Rosa's DopplerBench

Percentage Accurate: 73.4% → 98.2%

Time: 12.0s

Alternatives: 19

Speedup: 1.0×

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: 73.4% 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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. associate-/l* 69.7%

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

   2. distribute-lft-neg-out 69.7%

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

   3. distribute-rgt-neg-in 69.7%

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

   4. associate-/r* 79.3%

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

   5. distribute-neg-frac2 79.3%

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

3. Simplified 79.3%

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

   1. distribute-frac-neg2 79.3%

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

   2. distribute-rgt-neg-out 79.3%

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

   3. associate-/r* 69.7%

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

   4. distribute-lft-neg-out 69.7%

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

   5. associate-/l* 70.5%

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

   6. times-frac 98.4%

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

   7. frac-2neg 98.4%

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

   8. associate-*r/ 98.4%

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

   9. add-sqr-sqrt 41.2%

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

   10. sqrt-unprod 38.0%

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

   11. sqr-neg 38.0%

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

   12. sqrt-unprod 16.9%

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

   13. add-sqr-sqrt 32.7%

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

   14. add-sqr-sqrt 16.5%

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

   15. sqrt-unprod 57.5%

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

   16. sqr-neg 57.5%

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

   17. sqrt-prod 53.8%

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

   18. add-sqr-sqrt 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 2: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ t_2 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+157}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-176}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{-248}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 2.75 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (/ v (+ t1 u)) (- (- u) t1))))
        (t_2 (/ (- v) (+ t1 (* u 2.0)))))
   (if (<= t1 -2.5e+157)
     t_2
     (if (<= t1 -1.65e-176)
       t_1
       (if (<= t1 6e-248)
         (/ v (* (+ t1 u) (/ (- u) t1)))
         (if (<= t1 2.75e+107) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	double t_2 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -2.5e+157) {
		tmp = t_2;
	} else if (t1 <= -1.65e-176) {
		tmp = t_1;
	} else if (t1 <= 6e-248) {
		tmp = v / ((t1 + u) * (-u / t1));
	} else if (t1 <= 2.75e+107) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t1 * ((v / (t1 + u)) / (-u - t1))
    t_2 = -v / (t1 + (u * 2.0d0))
    if (t1 <= (-2.5d+157)) then
        tmp = t_2
    else if (t1 <= (-1.65d-176)) then
        tmp = t_1
    else if (t1 <= 6d-248) then
        tmp = v / ((t1 + u) * (-u / t1))
    else if (t1 <= 2.75d+107) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	double t_2 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -2.5e+157) {
		tmp = t_2;
	} else if (t1 <= -1.65e-176) {
		tmp = t_1;
	} else if (t1 <= 6e-248) {
		tmp = v / ((t1 + u) * (-u / t1));
	} else if (t1 <= 2.75e+107) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * ((v / (t1 + u)) / (-u - t1))
	t_2 = -v / (t1 + (u * 2.0))
	tmp = 0
	if t1 <= -2.5e+157:
		tmp = t_2
	elif t1 <= -1.65e-176:
		tmp = t_1
	elif t1 <= 6e-248:
		tmp = v / ((t1 + u) * (-u / t1))
	elif t1 <= 2.75e+107:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)))
	t_2 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -2.5e+157)
		tmp = t_2;
	elseif (t1 <= -1.65e-176)
		tmp = t_1;
	elseif (t1 <= 6e-248)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(Float64(-u) / t1)));
	elseif (t1 <= 2.75e+107)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	t_2 = -v / (t1 + (u * 2.0));
	tmp = 0.0;
	if (t1 <= -2.5e+157)
		tmp = t_2;
	elseif (t1 <= -1.65e-176)
		tmp = t_1;
	elseif (t1 <= 6e-248)
		tmp = v / ((t1 + u) * (-u / t1));
	elseif (t1 <= 2.75e+107)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.5e+157], t$95$2, If[LessEqual[t1, -1.65e-176], t$95$1, If[LessEqual[t1, 6e-248], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[((-u) / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.75e+107], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.49999999999999988e157 or 2.7500000000000002e107 < t1

   1. Initial program 44.2%

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

      1. associate-/l* 44.4%

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

      2. distribute-lft-neg-out 44.4%

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

      3. distribute-rgt-neg-in 44.4%

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

      4. associate-/r* 60.9%

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

      5. distribute-neg-frac2 60.9%

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

   3. Simplified 60.9%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 98.1%

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

      9. *-un-lft-identity 98.1%

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

      10. frac-2neg 98.1%

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

      11. sub-neg 98.1%

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

      12. distribute-neg-in 98.1%

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

      13. +-commutative 98.1%

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

      14. remove-double-neg 98.1%

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

      15. add-sqr-sqrt 40.0%

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

      16. sqrt-unprod 4.3%

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

      17. sqr-neg 4.3%

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

      18. sqrt-unprod 13.9%

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

      19. add-sqr-sqrt 30.7%

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

      20. add-sqr-sqrt 13.4%

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

      21. sqrt-unprod 46.9%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in u around 0 91.4%

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

      1. *-commutative 91.4%

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

   9. Simplified 91.4%

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

   if -2.49999999999999988e157 < t1 < -1.65000000000000006e-176 or 6.00000000000000027e-248 < t1 < 2.7500000000000002e107

   1. Initial program 84.2%

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

      1. associate-/l* 82.0%

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

      2. distribute-lft-neg-out 82.0%

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

      3. distribute-rgt-neg-in 82.0%

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

      4. associate-/r* 88.3%

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

      5. distribute-neg-frac2 88.3%

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

   3. Simplified 88.3%

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

   if -1.65000000000000006e-176 < t1 < 6.00000000000000027e-248

   1. Initial program 71.4%

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

      1. associate-/l* 74.2%

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

      2. distribute-lft-neg-out 74.2%

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

      3. distribute-rgt-neg-in 74.2%

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

      4. associate-/r* 81.9%

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

      5. distribute-neg-frac2 81.9%

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

   3. Simplified 81.9%

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

      1. associate-*r/ 89.5%

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

      2. +-commutative 89.5%

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

      3. distribute-neg-in 89.5%

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

      4. sub-neg 89.5%

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

      5. associate-*l/ 92.1%

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

      6. clear-num 92.0%

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

      7. frac-2neg 92.0%

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

      8. frac-times 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. frac-2neg 99.7%

