Rosa's DopplerBench

Percentage Accurate: 73.2% → 98.0%

Time: 18.4s

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: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{v}{-1 - \frac{u}{t1}}}{u + t1} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. associate-/l* 74.8%

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

   2. distribute-lft-neg-out 74.8%

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

   3. distribute-rgt-neg-in 74.8%

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

   4. associate-/r* 85.8%

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

   5. distribute-neg-frac2 85.8%

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

3. Simplified 85.8%

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

   1. distribute-frac-neg2 85.8%

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

   2. distribute-rgt-neg-out 85.8%

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

   3. associate-/r* 74.8%

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

   4. distribute-lft-neg-out 74.8%

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

   5. associate-/l* 74.4%

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

   6. times-frac 97.4%

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

   7. frac-2neg 97.4%

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

   8. associate-*r/ 97.8%

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

   9. add-sqr-sqrt 45.0%

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

   10. sqrt-unprod 43.7%

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

   11. sqr-neg 43.7%

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

   12. sqrt-unprod 22.0%

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

   13. add-sqr-sqrt 37.9%

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

   14. add-sqr-sqrt 14.8%

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

   15. sqrt-unprod 57.1%

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

   16. sqr-neg 57.1%

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

   17. sqrt-prod 50.9%

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

   18. add-sqr-sqrt 97.8%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in v around 0 85.7%

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

   1. mul-1-neg 85.7%

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

   2. associate-*r/ 98.1%

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

   3. *-commutative 98.1%

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

   4. associate-/r/ 97.8%

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

   5. distribute-neg-frac 97.8%

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

   6. *-lft-identity 97.8%

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

   7. associate-*l/ 97.7%

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

   8. +-commutative 97.7%

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

   9. distribute-rgt-in 97.7%

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

   10. rgt-mult-inverse 97.7%

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

   11. rgt-mult-inverse 97.6%

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

   12. fma-define 97.6%

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

   13. rgt-mult-inverse 97.7%

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

9. Simplified 97.7%

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

    1. fma-undefine 97.7%

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

    2. un-div-inv 97.8%

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

11. Applied egg-rr 97.8%

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

12. Final simplification 97.8%

    \[\leadsto \frac{\frac{v}{-1 - \frac{u}{t1}}}{u + t1} \]
13. Add Preprocessing

Alternative 2: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.6 \cdot 10^{+134}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+195}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u + t1}}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -3.6e+134)
   (/ v (- (- t1) (* u 2.0)))
   (if (<= t1 9e+195) (* t1 (/ (/ v (+ u t1)) (- (- t1) u))) (/ v (- t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.6e+134) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 9e+195) {
		tmp = t1 * ((v / (u + t1)) / (-t1 - 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 (t1 <= (-3.6d+134)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (t1 <= 9d+195) then
        tmp = t1 * ((v / (u + t1)) / (-t1 - u))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.6e+134) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 9e+195) {
		tmp = t1 * ((v / (u + t1)) / (-t1 - u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -3.6e+134:
		tmp = v / (-t1 - (u * 2.0))
	elif t1 <= 9e+195:
		tmp = t1 * ((v / (u + t1)) / (-t1 - u))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -3.6e+134)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (t1 <= 9e+195)
		tmp = Float64(t1 * Float64(Float64(v / Float64(u + t1)) / Float64(Float64(-t1) - u)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -3.6e+134)
		tmp = v / (-t1 - (u * 2.0));
	elseif (t1 <= 9e+195)
		tmp = t1 * ((v / (u + t1)) / (-t1 - u));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -3.6e+134], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 9e+195], N[(t1 * N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.59999999999999988e134

   1. Initial program 39.5%

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

      1. associate-/l* 38.2%

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

      2. distribute-lft-neg-out 38.2%

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

      3. distribute-rgt-neg-in 38.2%

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

      4. associate-/r* 72.1%

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

      5. distribute-neg-frac2 72.1%

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

   3. Simplified 72.1%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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/ 100.0%

