.4%

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

   6. clear-num 98.3%

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

   7. frac-2neg 98.3%

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

   8. frac-times 96.5%

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

   9. *-un-lft-identity 96.5%

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

   10. frac-2neg 96.5%

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

   11. sub-neg 96.5%

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

   12. distribute-neg-in 96.5%

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

   13. +-commutative 96.5%

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

   14. remove-double-neg 96.5%

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

   15. add-sqr-sqrt 51.7%

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

   16. sqrt-unprod 43.8%

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

   17. sqr-neg 43.8%

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

   18. sqrt-unprod 14.1%

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

   19. add-sqr-sqrt 32.7%

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

   20. add-sqr-sqrt 16.4%

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

   21. sqrt-unprod 53.4%

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

6. Applied egg-rr 96.5%

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

7. Taylor expanded in t1 around inf 96.5%

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

8. Taylor expanded in v around 0 96.5%

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

   1. distribute-lft-in 91.1%

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

   2. associate-*r/ 91.1%

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

   3. +-commutative 91.1%

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

   4. distribute-lft-in 96.5%

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

   5. associate-*r/ 96.5%

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

   6. neg-mul-1 96.5%

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

   7. distribute-neg-frac2 96.5%

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

   8. *-commutative 96.5%

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

   9. distribute-rgt-neg-in 96.5%

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

   10. distribute-neg-in 96.5%

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

   11. metadata-eval 96.5%

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

   12. unsub-neg 96.5%

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

10. Simplified 96.5%

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

11. Final simplification 96.5%

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

Alternative 13: 61.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. associate-/l* 72.5%

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

   2. distribute-lft-neg-out 72.5%

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

   3. distribute-rgt-neg-in 72.5%

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

   4. associate-/r* 83.4%

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

   5. distribute-neg-frac2 83.4%

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

3. Simplified 83.4%

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

   1. distribute-frac-neg2 83.4%

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

   2. distribute-rgt-neg-out 83.4%

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

   3. associate-/r* 72.5%

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

   4. distribute-lft-neg-out 72.5%

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

   5. associate-/l* 69.1%

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

   6. times-frac 98.4%

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

   7. frac-2neg 98.4%

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

   8. associate-*r/ 99.1%

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

   9. add-sqr-sqrt 52.5%

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

   10. sqrt-unprod 44.3%

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

   11. sqr-neg 44.3%

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

   12. sqrt-unprod 14.1%

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

   13. add-sqr-sqrt 32.7%

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

   14. add-sqr-sqrt 16.4%

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

   15. sqrt-unprod 53.4%

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

   16. sqr-neg 53.4%

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

   17. sqrt-prod 48.6%

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

   18. add-sqr-sqrt 99.1%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in t1 around inf 59.3%

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

   1. mul-1-neg 59.3%

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

9. Simplified 59.3%

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

Alternative 14: 14.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. associate-/l* 72.5%

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

   2. distribute-lft-neg-out 72.5%

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

   3. distribute-rgt-neg-in 72.5%

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

   4. associate-/r* 83.4%

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

   5. distribute-neg-frac2 83.4%

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

3. Simplified 83.4%

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

5. Taylor expanded in t1 around inf 51.8%

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

   1. associate-*r/ 51.8%

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

   2. neg-mul-1 51.8%

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

7. Simplified 51.8%

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

   1. neg-sub0 51.8%

      \[\leadsto \frac{\color{blue}{0 - v}}{t1} \]

   2. sub-neg 51.8%

      \[\leadsto \frac{\color{blue}{0 + \left(-v\right)}}{t1} \]

   3. add-sqr-sqrt 23.7%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

   4. sqrt-unprod 30.7%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1} \]

   5. sqr-neg 30.7%

      \[\leadsto \frac{0 + \sqrt{\color{blue}{v \cdot v}}}{t1} \]

   6. sqrt-unprod 5.4%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

   7. add-sqr-sqrt 11.1%

      \[\leadsto \frac{0 + \color{blue}{v}}{t1} \]

9. Applied egg-rr 11.1%

   \[\leadsto \frac{\color{blue}{0 + v}}{t1} \]
10. Step-by-step derivation

    1. +-lft-identity 11.1%

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

11. Simplified 11.1%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))