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

      11. sub-neg 99.7%

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

      12. distribute-neg-in 99.7%

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

      13. +-commutative 99.7%

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

      14. remove-double-neg 99.7%

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

      15. add-sqr-sqrt 59.2%

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

      16. sqrt-unprod 50.3%

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

      17. sqr-neg 50.3%

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

      18. sqrt-unprod 21.9%

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

      19. add-sqr-sqrt 49.5%

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

      20. add-sqr-sqrt 30.3%

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

      21. sqrt-unprod 60.5%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in t1 around 0 97.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-176}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{-248}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 2.75 \cdot 10^{+107}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -1.85 \cdot 10^{+158}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -1.46 \cdot 10^{-176}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{t\_1}\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 1.36 \cdot 10^{+104}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (/ (- v) (+ t1 (* u 2.0)))))
   (if (<= t1 -1.85e+158)
     t_2
     (if (<= t1 -1.46e-176)
       (* t1 (/ (/ v (+ t1 u)) t_1))
       (if (<= t1 2.8e-265)
         (/ v (* (+ t1 u) (/ (- u) t1)))
         (if (<= t1 1.36e+104) (* v (/ t1 (* (+ t1 u) t_1))) t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -1.85e+158) {
		tmp = t_2;
	} else if (t1 <= -1.46e-176) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else if (t1 <= 2.8e-265) {
		tmp = v / ((t1 + u) * (-u / t1));
	} else if (t1 <= 1.36e+104) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -u - t1
    t_2 = -v / (t1 + (u * 2.0d0))
    if (t1 <= (-1.85d+158)) then
        tmp = t_2
    else if (t1 <= (-1.46d-176)) then
        tmp = t1 * ((v / (t1 + u)) / t_1)
    else if (t1 <= 2.8d-265) then
        tmp = v / ((t1 + u) * (-u / t1))
    else if (t1 <= 1.36d+104) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -1.85e+158) {
		tmp = t_2;
	} else if (t1 <= -1.46e-176) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else if (t1 <= 2.8e-265) {
		tmp = v / ((t1 + u) * (-u / t1));
	} else if (t1 <= 1.36e+104) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	t_2 = -v / (t1 + (u * 2.0))
	tmp = 0
	if t1 <= -1.85e+158:
		tmp = t_2
	elif t1 <= -1.46e-176:
		tmp = t1 * ((v / (t1 + u)) / t_1)
	elif t1 <= 2.8e-265:
		tmp = v / ((t1 + u) * (-u / t1))
	elif t1 <= 1.36e+104:
		tmp = v * (t1 / ((t1 + u) * t_1))
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -1.85e+158)
		tmp = t_2;
	elseif (t1 <= -1.46e-176)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / t_1));
	elseif (t1 <= 2.8e-265)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(Float64(-u) / t1)));
	elseif (t1 <= 1.36e+104)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	t_2 = -v / (t1 + (u * 2.0));
	tmp = 0.0;
	if (t1 <= -1.85e+158)
		tmp = t_2;
	elseif (t1 <= -1.46e-176)
		tmp = t1 * ((v / (t1 + u)) / t_1);
	elseif (t1 <= 2.8e-265)
		tmp = v / ((t1 + u) * (-u / t1));
	elseif (t1 <= 1.36e+104)
		tmp = v * (t1 / ((t1 + u) * t_1));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.85e+158], t$95$2, If[LessEqual[t1, -1.46e-176], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.8e-265], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[((-u) / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.36e+104], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.85000000000000005e158 or 1.3599999999999999e104 < t1

   1. Initial program 44.5%

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

      1. associate-/l* 43.4%

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

      2. distribute-lft-neg-out 43.4%

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

      3. distribute-rgt-neg-in 43.4%

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

      4. associate-/r* 60.7%

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

      5. distribute-neg-frac2 60.7%

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

   3. Simplified 60.7%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 98.2%

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

      9. *-un-lft-identity 98.2%

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

      10. frac-2neg 98.2%

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

      11. sub-neg 98.2%

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

      12. distribute-neg-in 98.2%

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

      13. +-commutative 98.2%

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

      14. remove-double-neg 98.2%

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

      15. add-sqr-sqrt 39.0%

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

      16. sqrt-unprod 4.3%

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

      17. sqr-neg 4.3%

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

      18. sqrt-unprod 13.6%

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

      19. add-sqr-sqrt 30.0%

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

      20. add-sqr-sqrt 13.1%

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

      21. sqrt-unprod 47.1%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in u around 0 90.4%

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

      1. *-commutative 90.4%

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

   9. Simplified 90.4%

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

   if -1.85000000000000005e158 < t1 < -1.46e-176

   1. Initial program 81.7%

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

      1. associate-/l* 83.2%

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

      2. distribute-lft-neg-out 83.2%

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

      3. distribute-rgt-neg-in 83.2%

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

      4. associate-/r* 92.0%

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

      5. distribute-neg-frac2 92.0%

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

   3. Simplified 92.0%

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

   if -1.46e-176 < t1 < 2.80000000000000023e-265

   1. Initial program 65.9%

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

      1. associate-/l* 72.3%

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

      2. distribute-lft-neg-out 72.3%

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

      3. distribute-rgt-neg-in 72.3%

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

      4. associate-/r* 81.6%

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

      5. distribute-neg-frac2 81.6%

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

   3. Simplified 81.6%

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

      1. associate-*r/ 87.4%

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

      2. +-commutative 87.4%

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

      3. distribute-neg-in 87.4%

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

      4. sub-neg 87.4%

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

      5. associate-*l/ 90.7%

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

      6. clear-num 90.6%

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

      7. frac-2neg 90.6%

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

      8. frac-times 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. frac-2neg 99.7%

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

      11. sub-neg 99.7%

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

      12. distribute-neg-in 99.7%

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

      13. +-commutative 99.7%

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

      14. remove-double-neg 99.7%

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

      15. add-sqr-sqrt 70.6%

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

      16. sqrt-unprod 50.0%

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

      17. sqr-neg 50.0%

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

      18. sqrt-unprod 16.2%

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

      19. add-sqr-sqrt 49.2%

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

      20. add-sqr-sqrt 29.6%

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

      21. sqrt-unprod 59.1%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in t1 around 0 97.4%