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

      6. frac-2neg 100.0%

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

      7. clear-num 100.0%

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

      8. frac-times 100.0%

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

      9. *-un-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-neg-in 100.0%

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

      12. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 95.7%

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

      1. *-commutative 95.7%

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

   9. Simplified 95.7%

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

   if -3.59999999999999988e134 < t1 < 9.00000000000000018e195

   1. Initial program 83.2%

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

      1. associate-/l* 83.7%

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

      2. distribute-lft-neg-out 83.7%

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

      3. distribute-rgt-neg-in 83.7%

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

      4. associate-/r* 90.9%

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

      5. distribute-neg-frac2 90.9%

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

   3. Simplified 90.9%

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

   if 9.00000000000000018e195 < t1

   1. Initial program 53.6%

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

      1. associate-/l* 54.8%

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

      2. distribute-lft-neg-out 54.8%

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

      3. distribute-rgt-neg-in 54.8%

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

      4. associate-/r* 61.3%

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

      5. distribute-neg-frac2 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in t1 around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.6 \cdot 10^{+134}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+195}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u + t1}}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{-99} \lor \neg \left(t1 \leq 2 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{-t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.1e-99) (not (<= t1 2e-97)))
   (/ v (- (- t1) (* u 2.0)))
   (/ v (* u (/ u (- t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.1e-99) || !(t1 <= 2e-97)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = v / (u * (u / -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 ((t1 <= (-1.1d-99)) .or. (.not. (t1 <= 2d-97))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = v / (u * (u / -t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.1e-99) || !(t1 <= 2e-97)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = v / (u * (u / -t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.1e-99) or not (t1 <= 2e-97):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = v / (u * (u / -t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.1e-99) || !(t1 <= 2e-97))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(v / Float64(u * Float64(u / Float64(-t1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.1e-99) || ~((t1 <= 2e-97)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = v / (u * (u / -t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.1e-99], N[Not[LessEqual[t1, 2e-97]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[(u / (-t1)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.10000000000000002e-99 or 2.00000000000000007e-97 < t1

   1. Initial program 69.2%

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

      1. associate-/l* 70.4%

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

      2. distribute-lft-neg-out 70.4%

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

      3. distribute-rgt-neg-in 70.4%

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

      4. associate-/r* 84.0%

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

      5. distribute-neg-frac2 84.0%

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

   3. Simplified 84.0%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. clear-num 99.9%

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

      8. frac-times 96.1%

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

      9. *-un-lft-identity 96.1%

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

      10. +-commutative 96.1%

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

      11. distribute-neg-in 96.1%

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

      12. sub-neg 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in u around 0 80.8%

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

      1. *-commutative 80.8%

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

   9. Simplified 80.8%

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

   if -1.10000000000000002e-99 < t1 < 2.00000000000000007e-97

   1. Initial program 85.4%

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

      1. times-frac 92.1%

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

      2. distribute-frac-neg 92.1%

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

      3. distribute-neg-frac2 92.1%

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

      4. +-commutative 92.1%

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

      5. distribute-neg-in 92.1%

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

      6. unsub-neg 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in t1 around 0 85.4%

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

      1. associate-*r/ 85.4%

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

      2. mul-1-neg 85.4%

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

   7. Simplified 85.4%

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

      1. clear-num 85.3%

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

      2. frac-2neg 85.3%

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

      3. frac-times 88.7%

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

      4. *-un-lft-identity 88.7%

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

      5. add-sqr-sqrt 45.5%

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

      6. sqrt-unprod 52.4%

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

      7. sqr-neg 52.4%

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

      8. sqrt-unprod 19.2%

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

      9. add-sqr-sqrt 48.2%

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

      10. add-sqr-sqrt 20.3%

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

      11. sqrt-unprod 51.6%

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

      12. sqr-neg 51.6%

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

      13. sqrt-unprod 55.2%

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

      14. add-sqr-sqrt 88.7%

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

      15. distribute-neg-in 88.7%

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

      16. add-sqr-sqrt 33.2%

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

      17. sqrt-unprod 91.1%

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

      18. sqr-neg 91.1%

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

      19. sqrt-unprod 56.6%

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

      20. add-sqr-sqrt 88.7%

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

      21. sub-neg 88.7%

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

   9. Applied egg-rr 88.7%

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

   10. Taylor expanded in t1 around 0 91.2%

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

       1. mul-1-neg 91.2%

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

   12. Simplified 91.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{-99} \lor \neg \left(t1 \leq 2 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{-t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.8 \cdot 10^{-104} \lor \neg \left(t1 \leq 5.2 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{-t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.8e-104) (not (<= t1 5.2e-89)))
   (/ v (- u t1))
   (/ v (* u (/ u (- t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.8e-104) || !(t1 <= 5.2e-89)) {
		tmp = v / (u - t1);
	} else {
		tmp = v / (u * (u / -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 ((t1 <= (-9.8d-104)) .or. (.not. (t1 <= 5.2d-89))) then
        tmp = v / (u - t1)
    else
        tmp = v / (u * (u / -t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.8e-104) || !(t1 <= 5.2e-89)) {
		tmp = v / (u - t1);
	} else {
		tmp = v / (u * (u / -t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.8e-104) or not (t1 <= 5.2e-89):
		tmp = v / (u - t1)
	else:
		tmp = v / (u * (u / -t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.8e-104) || !(t1 <= 5.2e-89))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(v / Float64(u * Float64(u / Float64(-t1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9.8e-104) || ~((t1 <= 5.2e-89)))
		tmp = v / (u - t1);
	else
		tmp = v / (u * (u / -t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.8e-104], N[Not[LessEqual[t1, 5.2e-89]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[(u / (-t1)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.8000000000000006e-104 or 5.1999999999999997e-89 < t1