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

   if 2.80000000000000023e-265 < t1 < 1.3599999999999999e104

   1. Initial program 87.4%

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

      1. associate-*l/ 89.1%

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

      2. *-commutative 89.1%

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

   3. Simplified 89.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{+158}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq -1.46 \cdot 10^{-176}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 1.36 \cdot 10^{+104}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ t_2 := \frac{t1}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.2 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -8.6 \cdot 10^{-118}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1.7 \cdot 10^{-159}:\\ \;\;\;\;t\_2 \cdot \frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-15}:\\ \;\;\;\;t\_2 \cdot \frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))) (t_2 (/ t1 (+ t1 u))))
   (if (<= t1 -3.2e-31)
     t_1
     (if (<= t1 -8.6e-118)
       (* t1 (/ (/ v u) (- (- u) t1)))
       (if (<= t1 -1.7e-159)
         (* t_2 (/ v (- t1)))
         (if (<= t1 1.35e-15) (* t_2 (/ v (- u))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = t1 / (t1 + u);
	double tmp;
	if (t1 <= -3.2e-31) {
		tmp = t_1;
	} else if (t1 <= -8.6e-118) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (t1 <= -1.7e-159) {
		tmp = t_2 * (v / -t1);
	} else if (t1 <= 1.35e-15) {
		tmp = t_2 * (v / -u);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    t_2 = t1 / (t1 + u)
    if (t1 <= (-3.2d-31)) then
        tmp = t_1
    else if (t1 <= (-8.6d-118)) then
        tmp = t1 * ((v / u) / (-u - t1))
    else if (t1 <= (-1.7d-159)) then
        tmp = t_2 * (v / -t1)
    else if (t1 <= 1.35d-15) then
        tmp = t_2 * (v / -u)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double t_2 = t1 / (t1 + u);
	double tmp;
	if (t1 <= -3.2e-31) {
		tmp = t_1;
	} else if (t1 <= -8.6e-118) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (t1 <= -1.7e-159) {
		tmp = t_2 * (v / -t1);
	} else if (t1 <= 1.35e-15) {
		tmp = t_2 * (v / -u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	t_2 = t1 / (t1 + u)
	tmp = 0
	if t1 <= -3.2e-31:
		tmp = t_1
	elif t1 <= -8.6e-118:
		tmp = t1 * ((v / u) / (-u - t1))
	elif t1 <= -1.7e-159:
		tmp = t_2 * (v / -t1)
	elif t1 <= 1.35e-15:
		tmp = t_2 * (v / -u)
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	t_2 = Float64(t1 / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -3.2e-31)
		tmp = t_1;
	elseif (t1 <= -8.6e-118)
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(Float64(-u) - t1)));
	elseif (t1 <= -1.7e-159)
		tmp = Float64(t_2 * Float64(v / Float64(-t1)));
	elseif (t1 <= 1.35e-15)
		tmp = Float64(t_2 * Float64(v / Float64(-u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	t_2 = t1 / (t1 + u);
	tmp = 0.0;
	if (t1 <= -3.2e-31)
		tmp = t_1;
	elseif (t1 <= -8.6e-118)
		tmp = t1 * ((v / u) / (-u - t1));
	elseif (t1 <= -1.7e-159)
		tmp = t_2 * (v / -t1);
	elseif (t1 <= 1.35e-15)
		tmp = t_2 * (v / -u);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.2e-31], t$95$1, If[LessEqual[t1, -8.6e-118], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.7e-159], N[(t$95$2 * N[(v / (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.35e-15], N[(t$95$2 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -3.20000000000000018e-31 or 1.35000000000000005e-15 < t1

   1. Initial program 60.9%

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

      1. associate-/l* 59.3%

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

      2. distribute-lft-neg-out 59.3%

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

      3. distribute-rgt-neg-in 59.3%

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

      4. associate-/r* 72.6%

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

      5. distribute-neg-frac2 72.6%

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

   3. Simplified 72.6%

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

      1. associate-*r/ 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. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 95.3%

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

      9. *-un-lft-identity 95.3%

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

      10. frac-2neg 95.3%

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

      11. sub-neg 95.3%

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

      12. distribute-neg-in 95.3%

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

      13. +-commutative 95.3%

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

      14. remove-double-neg 95.3%

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

      15. add-sqr-sqrt 42.3%

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

      16. sqrt-unprod 22.3%

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

      17. sqr-neg 22.3%

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

      18. sqrt-unprod 10.2%

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

      19. add-sqr-sqrt 26.0%

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

      20. add-sqr-sqrt 10.9%

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

      21. sqrt-unprod 50.3%

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

   6. Applied egg-rr 95.3%

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

   7. Taylor expanded in u around 0 81.9%

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

      1. *-commutative 81.9%

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

   9. Simplified 81.9%

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

   if -3.20000000000000018e-31 < t1 < -8.60000000000000036e-118

   1. Initial program 84.4%

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

      1. associate-/l* 88.9%

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

      2. distribute-lft-neg-out 88.9%

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

      3. distribute-rgt-neg-in 88.9%

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

      4. associate-/r* 99.9%

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

      5. distribute-neg-frac2 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 82.6%

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

   if -8.60000000000000036e-118 < t1 < -1.69999999999999992e-159

   1. Initial program 73.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 96.5%

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

   if -1.69999999999999992e-159 < t1 < 1.35000000000000005e-15

   1. Initial program 80.6%

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

      1. times-frac 96.0%

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

      2. distribute-frac-neg 96.0%

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

      3. distribute-neg-frac2 96.0%

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

      4. +-commutative 96.0%

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

      5. distribute-neg-in 96.0%

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

      6. unsub-neg 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t1 around 0 76.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq -8.6 \cdot 10^{-118}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1.7 \cdot 10^{-159}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-15}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -2.05 \cdot 10^{-117}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1 \cdot 10^{-159}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))))
   (if (<= t1 -1.2e-31)
     t_1
     (if (<= t1 -2.05e-117)
       (* t1 (/ (/ v u) (- (- u) t1)))
       (if (<= t1 -1e-159)
         (* (/ t1 (+ t1 u)) (/ v (- t1)))
         (if (<= t1 1.9e-16) (/ v (* (+ t1 u) (/ (- u) t1))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -1.2e-31) {
		tmp = t_1;
	} else if (t1 <= -2.05e-117) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (t1 <= -1e-159) {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	} else if (t1 <= 1.9e-16) {
		tmp = v / ((t1 + u) * (-u / t1));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    if (t1 <= (-1.2d-31)) then
        tmp = t_1
    else if (t1 <= (-2.05d-117)) then
        tmp = t1 * ((v / u) / (-u - t1))
    else if (t1 <= (-1d-159)) then
        tmp = (t1 / (t1 + u)) * (v / -t1)
    else if (t1 <= 1.9d-16) then
        tmp = v / ((t1 + u) * (-u / t1))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -1.2e-31) {
		tmp = t_1;
	} else if (t1 <= -2.05e-117) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (t1 <= -1e-159) {
		tmp = (t1 / (t1 + u)) * (v / -t1);
	} else if (t1 <= 1.9e-16) {
		tmp = v / ((t1 + u) * (-u / t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	tmp = 0
	if t1 <= -1.2e-31:
		tmp = t_1
	elif t1 <= -2.05e-117:
		tmp = t1 * ((v / u) / (-u - t1))
	elif t1 <= -1e-159:
		tmp = (t1 / (t1 + u)) * (v / -t1)
	elif t1 <= 1.9e-16:
		tmp = v / ((t1 + u) * (-u / t1))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -1.2e-31)
		tmp = t_1;
	elseif (t1 <= -2.05e-117)
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(Float64(-u) - t1)));
	elseif (t1 <= -1e-159)
		tmp = Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(-t1)));
	elseif (t1 <= 1.9e-16)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(Float64(-u) / t1)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	tmp = 0.0;
	if (t1 <= -1.2e-31)
		tmp = t_1;
	elseif (t1 <= -2.05e-117)
		tmp = t1 * ((v / u) / (-u - t1));
	elseif (t1 <= -1e-159)
		tmp = (t1 / (t1 + u)) * (v / -t1);
	elseif (t1 <= 1.9e-16)
		tmp = v / ((t1 + u) * (-u / t1));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.2e-31], t$95$1, If[LessEqual[t1, -2.05e-117], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1e-159], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.9e-16], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[((-u) / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.2e-31 or 1.90000000000000006e-16 < t1

   1. Initial program 60.9%

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

      1. associate-/l* 59.3%

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

      2. distribute-lft-neg-out 59.3%

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

      3. distribute-rgt-neg-in 59.3%

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

      4. associate-/r* 72.6%

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

      5. distribute-neg-frac2 72.6%

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

   3. Simplified 72.6%

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

      1. associate-*r/ 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. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 95.3%

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

      9. *-un-lft-identity 95.3%

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

      10. frac-2neg 95.3%

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

      11. sub-neg 95.3%

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

      12. distribute-neg-in 95.3%

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

      13. +-commutative 95.3%

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

      14. remove-double-neg 95.3%

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

      15. add-sqr-sqrt 42.3%

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

      16. sqrt-unprod 22.3%

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

      17. sqr-neg 22.3%

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

      18. sqrt-unprod 10.2%

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

      19. add-sqr-sqrt 26.0%

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

      20. add-sqr-sqrt 10.9%

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

      21. sqrt-unprod 50.3%

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

   6. Applied egg-rr 95.3%

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

   7. Taylor expanded in u around 0 81.9%

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

      1. *-commutative 81.9%

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

   9. Simplified 81.9%

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

   if -1.2e-31 < t1 < -2.05000000000000016e-117

   1. Initial program 84.4%

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

      1. associate-/l* 88.9%

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

      2. distribute-lft-neg-out 88.9%

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

      3. distribute-rgt-neg-in 88.9%

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

      4. associate-/r* 99.9%

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

      5. distribute-neg-frac2 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 82.6%

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

   if -2.05000000000000016e-117 < t1 < -9.99999999999999989e-160

   1. Initial program 73.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 96.5%

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

   if -9.99999999999999989e-160 < t1 < 1.90000000000000006e-16

   1. Initial program 80.6%

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

      1. associate-/l* 79.3%

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

      2. distribute-lft-neg-out 79.3%

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

      3. distribute-rgt-neg-in 79.3%

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

      4. associate-/r* 84.6%

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

      5. distribute-neg-frac2 84.6%

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

   3. Simplified 84.6%

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

      1. associate-*r/ 95.0%

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

      2. +-commutative 95.0%

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

      3. distribute-neg-in 95.0%

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

      4. sub-neg 95.0%

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

      5. associate-*l/ 96.0%

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

      6. clear-num 95.9%

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

      7. frac-2neg 95.9%

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

      8. frac-times 97.0%

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

      9. *-un-lft-identity 97.0%

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

      10. frac-2neg 97.0%

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

      11. sub-neg 97.0%

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

      12. distribute-neg-in 97.0%

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

      13. +-commutative 97.0%

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

      14. remove-double-neg 97.0%

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

      15. add-sqr-sqrt 24.1%

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

      16. sqrt-unprod 42.0%

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

      17. sqr-neg 42.0%

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

      18. sqrt-unprod 30.2%

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

      19. add-sqr-sqrt 41.5%

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

      20. add-sqr-sqrt 23.7%

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

      21. sqrt-unprod 70.2%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in t1 around 0 77.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq -2.05 \cdot 10^{-117}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1 \cdot 10^{-159}:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -3.4 \cdot 10^{-118}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1}{-v}}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))))
   (if (<= t1 -9.2e-32)
     t_1
     (if (<= t1 -3.4e-118)
       (* t1 (/ (/ v u) (- (- u) t1)))
       (if (<= t1 -1.6e-159)
         (/ (/ t1 (+ t1 u)) (/ t1 (- v)))
         (if (<= t1 1.9e-16) (/ v (* (+ t1 u) (/ (- u) t1))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -9.2e-32) {
		tmp = t_1;
	} else if (t1 <= -3.4e-118) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (t1 <= -1.6e-159) {
		tmp = (t1 / (t1 + u)) / (t1 / -v);
	} else if (t1 <= 1.9e-16) {
		tmp = v / ((t1 + u) * (-u / t1));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    if (t1 <= (-9.2d-32)) then
        tmp = t_1
    else if (t1 <= (-3.4d-118)) then
        tmp = t1 * ((v / u) / (-u - t1))
    else if (t1 <= (-1.6d-159)) then
        tmp = (t1 / (t1 + u)) / (t1 / -v)
    else if (t1 <= 1.9d-16) then
        tmp = v / ((t1 + u) * (-u / t1))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -9.2e-32) {
		tmp = t_1;
	} else if (t1 <= -3.4e-118) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (t1 <= -1.6e-159) {
		tmp = (t1 / (t1 + u)) / (t1 / -v);
	} else if (t1 <= 1.9e-16) {
		tmp = v / ((t1 + u) * (-u / t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	tmp = 0
	if t1 <= -9.2e-32:
		tmp = t_1
	elif t1 <= -3.4e-118:
		tmp = t1 * ((v / u) / (-u - t1))
	elif t1 <= -1.6e-159:
		tmp = (t1 / (t1 + u)) / (t1 / -v)
	elif t1 <= 1.9e-16:
		tmp = v / ((t1 + u) * (-u / t1))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -9.2e-32)
		tmp = t_1;
	elseif (t1 <= -3.4e-118)
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(Float64(-u) - t1)));
	elseif (t1 <= -1.6e-159)
		tmp = Float64(Float64(t1 / Float64(t1 + u)) / Float64(t1 / Float64(-v)));
	elseif (t1 <= 1.9e-16)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(Float64(-u) / t1)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	tmp = 0.0;
	if (t1 <= -9.2e-32)
		tmp = t_1;
	elseif (t1 <= -3.4e-118)
		tmp = t1 * ((v / u) / (-u - t1));
	elseif (t1 <= -1.6e-159)
		tmp = (t1 / (t1 + u)) / (t1 / -v);
	elseif (t1 <= 1.9e-16)
		tmp = v / ((t1 + u) * (-u / t1));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -9.2e-32], t$95$1, If[LessEqual[t1, -3.4e-118], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.6e-159], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 / (-v)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.9e-16], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[((-u) / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -9.2000000000000002e-32 or 1.90000000000000006e-16 < t1

   1. Initial program 60.9%

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

      1. associate-/l* 59.3%

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

      2. distribute-lft-neg-out 59.3%

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

      3. distribute-rgt-neg-in 59.3%

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

      4. associate-/r* 72.6%

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

      5. distribute-neg-frac2 72.6%

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

   3. Simplified 72.6%

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

      1. associate-*r/ 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. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 95.3%

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

      9. *-un-lft-identity 95.3%

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

      10. frac-2neg 95.3%

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

      11. sub-neg 95.3%

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

      12. distribute-neg-in 95.3%

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

      13. +-commutative 95.3%

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

      14. remove-double-neg 95.3%

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

      15. add-sqr-sqrt 42.3%

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

      16. sqrt-unprod 22.3%

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

      17. sqr-neg 22.3%

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

      18. sqrt-unprod 10.2%

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

      19. add-sqr-sqrt 26.0%

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

      20. add-sqr-sqrt 10.9%

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

      21. sqrt-unprod 50.3%

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

   6. Applied egg-rr 95.3%

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

   7. Taylor expanded in u around 0 81.9%

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

      1. *-commutative 81.9%

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

   9. Simplified 81.9%

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

   if -9.2000000000000002e-32 < t1 < -3.39999999999999991e-118

   1. Initial program 84.4%

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

      1. associate-/l* 88.9%

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

      2. distribute-lft-neg-out 88.9%

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

      3. distribute-rgt-neg-in 88.9%

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

      4. associate-/r* 99.9%

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

      5. distribute-neg-frac2 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 82.6%