   1. Initial program 68.8%

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

      1. associate-/l* 70.1%

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

      2. distribute-lft-neg-out 70.1%

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

      3. distribute-rgt-neg-in 70.1%

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

      4. associate-/r* 83.8%

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

      5. distribute-neg-frac2 83.8%

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

   3. Simplified 83.8%

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

      1. distribute-frac-neg2 83.8%

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

      2. distribute-rgt-neg-out 83.8%

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

      3. associate-/r* 70.1%

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

      4. distribute-lft-neg-out 70.1%

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

      5. associate-/l* 68.8%

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

      6. times-frac 99.9%

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

      7. frac-2neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 50.6%

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

      10. sqrt-unprod 41.7%

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

      11. sqr-neg 41.7%

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

      12. sqrt-unprod 20.0%

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

      13. add-sqr-sqrt 34.5%

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

      14. add-sqr-sqrt 13.0%

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

      15. sqrt-unprod 52.8%

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

      16. sqr-neg 52.8%

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

      17. sqrt-prod 50.7%

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

      18. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 80.8%

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

      1. mul-1-neg 80.8%

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

   9. Simplified 80.8%

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

       1. distribute-frac-neg 80.8%

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

       2. neg-sub0 80.8%

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

       3. frac-2neg 80.8%

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

       4. add-sqr-sqrt 40.7%

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

       5. sqrt-unprod 47.3%

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

       6. sqr-neg 47.3%

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

       7. sqrt-unprod 14.6%

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

       8. add-sqr-sqrt 28.9%

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

       9. distribute-neg-in 28.9%

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

       10. add-sqr-sqrt 12.2%

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

       11. sqrt-unprod 44.5%

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

       12. sqr-neg 44.5%

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

       13. sqrt-unprod 38.9%

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

       14. add-sqr-sqrt 80.9%

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

       15. sub-neg 80.9%

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

   11. Applied egg-rr 80.9%

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

       1. neg-sub0 80.9%

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

       2. distribute-neg-frac 80.9%

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

   13. Simplified 80.9%

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

       1. frac-2neg 80.9%

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

       2. div-inv 80.7%

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

       3. remove-double-neg 80.7%

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

       4. sub-neg 80.7%

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

       5. distribute-neg-in 80.7%

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

       6. add-sqr-sqrt 40.1%

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

       7. sqrt-unprod 82.6%

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

       8. sqr-neg 82.6%

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

       9. sqrt-unprod 40.5%

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

       10. add-sqr-sqrt 80.6%

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

       11. distribute-neg-in 80.6%

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

       12. +-commutative 80.6%

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

       13. distribute-neg-in 80.6%

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

       14. add-sqr-sqrt 40.1%

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

       15. sqrt-unprod 82.7%

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

       16. sqr-neg 82.7%

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

       17. sqrt-unprod 40.6%

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

       18. add-sqr-sqrt 80.7%

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

   15. Applied egg-rr 80.7%

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

       1. sub-neg 80.7%

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

       2. associate-*r/ 80.9%

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

       3. *-rgt-identity 80.9%

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

   17. Simplified 80.9%

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

   if -9.8000000000000006e-104 < t1 < 5.1999999999999997e-89

   1. Initial program 85.7%

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

      1. times-frac 92.3%

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

      2. distribute-frac-neg 92.3%

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

      3. distribute-neg-frac2 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-neg-in 92.3%

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

      6. unsub-neg 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in t1 around 0 84.7%

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

      1. associate-*r/ 84.7%

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

      2. mul-1-neg 84.7%

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

   7. Simplified 84.7%

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

      1. clear-num 84.6%

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

      2. frac-2neg 84.6%

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

      3. frac-times 87.8%

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

      4. *-un-lft-identity 87.8%

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

      5. add-sqr-sqrt 45.6%

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

      6. sqrt-unprod 52.3%

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

      7. sqr-neg 52.3%

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

      8. sqrt-unprod 18.7%

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

      9. add-sqr-sqrt 47.2%

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

      10. add-sqr-sqrt 19.9%

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

      11. sqrt-unprod 51.6%

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

      12. sqr-neg 51.6%

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

      13. sqrt-unprod 55.1%

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

      14. add-sqr-sqrt 87.8%

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

      15. distribute-neg-in 87.8%

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

      16. add-sqr-sqrt 32.5%

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

      17. sqrt-unprod 90.1%

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

      18. sqr-neg 90.1%

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

      19. sqrt-unprod 56.5%

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

      20. add-sqr-sqrt 87.8%

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

      21. sub-neg 87.8%

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

   9. Applied egg-rr 87.8%

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

   10. Taylor expanded in t1 around 0 90.3%

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

       1. mul-1-neg 90.3%

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

   12. Simplified 90.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.8 \cdot 10^{-104} \lor \neg \left(t1 \leq 5.2 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{-t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{-60} \lor \neg \left(t1 \leq 2.9 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.5e-60) (not (<= t1 2.9e-19)))
   (/ v (- u t1))
   (* (/ (- t1) u) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.5e-60) || !(t1 <= 2.9e-19)) {
		tmp = v / (u - 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 <= (-9.5d-60)) .or. (.not. (t1 <= 2.9d-19))) then
        tmp = v / (u - t1)
    else
        tmp = (-t1 / u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.5e-60) || !(t1 <= 2.9e-19)) {
		tmp = v / (u - t1);
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.5e-60) or not (t1 <= 2.9e-19):
		tmp = v / (u - t1)
	else:
		tmp = (-t1 / u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.5e-60) || !(t1 <= 2.9e-19))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9.5e-60) || ~((t1 <= 2.9e-19)))
		tmp = v / (u - t1);
	else
		tmp = (-t1 / u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.5e-60], N[Not[LessEqual[t1, 2.9e-19]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.49999999999999958e-60 or 2.9e-19 < t1