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

   if -3.39999999999999991e-118 < t1 < -1.6e-159

   1. Initial program 73.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 96.5%

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

      1. *-commutative 96.5%

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

      2. clear-num 96.5%

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

      3. frac-2neg 96.5%

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

      4. frac-times 65.9%

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

      5. *-un-lft-identity 65.9%

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

      6. neg-sub0 65.9%

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

      7. add-sqr-sqrt 34.4%

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

      8. sqrt-unprod 75.7%

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

      9. sqr-neg 75.7%

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

      10. sqrt-unprod 31.2%

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

      11. add-sqr-sqrt 65.6%

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

      12. associate-+l- 65.6%

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

      13. neg-sub0 65.6%

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

      14. add-sqr-sqrt 34.4%

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

      15. sqrt-unprod 75.7%

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

      16. sqr-neg 75.7%

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

      17. sqrt-unprod 31.5%

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

      18. add-sqr-sqrt 65.9%

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

      19. +-commutative 65.9%

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

   7. Applied egg-rr 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-/r* 96.5%

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

   9. Simplified 96.5%

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

   if -1.6e-159 < t1 < 1.90000000000000006e-16

   1. Initial program 80.6%

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

      1. associate-/l* 79.3%

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

      2. distribute-lft-neg-out 79.3%

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

      3. distribute-rgt-neg-in 79.3%

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

      4. associate-/r* 84.6%

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

      5. distribute-neg-frac2 84.6%

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

   3. Simplified 84.6%

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

      1. associate-*r/ 95.0%

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

      2. +-commutative 95.0%

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

      3. distribute-neg-in 95.0%

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

      4. sub-neg 95.0%

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

      5. associate-*l/ 96.0%

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

      6. clear-num 95.9%

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

      7. frac-2neg 95.9%

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

      8. frac-times 97.0%

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

      9. *-un-lft-identity 97.0%

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

      10. frac-2neg 97.0%

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

      11. sub-neg 97.0%

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

      12. distribute-neg-in 97.0%

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

      13. +-commutative 97.0%

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

      14. remove-double-neg 97.0%

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

      15. add-sqr-sqrt 24.1%

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

      16. sqrt-unprod 42.0%

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

      17. sqr-neg 42.0%

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

      18. sqrt-unprod 30.2%

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

      19. add-sqr-sqrt 41.5%

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

      20. add-sqr-sqrt 23.7%

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

      21. sqrt-unprod 70.2%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in t1 around 0 77.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq -3.4 \cdot 10^{-118}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1}{-v}}\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq -1750000:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \mathbf{elif}\;u \leq -7.6 \cdot 10^{-51}:\\ \;\;\;\;t1 \cdot \left(\frac{v}{u} \cdot \frac{-1}{u - t1}\right)\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+21}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1e+80)
   (/ (/ t1 (/ u v)) (- t1 u))
   (if (<= u -1750000.0)
     (/ v (* t1 (/ (- u t1) t1)))
     (if (<= u -7.6e-51)
       (* t1 (* (/ v u) (/ -1.0 (- u t1))))
       (if (<= u 1.25e+21)
         (/ (- v) (+ t1 (* u 2.0)))
         (/ (* t1 (/ v (- u))) (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e+80) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else if (u <= -1750000.0) {
		tmp = v / (t1 * ((u - t1) / t1));
	} else if (u <= -7.6e-51) {
		tmp = t1 * ((v / u) * (-1.0 / (u - t1)));
	} else if (u <= 1.25e+21) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (t1 * (v / -u)) / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1d+80)) then
        tmp = (t1 / (u / v)) / (t1 - u)
    else if (u <= (-1750000.0d0)) then
        tmp = v / (t1 * ((u - t1) / t1))
    else if (u <= (-7.6d-51)) then
        tmp = t1 * ((v / u) * ((-1.0d0) / (u - t1)))
    else if (u <= 1.25d+21) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = (t1 * (v / -u)) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e+80) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else if (u <= -1750000.0) {
		tmp = v / (t1 * ((u - t1) / t1));
	} else if (u <= -7.6e-51) {
		tmp = t1 * ((v / u) * (-1.0 / (u - t1)));
	} else if (u <= 1.25e+21) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (t1 * (v / -u)) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1e+80:
		tmp = (t1 / (u / v)) / (t1 - u)
	elif u <= -1750000.0:
		tmp = v / (t1 * ((u - t1) / t1))
	elif u <= -7.6e-51:
		tmp = t1 * ((v / u) * (-1.0 / (u - t1)))
	elif u <= 1.25e+21:
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (t1 * (v / -u)) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1e+80)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(t1 - u));
	elseif (u <= -1750000.0)
		tmp = Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)));
	elseif (u <= -7.6e-51)
		tmp = Float64(t1 * Float64(Float64(v / u) * Float64(-1.0 / Float64(u - t1))));
	elseif (u <= 1.25e+21)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1e+80)
		tmp = (t1 / (u / v)) / (t1 - u);
	elseif (u <= -1750000.0)
		tmp = v / (t1 * ((u - t1) / t1));
	elseif (u <= -7.6e-51)
		tmp = t1 * ((v / u) * (-1.0 / (u - t1)));
	elseif (u <= 1.25e+21)
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (t1 * (v / -u)) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1e+80], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1750000.0], N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -7.6e-51], N[(t1 * N[(N[(v / u), $MachinePrecision] * N[(-1.0 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.25e+21], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -1e80

   1. Initial program 67.9%

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

      1. associate-/l* 70.8%

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

      2. distribute-lft-neg-out 70.8%

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

      3. distribute-rgt-neg-in 70.8%

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

      4. associate-/r* 86.7%

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

      5. distribute-neg-frac2 86.7%

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

   3. Simplified 86.7%

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

      1. distribute-frac-neg2 86.7%

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

      2. distribute-rgt-neg-out 86.7%

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

      3. associate-/r* 70.8%

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

      4. distribute-lft-neg-out 70.8%

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

      5. associate-/l* 67.9%

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

      6. times-frac 97.9%

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

      7. frac-2neg 97.9%

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

      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 53.8%

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

      10. sqrt-unprod 63.2%

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

      11. sqr-neg 63.2%

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

      12. sqrt-unprod 28.2%

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

      13. add-sqr-sqrt 56.3%

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

      14. add-sqr-sqrt 53.6%

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

      15. sqrt-unprod 56.8%

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

      16. sqr-neg 56.8%

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

      17. sqrt-prod 8.3%

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

      18. add-sqr-sqrt 97.8%

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

   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 0 72.7%

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

      1. mul-1-neg 72.7%

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

      2. associate-/l* 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

      4. distribute-neg-frac2 85.0%

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

   9. Simplified 85.0%

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

       1. frac-2neg 85.0%

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

       2. div-inv 84.8%

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

       3. distribute-rgt-neg-in 84.8%

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

       4. distribute-frac-neg 84.8%

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

       5. frac-2neg 84.8%

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

       6. clear-num 86.1%

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

       7. un-div-inv 86.1%

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

       8. +-commutative 86.1%

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

       9. distribute-neg-in 86.1%

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

       10. add-sqr-sqrt 86.0%

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

       11. sqrt-unprod 69.2%

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

       12. sqr-neg 69.2%

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

       13. sqrt-unprod 0.0%

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

       14. add-sqr-sqrt 55.9%

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

   11. Applied egg-rr 55.9%

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

       1. associate-*r/ 55.9%

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

       2. *-rgt-identity 55.9%

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

       3. associate-/r/ 54.6%

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

       4. /-rgt-identity 54.6%

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

       5. times-frac 54.2%

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

       6. *-rgt-identity 54.2%

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

       7. associate-/l* 54.6%

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

       8. sub-neg 54.6%

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

   13. Simplified 54.6%

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

       1. associate-*r/ 54.2%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 65.7%