   1. Initial program 67.4%

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

      1. associate-/l* 68.8%

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

      2. distribute-lft-neg-out 68.8%

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

      3. distribute-rgt-neg-in 68.8%

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

      4. associate-/r* 83.0%

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

      5. distribute-neg-frac2 83.0%

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

   3. Simplified 83.0%

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

      1. distribute-frac-neg2 83.0%

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

      2. distribute-rgt-neg-out 83.0%

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

      3. associate-/r* 68.8%

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

      4. distribute-lft-neg-out 68.8%

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

      5. associate-/l* 67.4%

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

      6. times-frac 99.9%

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

      7. frac-2neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 52.3%

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

      10. sqrt-unprod 40.1%

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

      11. sqr-neg 40.1%

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

      12. sqrt-unprod 20.4%

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

      13. add-sqr-sqrt 35.3%

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

      14. add-sqr-sqrt 13.2%

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

      15. sqrt-unprod 51.4%

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

      16. sqr-neg 51.4%

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

      17. sqrt-prod 49.1%

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

      18. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 83.6%

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

      1. mul-1-neg 83.6%

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

   9. Simplified 83.6%

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

       1. distribute-frac-neg 83.6%

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

       2. neg-sub0 83.6%

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

       3. frac-2neg 83.6%

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

       4. add-sqr-sqrt 41.7%

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

       5. sqrt-unprod 49.9%

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

       6. sqr-neg 49.9%

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

       7. sqrt-unprod 15.9%

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

       8. add-sqr-sqrt 30.5%

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

       9. distribute-neg-in 30.5%

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

       10. add-sqr-sqrt 13.2%

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

       11. sqrt-unprod 44.7%

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

       12. sqr-neg 44.7%

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

       13. sqrt-unprod 39.0%

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

       14. add-sqr-sqrt 83.7%

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

       15. sub-neg 83.7%

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

   11. Applied egg-rr 83.7%

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

       1. neg-sub0 83.7%

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

       2. distribute-neg-frac 83.7%

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

   13. Simplified 83.7%

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

       1. frac-2neg 83.7%

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

       2. div-inv 83.5%

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

       3. remove-double-neg 83.5%

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

       4. sub-neg 83.5%

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

       5. distribute-neg-in 83.5%

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

       6. add-sqr-sqrt 41.0%

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

       7. sqrt-unprod 84.1%

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

       8. sqr-neg 84.1%

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

       9. sqrt-unprod 42.4%

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

       10. add-sqr-sqrt 83.4%

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

       11. distribute-neg-in 83.4%

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

       12. +-commutative 83.4%

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

       13. distribute-neg-in 83.4%

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

       14. add-sqr-sqrt 41.0%

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

       15. sqrt-unprod 84.3%

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

       16. sqr-neg 84.3%

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

       17. sqrt-unprod 42.5%

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

       18. add-sqr-sqrt 83.5%

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

   15. Applied egg-rr 83.5%

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

       1. sub-neg 83.5%

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

       2. associate-*r/ 83.7%

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

       3. *-rgt-identity 83.7%

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

   17. Simplified 83.7%

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

   if -9.49999999999999958e-60 < t1 < 2.9e-19

   1. Initial program 84.7%

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

      1. times-frac 93.7%

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

      2. distribute-frac-neg 93.7%

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

      3. distribute-neg-frac2 93.7%

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

      4. +-commutative 93.7%

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

      5. distribute-neg-in 93.7%

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

      6. unsub-neg 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in t1 around 0 79.9%

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

      1. associate-*r/ 79.9%

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

      2. mul-1-neg 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in t1 around 0 82.1%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{-60} \lor \neg \left(t1 \leq 2.9 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-55} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.12e-55) (not (<= t1 1.35e-19)))
   (/ v (- u t1))
   (* t1 (/ (/ v u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.12e-55) || !(t1 <= 1.35e-19)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / 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 <= (-1.12d-55)) .or. (.not. (t1 <= 1.35d-19))) then
        tmp = v / (u - t1)
    else
        tmp = t1 * ((v / u) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.12e-55) || !(t1 <= 1.35e-19)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.12e-55) or not (t1 <= 1.35e-19):
		tmp = v / (u - t1)
	else:
		tmp = t1 * ((v / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.12e-55) || !(t1 <= 1.35e-19))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.12e-55) || ~((t1 <= 1.35e-19)))
		tmp = v / (u - t1);
	else
		tmp = t1 * ((v / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.12e-55], N[Not[LessEqual[t1, 1.35e-19]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.11999999999999997e-55 or 1.35e-19 < t1

   1. Initial program 67.4%

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

      1. associate-/l* 68.8%

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

      2. distribute-lft-neg-out 68.8%

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

      3. distribute-rgt-neg-in 68.8%

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

      4. associate-/r* 83.0%

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

      5. distribute-neg-frac2 83.0%

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

   3. Simplified 83.0%

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

      1. distribute-frac-neg2 83.0%

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

      2. distribute-rgt-neg-out 83.0%

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

      3. associate-/r* 68.8%

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

      4. distribute-lft-neg-out 68.8%

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

      5. associate-/l* 67.4%

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

      6. times-frac 99.9%

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

      7. frac-2neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 52.3%

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

      10. sqrt-unprod 40.1%

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

      11. sqr-neg 40.1%

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

      12. sqrt-unprod 20.4%

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

      13. add-sqr-sqrt 35.3%

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

      14. add-sqr-sqrt 13.2%

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

      15. sqrt-unprod 51.4%

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

      16. sqr-neg 51.4%

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

      17. sqrt-prod 49.1%

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

      18. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 83.6%

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

      1. mul-1-neg 83.6%

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

   9. Simplified 83.6%

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

       1. distribute-frac-neg 83.6%

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

       2. neg-sub0 83.6%

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

       3. frac-2neg 83.6%

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

       4. add-sqr-sqrt 41.7%

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

       5. sqrt-unprod 49.9%

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

       6. sqr-neg 49.9%

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

       7. sqrt-unprod 15.9%

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

       8. add-sqr-sqrt 30.5%

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

       9. distribute-neg-in 30.5%

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

       10. add-sqr-sqrt 13.2%

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

       11. sqrt-unprod 44.7%

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

       12. sqr-neg 44.7%

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

       13. sqrt-unprod 39.0%

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

       14. add-sqr-sqrt 83.7%

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

       15. sub-neg 83.7%

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

   11. Applied egg-rr 83.7%

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

       1. neg-sub0 83.7%

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

       2. distribute-neg-frac 83.7%

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

   13. Simplified 83.7%

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

       1. frac-2neg 83.7%

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

       2. div-inv 83.5%

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

       3. remove-double-neg 83.5%

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

       4. sub-neg 83.5%

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

       5. distribute-neg-in 83.5%

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

       6. add-sqr-sqrt 41.0%

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

       7. sqrt-unprod 84.1%

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

       8. sqr-neg 84.1%

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

       9. sqrt-unprod 42.4%

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

       10. add-sqr-sqrt 83.4%

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

       11. distribute-neg-in 83.4%

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

       12. +-commutative 83.4%

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

       13. distribute-neg-in 83.4%

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

       14. add-sqr-sqrt 41.0%

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

       15. sqrt-unprod 84.3%

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

       16. sqr-neg 84.3%

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

       17. sqrt-unprod 42.5%

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

       18. add-sqr-sqrt 83.5%

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

   15. Applied egg-rr 83.5%

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

       1. sub-neg 83.5%

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

       2. associate-*r/ 83.7%

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

       3. *-rgt-identity 83.7%

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

   17. Simplified 83.7%

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

   if -1.11999999999999997e-55 < t1 < 1.35e-19

   1. Initial program 84.7%

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

      1. associate-/l* 83.5%

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

      2. distribute-lft-neg-out 83.5%

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

      3. distribute-rgt-neg-in 83.5%

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

      4. associate-/r* 89.9%

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

      5. distribute-neg-frac2 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in t1 around 0 78.8%

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

      1. mul-1-neg 78.8%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in t1 around 0 81.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-55} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.65 \cdot 10^{+75} \lor \neg \left(u \leq 4.2 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.65e+75) (not (<= u 4.2e+66)))
   (/ t1 (* u (/ u v)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.65e+75) || !(u <= 4.2e+66)) {
		tmp = t1 / (u * (u / v));
	} 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 <= (-1.65d+75)) .or. (.not. (u <= 4.2d+66))) then
        tmp = t1 / (u * (u / v))
    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 <= -1.65e+75) || !(u <= 4.2e+66)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.65e+75) or not (u <= 4.2e+66):
		tmp = t1 / (u * (u / v))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.65e+75) || !(u <= 4.2e+66))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.65e+75) || ~((u <= 4.2e+66)))
		tmp = t1 / (u * (u / v));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.65e+75], N[Not[LessEqual[u, 4.2e+66]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.64999999999999999e75 or 4.20000000000000011e66 < u

   1. Initial program 76.5%

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

      1. associate-/l* 76.6%

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

      2. distribute-lft-neg-out 76.6%

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

      3. distribute-rgt-neg-in 76.6%

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

      4. associate-/r* 91.6%

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

      5. distribute-neg-frac2 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in t1 around 0 84.8%

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

      1. mul-1-neg 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in t1 around 0 83.8%