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

       4. sqr-neg 65.7%

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

       5. sqrt-unprod 72.6%

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

       6. add-sqr-sqrt 72.7%

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

       7. distribute-frac-neg2 72.7%

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

       8. associate-*r/ 85.0%

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

       9. clear-num 86.2%

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

       10. un-div-inv 86.2%

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

   15. Applied egg-rr 86.2%

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

   if -1e80 < u < -1.75e6

   1. Initial program 67.0%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 76.7%

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

      1. clear-num 89.8%

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

      2. frac-times 89.8%

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

      3. *-un-lft-identity 89.8%

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

      4. add-sqr-sqrt 89.8%

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

      5. sqrt-unprod 89.8%

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

      6. sqr-neg 89.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 89.3%

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

   7. Applied egg-rr 89.3%

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

   if -1.75e6 < u < -7.60000000000000006e-51

   1. Initial program 94.0%

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

      1. associate-/l* 99.8%

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

      2. distribute-lft-neg-out 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. associate-/r* 99.7%

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

      5. distribute-neg-frac2 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-frac-neg2 99.7%

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

      2. distribute-rgt-neg-out 99.7%

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

      3. associate-/r* 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. associate-/l* 94.0%

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

      6. times-frac 99.6%

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

      7. frac-2neg 99.6%

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

      8. associate-*r/ 98.2%

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

      9. add-sqr-sqrt 30.9%

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

      10. sqrt-unprod 45.3%

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

      11. sqr-neg 45.3%

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

      12. sqrt-unprod 13.8%

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

      13. add-sqr-sqrt 26.4%

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

      14. add-sqr-sqrt 26.3%

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

      15. sqrt-unprod 38.8%

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

      16. sqr-neg 38.8%

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

      17. sqrt-prod 12.4%

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

      18. add-sqr-sqrt 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in t1 around 0 75.7%

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

      1. mul-1-neg 75.7%

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

      2. associate-/l* 80.0%

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

      3. distribute-rgt-neg-in 80.0%

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

      4. distribute-neg-frac2 80.0%

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

   9. Simplified 80.0%

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

       1. frac-2neg 80.0%

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

       2. div-inv 79.7%

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

       3. distribute-rgt-neg-in 79.7%

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

       4. distribute-frac-neg 79.7%

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

       5. frac-2neg 79.7%

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

       6. clear-num 79.4%

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

       7. un-div-inv 79.6%

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

       8. +-commutative 79.6%

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

       9. distribute-neg-in 79.6%

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

       10. add-sqr-sqrt 79.6%

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

       11. sqrt-unprod 79.6%

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

       12. sqr-neg 79.6%

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

       13. sqrt-unprod 0.0%

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

       14. add-sqr-sqrt 20.6%

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

   11. Applied egg-rr 20.6%

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

       1. associate-*r/ 20.6%

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

       2. *-rgt-identity 20.6%

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

       3. associate-/r/ 20.6%

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

       4. /-rgt-identity 20.6%

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

       5. times-frac 20.7%

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

       6. *-rgt-identity 20.7%

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

       7. associate-/l* 20.6%

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

       8. sub-neg 20.6%

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

   13. Simplified 20.6%

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

       1. frac-2neg 20.6%

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

       2. div-inv 20.6%

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

       3. distribute-lft-neg-in 20.6%

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

       4. add-sqr-sqrt 6.8%

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

       5. sqrt-unprod 40.0%

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

       6. sqr-neg 40.0%

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

       7. sqrt-unprod 55.0%

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

       8. add-sqr-sqrt 80.6%

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

       9. sub-neg 80.6%

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

       10. distribute-neg-in 80.6%

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

       11. add-sqr-sqrt 25.2%

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

       12. sqrt-unprod 79.7%

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

       13. sqr-neg 79.7%

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

       14. sqrt-unprod 54.5%

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

       15. add-sqr-sqrt 79.7%

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

       16. distribute-neg-in 79.7%

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

       17. +-commutative 79.7%

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

       18. distribute-neg-in 79.7%

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

       19. add-sqr-sqrt 25.2%

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

       20. sqrt-unprod 80.6%

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

       21. sqr-neg 80.6%

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

       22. sqrt-unprod 55.4%

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

       23. add-sqr-sqrt 80.6%

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

       24. sub-neg 80.6%

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

   15. Applied egg-rr 80.6%

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

       1. associate-*l* 82.3%

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

   17. Simplified 82.3%

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

   if -7.60000000000000006e-51 < u < 1.25e21

   1. Initial program 66.8%

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

      1. associate-/l* 64.5%

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

      2. distribute-lft-neg-out 64.5%

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

      3. distribute-rgt-neg-in 64.5%

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

      4. associate-/r* 71.8%

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

      5. distribute-neg-frac2 71.8%

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

   3. Simplified 71.8%

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

      1. associate-*r/ 95.8%

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

      2. +-commutative 95.8%

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

      3. distribute-neg-in 95.8%

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

      4. sub-neg 95.8%

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

      5. associate-*l/ 97.7%

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

      6. clear-num 97.6%

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

      7. frac-2neg 97.6%

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

      8. frac-times 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. frac-2neg 99.8%

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

      11. sub-neg 99.8%

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

      12. distribute-neg-in 99.8%

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

      13. +-commutative 99.8%

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

      14. remove-double-neg 99.8%

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

      15. add-sqr-sqrt 42.7%

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

      16. sqrt-unprod 21.6%

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

      17. sqr-neg 21.6%

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

      18. sqrt-unprod 5.7%

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

      19. add-sqr-sqrt 14.4%

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

      20. add-sqr-sqrt 6.9%

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

      21. sqrt-unprod 55.0%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in u around 0 78.8%