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

      1. clear-num 83.0%

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

      2. un-div-inv 83.0%

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

      3. div-inv 83.0%

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

      4. clear-num 83.1%

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

      5. add-sqr-sqrt 32.4%

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

      6. sqrt-unprod 70.0%

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

      7. sqr-neg 70.0%

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

      8. sqrt-unprod 41.0%

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

      9. add-sqr-sqrt 64.8%

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

   10. Applied egg-rr 64.8%

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

   if -1.64999999999999999e75 < u < 4.20000000000000011e66

   1. Initial program 73.1%

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

      1. associate-/l* 73.6%

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

      2. distribute-lft-neg-out 73.6%

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

      3. distribute-rgt-neg-in 73.6%

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

      4. associate-/r* 82.0%

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

      5. distribute-neg-frac2 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in t1 around inf 74.4%

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

      1. associate-*r/ 74.4%

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

      2. neg-mul-1 74.4%

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

   7. Simplified 74.4%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.65 \cdot 10^{+75} \lor \neg \left(u \leq 4.2 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 23.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.75 \cdot 10^{+132} \lor \neg \left(t1 \leq 2.3 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.75e+132) (not (<= t1 2.3e+79))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.75e+132) || !(t1 <= 2.3e+79)) {
		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 <= (-1.75d+132)) .or. (.not. (t1 <= 2.3d+79))) 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 <= -1.75e+132) || !(t1 <= 2.3e+79)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.75e+132) or not (t1 <= 2.3e+79):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.75e+132) || !(t1 <= 2.3e+79))
		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 <= -1.75e+132) || ~((t1 <= 2.3e+79)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.75e+132], N[Not[LessEqual[t1, 2.3e+79]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.7500000000000001e132 or 2.3e79 < t1

   1. Initial program 51.5%

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

      1. associate-/l* 50.2%

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

      2. distribute-lft-neg-out 50.2%

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

      3. distribute-rgt-neg-in 50.2%

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

      4. associate-/r* 72.5%

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

      5. distribute-neg-frac2 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in t1 around inf 88.8%

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

      1. associate-*r/ 88.8%

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

      2. neg-mul-1 88.8%

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

   7. Simplified 88.8%

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

      1. neg-sub0 88.8%

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

      2. sub-neg 88.8%

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

      3. add-sqr-sqrt 42.4%

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

      4. sqrt-unprod 56.2%

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

      5. sqr-neg 56.2%

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

      6. sqrt-unprod 20.9%

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

      7. add-sqr-sqrt 39.0%

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

   9. Applied egg-rr 39.0%

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

       1. +-lft-identity 39.0%

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

   11. Simplified 39.0%

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

   if -1.7500000000000001e132 < t1 < 2.3e79

   1. Initial program 85.4%

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

      1. associate-/l* 86.6%

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

      2. distribute-lft-neg-out 86.6%

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

      3. distribute-rgt-neg-in 86.6%

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

      4. associate-/r* 92.2%

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

      5. distribute-neg-frac2 92.2%

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

   3. Simplified 92.2%

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

      1. distribute-frac-neg2 92.2%

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

      2. distribute-rgt-neg-out 92.2%

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

      3. associate-/r* 86.6%

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

      4. distribute-lft-neg-out 86.6%

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

      5. associate-/l* 85.4%

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

      6. times-frac 96.2%

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

      7. frac-2neg 96.2%

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

      8. associate-*r/ 96.7%

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

      9. add-sqr-sqrt 45.4%

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

      10. sqrt-unprod 58.7%

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

      11. sqr-neg 58.7%

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

      12. sqrt-unprod 19.2%

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

      13. add-sqr-sqrt 35.2%

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

      14. add-sqr-sqrt 14.5%

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

      15. sqrt-unprod 59.8%

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

      16. sqr-neg 59.8%

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

      17. sqrt-prod 50.0%

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

      18. add-sqr-sqrt 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in t1 around inf 47.2%

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

      1. mul-1-neg 47.2%

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

   9. Simplified 47.2%

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

       1. distribute-frac-neg 47.2%

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

       2. neg-sub0 47.2%

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

       3. frac-2neg 47.2%

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

       4. add-sqr-sqrt 24.9%

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

       5. sqrt-unprod 30.3%

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

       6. sqr-neg 30.3%

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

       7. sqrt-unprod 7.4%

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

       8. add-sqr-sqrt 16.2%

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

       9. distribute-neg-in 16.2%

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

       10. add-sqr-sqrt 7.7%

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

       11. sqrt-unprod 29.9%

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

       12. sqr-neg 29.9%

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

       13. sqrt-unprod 22.1%

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

       14. add-sqr-sqrt 48.4%

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

       15. sub-neg 48.4%

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

   11. Applied egg-rr 48.4%

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

       1. neg-sub0 48.4%

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

       2. distribute-neg-frac 48.4%

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

   13. Simplified 48.4%

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

   14. Taylor expanded in t1 around 0 17.3%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.75 \cdot 10^{+132} \lor \neg \left(t1 \leq 2.3 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 + t1))) / (u + t1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. associate-/l* 74.8%

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

   2. distribute-lft-neg-out 74.8%

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

   3. distribute-rgt-neg-in 74.8%

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

   4. associate-/r* 85.8%

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

   5. distribute-neg-frac2 85.8%

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

3. Simplified 85.8%

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

   1. distribute-frac-neg2 85.8%

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

   2. distribute-rgt-neg-out 85.8%

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

   3. associate-/r* 74.8%

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

   4. distribute-lft-neg-out 74.8%

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

   5. associate-/l* 74.4%

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

   6. times-frac 97.4%

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

   7. frac-2neg 97.4%

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

   8. associate-*r/ 97.8%

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

   9. add-sqr-sqrt 45.0%

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

   10. sqrt-unprod 43.7%

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

   11. sqr-neg 43.7%

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

   12. sqrt-unprod 22.0%

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

   13. add-sqr-sqrt 37.9%

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

   14. add-sqr-sqrt 14.8%

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

   15. sqrt-unprod 57.1%

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

   16. sqr-neg 57.1%

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

   17. sqrt-prod 50.9%

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

   18. add-sqr-sqrt 97.8%

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

6. Applied egg-rr 97.8%

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

7. Final simplification 97.8%

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

Alternative 10: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. times-frac 97.4%

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

   2. distribute-frac-neg 97.4%

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

   3. distribute-neg-frac2 97.4%

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

   4. +-commutative 97.4%

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

   5. distribute-neg-in 97.4%

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

   6. unsub-neg 97.4%

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

3. Simplified 97.4%

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

5. Final simplification 97.4%

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

Alternative 11: 56.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 3e+145) (/ v (- t1)) (/ 1.0 (/ u v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3e+145) {
		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+145) 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+145) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3e+145:
		tmp = v / -t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3e+145)
		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+145)
		tmp = v / -t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3e+145], N[(v / (-t1)), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.0000000000000002e145

   1. Initial program 73.6%

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

      1. associate-/l* 74.0%

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

      2. distribute-lft-neg-out 74.0%

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

      3. distribute-rgt-neg-in 74.0%

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

      4. associate-/r* 84.3%

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

      5. distribute-neg-frac2 84.3%

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

   3. Simplified 84.3%

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

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

   if 3.0000000000000002e145 < u

   1. Initial program 79.5%

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

      1. associate-/l* 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

      4. associate-/r* 94.5%

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

      5. distribute-neg-frac2 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in t1 around 0 94.5%

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

      1. mul-1-neg 94.5%

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

   7. Simplified 94.5%

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

   8. Taylor expanded in t1 around inf 45.3%

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

      1. associate-*r/ 45.3%

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

      2. mul-1-neg 45.3%

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

   10. Simplified 45.3%

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

       1. add-sqr-sqrt 26.7%

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

       2. sqrt-unprod 45.0%

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

       3. sqr-neg 45.0%

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

       4. sqrt-unprod 18.7%

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

       5. add-sqr-sqrt 45.4%

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

       6. clear-num 46.4%

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

       7. inv-pow 46.4%

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

   12. Applied egg-rr 46.4%

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

       1. unpow-1 46.4%

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

   14. Simplified 46.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 3 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 5.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u 5.5e+144) (/ v (- t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 5.5e+144) {
		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 <= 5.5d+144) 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 <= 5.5e+144) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 5.5e+144:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 5.5e+144)
		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 <= 5.5e+144)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 5.5e+144], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 5.50000000000000022e144

   1. Initial program 73.6%

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

      1. associate-/l* 74.0%

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

      2. distribute-lft-neg-out 74.0%

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

      3. distribute-rgt-neg-in 74.0%

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

      4. associate-/r* 84.3%

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

      5. distribute-neg-frac2 84.3%

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

   3. Simplified 84.3%

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

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

   if 5.50000000000000022e144 < u

   1. Initial program 79.5%

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

      1. associate-/l* 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

      4. associate-/r* 94.5%

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

      5. distribute-neg-frac2 94.5%

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

   3. Simplified 94.5%

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

      1. distribute-frac-neg2 94.5%

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

      2. distribute-rgt-neg-out 94.5%

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

      3. associate-/r* 79.6%

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

      4. distribute-lft-neg-out 79.6%

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

      5. associate-/l* 79.5%

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

      6. times-frac 97.4%

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

      7. frac-2neg 97.4%

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

      8. associate-*r/ 97.4%

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

      9. add-sqr-sqrt 38.0%

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

      10. sqrt-unprod 68.6%

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

      11. sqr-neg 68.6%

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

      12. sqrt-unprod 46.6%

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

      13. add-sqr-sqrt 79.3%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 79.6%

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

      16. sqr-neg 79.6%

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

      17. sqrt-prod 97.3%

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

      18. add-sqr-sqrt 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in t1 around inf 45.3%