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

      1. *-commutative 78.8%

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

   9. Simplified 78.8%

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

   if 1.25e21 < u

   1. Initial program 76.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 88.0%

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

      5. distribute-neg-frac2 88.0%

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

   3. Simplified 88.0%

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

      1. distribute-frac-neg2 88.0%

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

      2. distribute-rgt-neg-out 88.0%

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

      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* 76.0%

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

      6. times-frac 99.8%

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

      7. frac-2neg 99.8%

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

      8. associate-*r/ 99.8%

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

      9. add-sqr-sqrt 35.9%

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

      10. sqrt-unprod 55.3%

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

      11. sqr-neg 55.3%

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

      12. sqrt-unprod 33.4%

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

      13. add-sqr-sqrt 59.0%

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

      14. add-sqr-sqrt 0.4%

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

      15. sqrt-unprod 74.8%

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

      16. sqr-neg 74.8%

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

      17. sqrt-prod 93.5%

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

      18. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t1 around 0 79.0%

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

      1. mul-1-neg 79.0%

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

      2. associate-/l* 82.5%

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

      3. distribute-rgt-neg-in 82.5%

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

      4. distribute-neg-frac2 82.5%

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

   9. Simplified 82.5%

      \[\leadsto \frac{\color{blue}{t1 \cdot \frac{v}{-u}}}{t1 + u} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{elif}\;u \leq -1750000:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \mathbf{elif}\;u \leq -7.6 \cdot 10^{-51}:\\ \;\;\;\;t1 \cdot \left(\frac{v}{u} \cdot \frac{-1}{u - t1}\right)\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+21}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{if}\;u \leq -1.85 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq -1350000:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \mathbf{elif}\;u \leq -7.4 \cdot 10^{-54} \lor \neg \left(u \leq 2.6 \cdot 10^{+21}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* t1 (/ v u)) (- t1 u))))
   (if (<= u -1.85e+79)
     t_1
     (if (<= u -1350000.0)
       (/ v (* t1 (/ (- u t1) t1)))
       (if (or (<= u -7.4e-54) (not (<= u 2.6e+21)))
         t_1
         (/ (- v) (+ t1 (* u 2.0))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 * (v / u)) / (t1 - u);
	double tmp;
	if (u <= -1.85e+79) {
		tmp = t_1;
	} else if (u <= -1350000.0) {
		tmp = v / (t1 * ((u - t1) / t1));
	} else if ((u <= -7.4e-54) || !(u <= 2.6e+21)) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (t1 * (v / u)) / (t1 - u)
    if (u <= (-1.85d+79)) then
        tmp = t_1
    else if (u <= (-1350000.0d0)) then
        tmp = v / (t1 * ((u - t1) / t1))
    else if ((u <= (-7.4d-54)) .or. (.not. (u <= 2.6d+21))) then
        tmp = t_1
    else
        tmp = -v / (t1 + (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 * (v / u)) / (t1 - u);
	double tmp;
	if (u <= -1.85e+79) {
		tmp = t_1;
	} else if (u <= -1350000.0) {
		tmp = v / (t1 * ((u - t1) / t1));
	} else if ((u <= -7.4e-54) || !(u <= 2.6e+21)) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 * (v / u)) / (t1 - u)
	tmp = 0
	if u <= -1.85e+79:
		tmp = t_1
	elif u <= -1350000.0:
		tmp = v / (t1 * ((u - t1) / t1))
	elif (u <= -7.4e-54) or not (u <= 2.6e+21):
		tmp = t_1
	else:
		tmp = -v / (t1 + (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u))
	tmp = 0.0
	if (u <= -1.85e+79)
		tmp = t_1;
	elseif (u <= -1350000.0)
		tmp = Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)));
	elseif ((u <= -7.4e-54) || !(u <= 2.6e+21))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 * (v / u)) / (t1 - u);
	tmp = 0.0;
	if (u <= -1.85e+79)
		tmp = t_1;
	elseif (u <= -1350000.0)
		tmp = v / (t1 * ((u - t1) / t1));
	elseif ((u <= -7.4e-54) || ~((u <= 2.6e+21)))
		tmp = t_1;
	else
		tmp = -v / (t1 + (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.85e+79], t$95$1, If[LessEqual[u, -1350000.0], N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -7.4e-54], N[Not[LessEqual[u, 2.6e+21]], $MachinePrecision]], t$95$1, N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.85000000000000005e79 or -1.35e6 < u < -7.4000000000000006e-54 or 2.6e21 < u

   1. Initial program 75.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 89.1%

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

      5. distribute-neg-frac2 89.1%

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

   3. Simplified 89.1%

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

      1. distribute-frac-neg2 89.1%

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

      2. distribute-rgt-neg-out 89.1%

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

      3. associate-/r* 76.6%

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

      4. distribute-lft-neg-out 76.6%

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

      5. associate-/l* 75.1%

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

      6. times-frac 99.0%

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

      7. frac-2neg 99.0%

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

      8. associate-*r/ 98.7%

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

      9. add-sqr-sqrt 42.7%

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

      10. sqrt-unprod 57.2%

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

      11. sqr-neg 57.2%

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

      12. sqrt-unprod 28.4%

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

      13. add-sqr-sqrt 53.3%

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

      14. add-sqr-sqrt 26.4%

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

      15. sqrt-unprod 62.2%

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

      16. sqr-neg 62.2%

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

      17. sqrt-prod 46.2%

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

      18. add-sqr-sqrt 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in t1 around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-/l* 83.2%

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

      3. distribute-rgt-neg-in 83.2%

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

      4. distribute-neg-frac2 83.2%

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

   9. Simplified 83.2%

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

       1. frac-2neg 83.2%

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

       2. div-inv 83.1%

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

       3. distribute-rgt-neg-in 83.1%

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

       4. distribute-frac-neg 83.1%

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

       5. frac-2neg 83.1%

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

       6. clear-num 83.5%

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

       7. un-div-inv 83.5%

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

       8. +-commutative 83.5%

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

       9. distribute-neg-in 83.5%

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

       10. add-sqr-sqrt 47.4%

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

       11. sqrt-unprod 66.4%

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

       12. sqr-neg 66.4%

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

       13. sqrt-unprod 26.1%

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

       14. add-sqr-sqrt 52.5%

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

   11. Applied egg-rr 52.5%

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

       1. associate-*r/ 52.5%

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

       2. *-rgt-identity 52.5%

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

       3. associate-/r/ 52.0%

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

       4. /-rgt-identity 52.0%

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

       5. times-frac 51.7%

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

       6. *-rgt-identity 51.7%

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

       7. associate-/l* 52.0%

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

       8. sub-neg 52.0%

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

   13. Simplified 52.0%

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

       1. frac-2neg 52.0%

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

       2. div-inv 52.0%

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

       3. distribute-lft-neg-in 52.0%

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

       4. add-sqr-sqrt 23.8%

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

       5. sqrt-unprod 58.0%

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

       6. sqr-neg 58.0%

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

       7. sqrt-unprod 47.0%

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

       8. add-sqr-sqrt 83.0%

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

       9. sub-neg 83.0%

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

       10. distribute-neg-in 83.0%

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

       11. add-sqr-sqrt 35.8%

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

       12. sqrt-unprod 79.6%

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

       13. sqr-neg 79.6%

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

       14. sqrt-unprod 47.3%

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

       15. add-sqr-sqrt 83.1%

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

       16. distribute-neg-in 83.1%

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

       17. +-commutative 83.1%

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

       18. distribute-neg-in 83.1%

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

       19. add-sqr-sqrt 35.8%

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

       20. sqrt-unprod 79.6%

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

       21. sqr-neg 79.6%

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

       22. sqrt-unprod 47.2%

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

       23. add-sqr-sqrt 83.0%

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

       24. sub-neg 83.0%

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

   15. Applied egg-rr 83.0%

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

       1. associate-*r/ 83.2%

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

       2. *-rgt-identity 83.2%

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

   17. Simplified 83.2%

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

   if -1.85000000000000005e79 < u < -1.35e6

   1. Initial program 67.0%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 76.7%

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

      1. clear-num 89.8%

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

      2. frac-times 89.8%

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

      3. *-un-lft-identity 89.8%

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

      4. add-sqr-sqrt 89.8%

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

      5. sqrt-unprod 89.8%

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

      6. sqr-neg 89.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 89.3%

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

   7. Applied egg-rr 89.3%

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

   if -7.4000000000000006e-54 < u < 2.6e21

   1. Initial program 66.8%

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

      1. associate-/l* 64.5%

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

      2. distribute-lft-neg-out 64.5%

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

      3. distribute-rgt-neg-in 64.5%

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

      4. associate-/r* 71.8%

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

      5. distribute-neg-frac2 71.8%

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

   3. Simplified 71.8%

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

      1. associate-*r/ 95.8%

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

      2. +-commutative 95.8%

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

      3. distribute-neg-in 95.8%

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

      4. sub-neg 95.8%

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

      5. associate-*l/ 97.7%

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

      6. clear-num 97.6%

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

      7. frac-2neg 97.6%

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

      8. frac-times 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. frac-2neg 99.8%