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

      1. mul-1-neg 45.3%

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

   9. Simplified 45.3%

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

       1. distribute-frac-neg 45.3%

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

       2. neg-sub0 45.3%

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

       3. frac-2neg 45.3%

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

       4. add-sqr-sqrt 26.7%

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

       5. sqrt-unprod 45.0%

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

       6. sqr-neg 45.0%

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

       7. sqrt-unprod 18.7%

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

       8. add-sqr-sqrt 45.4%

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

       9. distribute-neg-in 45.4%

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

       10. add-sqr-sqrt 17.0%

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

       11. sqrt-unprod 47.7%

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

       12. sqr-neg 47.7%

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

       13. sqrt-unprod 28.4%

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

       14. add-sqr-sqrt 45.4%

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

       15. sub-neg 45.4%

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

   11. Applied egg-rr 45.4%

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

       1. neg-sub0 45.4%

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

       2. distribute-neg-frac 45.4%

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

   13. Simplified 45.4%

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

   14. Taylor expanded in t1 around 0 45.4%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 5.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. associate-/l* 74.8%

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

   2. distribute-lft-neg-out 74.8%

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

   3. distribute-rgt-neg-in 74.8%

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

   4. associate-/r* 85.8%

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

   5. distribute-neg-frac2 85.8%

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

3. Simplified 85.8%

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

   1. distribute-frac-neg2 85.8%

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

   2. distribute-rgt-neg-out 85.8%

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

   3. associate-/r* 74.8%

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

   4. distribute-lft-neg-out 74.8%

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

   5. associate-/l* 74.4%

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

   6. times-frac 97.4%

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

   7. frac-2neg 97.4%

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

   8. associate-*r/ 97.8%

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

   9. add-sqr-sqrt 45.0%

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

   10. sqrt-unprod 43.7%

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

   11. sqr-neg 43.7%

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

   12. sqrt-unprod 22.0%

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

   13. add-sqr-sqrt 37.9%

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

   14. add-sqr-sqrt 14.8%

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

   15. sqrt-unprod 57.1%

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

   16. sqr-neg 57.1%

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

   17. sqrt-prod 50.9%

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

   18. add-sqr-sqrt 97.8%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in t1 around inf 61.4%

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

   1. mul-1-neg 61.4%

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

9. Simplified 61.4%

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

    1. distribute-frac-neg 61.4%

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

    2. neg-sub0 61.4%

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

    3. frac-2neg 61.4%

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

    4. add-sqr-sqrt 30.5%

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

    5. sqrt-unprod 39.4%

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

    6. sqr-neg 39.4%

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

    7. sqrt-unprod 12.5%

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

    8. add-sqr-sqrt 24.4%

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

    9. distribute-neg-in 24.4%

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

    10. add-sqr-sqrt 10.2%

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

    11. sqrt-unprod 36.4%

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

    12. sqr-neg 36.4%

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

    13. sqrt-unprod 30.6%

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

    14. add-sqr-sqrt 62.2%

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

    15. sub-neg 62.2%

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

11. Applied egg-rr 62.2%

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

    1. neg-sub0 62.2%

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

    2. distribute-neg-frac 62.2%

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

13. Simplified 62.2%

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

    1. frac-2neg 62.2%

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

    2. div-inv 62.1%

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

    3. remove-double-neg 62.1%

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

    4. sub-neg 62.1%

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

    5. distribute-neg-in 62.1%

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

    6. add-sqr-sqrt 30.9%

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

    7. sqrt-unprod 67.3%

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

    8. sqr-neg 67.3%

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

    9. sqrt-unprod 30.9%

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

    10. add-sqr-sqrt 61.2%

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

    11. distribute-neg-in 61.2%

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

    12. +-commutative 61.2%

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

    13. distribute-neg-in 61.2%

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

    14. add-sqr-sqrt 30.4%

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

    15. sqrt-unprod 67.2%

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

    16. sqr-neg 67.2%

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

    17. sqrt-unprod 31.2%

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

    18. add-sqr-sqrt 62.1%

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

15. Applied egg-rr 62.1%

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

    1. sub-neg 62.1%

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

    2. associate-*r/ 62.2%

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

    3. *-rgt-identity 62.2%

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

17. Simplified 62.2%

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

Alternative 14: 13.6% 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 74.4%

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

   1. associate-/l* 74.8%

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

   2. distribute-lft-neg-out 74.8%

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

   3. distribute-rgt-neg-in 74.8%

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

   4. associate-/r* 85.8%

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

   5. distribute-neg-frac2 85.8%

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

3. Simplified 85.8%

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

5. Taylor expanded in t1 around inf 54.0%

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

   1. associate-*r/ 54.0%

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

   2. neg-mul-1 54.0%

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

7. Simplified 54.0%

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

   1. neg-sub0 54.0%

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

   2. sub-neg 54.0%

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

   3. add-sqr-sqrt 26.7%

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

   4. sqrt-unprod 35.4%

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

   5. sqr-neg 35.4%

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

   6. sqrt-unprod 7.8%

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

   7. add-sqr-sqrt 14.7%

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

9. Applied egg-rr 14.7%

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

    1. +-lft-identity 14.7%

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

11. Simplified 14.7%

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

Reproduce

herbie shell --seed 2024112 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))