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

      11. sub-neg 99.8%

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

      12. distribute-neg-in 99.8%

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

      13. +-commutative 99.8%

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

      14. remove-double-neg 99.8%

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

      15. add-sqr-sqrt 42.7%

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

      16. sqrt-unprod 21.6%

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

      17. sqr-neg 21.6%

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

      18. sqrt-unprod 5.7%

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

      19. add-sqr-sqrt 14.4%

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

      20. add-sqr-sqrt 6.9%

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

      21. sqrt-unprod 55.0%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in u around 0 78.8%

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

      1. *-commutative 78.8%

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

   9. Simplified 78.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{+79}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq -1350000:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \mathbf{elif}\;u \leq -7.4 \cdot 10^{-54} \lor \neg \left(u \leq 2.6 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{if}\;u \leq -1.9 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq -2200000:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \mathbf{elif}\;u \leq -2.65 \cdot 10^{-51}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* t1 (/ v u)) (- t1 u))))
   (if (<= u -1.9e+79)
     t_1
     (if (<= u -2200000.0)
       (/ v (* t1 (/ (- u t1) t1)))
       (if (<= u -2.65e-51)
         (* t1 (/ (/ v u) (- (- u) t1)))
         (if (<= u 2.2e+25) (/ (- v) (+ t1 (* u 2.0))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 * (v / u)) / (t1 - u);
	double tmp;
	if (u <= -1.9e+79) {
		tmp = t_1;
	} else if (u <= -2200000.0) {
		tmp = v / (t1 * ((u - t1) / t1));
	} else if (u <= -2.65e-51) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (u <= 2.2e+25) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (t1 * (v / u)) / (t1 - u)
    if (u <= (-1.9d+79)) then
        tmp = t_1
    else if (u <= (-2200000.0d0)) then
        tmp = v / (t1 * ((u - t1) / t1))
    else if (u <= (-2.65d-51)) then
        tmp = t1 * ((v / u) / (-u - t1))
    else if (u <= 2.2d+25) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 * (v / u)) / (t1 - u);
	double tmp;
	if (u <= -1.9e+79) {
		tmp = t_1;
	} else if (u <= -2200000.0) {
		tmp = v / (t1 * ((u - t1) / t1));
	} else if (u <= -2.65e-51) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (u <= 2.2e+25) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 * (v / u)) / (t1 - u)
	tmp = 0
	if u <= -1.9e+79:
		tmp = t_1
	elif u <= -2200000.0:
		tmp = v / (t1 * ((u - t1) / t1))
	elif u <= -2.65e-51:
		tmp = t1 * ((v / u) / (-u - t1))
	elif u <= 2.2e+25:
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u))
	tmp = 0.0
	if (u <= -1.9e+79)
		tmp = t_1;
	elseif (u <= -2200000.0)
		tmp = Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)));
	elseif (u <= -2.65e-51)
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(Float64(-u) - t1)));
	elseif (u <= 2.2e+25)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 * (v / u)) / (t1 - u);
	tmp = 0.0;
	if (u <= -1.9e+79)
		tmp = t_1;
	elseif (u <= -2200000.0)
		tmp = v / (t1 * ((u - t1) / t1));
	elseif (u <= -2.65e-51)
		tmp = t1 * ((v / u) / (-u - t1));
	elseif (u <= 2.2e+25)
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.9e+79], t$95$1, If[LessEqual[u, -2200000.0], N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -2.65e-51], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.2e+25], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.9000000000000001e79 or 2.2000000000000001e25 < u

   1. Initial program 72.0%

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

      1. associate-/l* 72.8%

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

      2. distribute-lft-neg-out 72.8%

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

      3. distribute-rgt-neg-in 72.8%

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

      4. associate-/r* 87.3%

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

      5. distribute-neg-frac2 87.3%

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

   3. Simplified 87.3%

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

      1. distribute-frac-neg2 87.3%

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

      2. distribute-rgt-neg-out 87.3%

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

      3. associate-/r* 72.8%

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

      4. distribute-lft-neg-out 72.8%

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

      5. associate-/l* 72.0%

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

      6. times-frac 98.9%

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

      7. frac-2neg 98.9%

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

      8. associate-*r/ 98.8%

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

      9. add-sqr-sqrt 44.7%

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

      10. sqrt-unprod 59.2%

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

      11. sqr-neg 59.2%

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

      12. sqrt-unprod 30.8%

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

      13. add-sqr-sqrt 57.7%

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

      14. add-sqr-sqrt 26.5%

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

      15. sqrt-unprod 66.0%

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

      16. sqr-neg 66.0%

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

      17. sqrt-prod 51.8%

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

      18. add-sqr-sqrt 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in t1 around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-/l* 83.7%

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

      3. distribute-rgt-neg-in 83.7%

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

      4. distribute-neg-frac2 83.7%

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

   9. Simplified 83.7%

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

       1. frac-2neg 83.7%

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

       2. div-inv 83.6%

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

       3. distribute-rgt-neg-in 83.6%

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

       4. distribute-frac-neg 83.6%

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

       5. frac-2neg 83.6%

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

       6. clear-num 84.2%

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

       7. un-div-inv 84.2%

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

       8. +-commutative 84.2%

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

       9. distribute-neg-in 84.2%

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

       10. add-sqr-sqrt 42.1%

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

       11. sqrt-unprod 64.3%

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

       12. sqr-neg 64.3%

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

       13. sqrt-unprod 30.4%

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

       14. add-sqr-sqrt 57.7%

           \[\leadsto \frac{t1}{\frac{u}{v}} \cdot \frac{1}{\color{blue}{u} + \left(-t1\right)} \]

   11. Applied egg-rr 57.7%

       \[\leadsto \color{blue}{\frac{t1}{\frac{u}{v}} \cdot \frac{1}{u + \left(-t1\right)}} \]
   12. Step-by-step derivation

       1. associate-*r/ 57.7%

          \[\leadsto \color{blue}{\frac{\frac{t1}{\frac{u}{v}} \cdot 1}{u + \left(-t1\right)}} \]

       2. *-rgt-identity 57.7%

          \[\leadsto \frac{\color{blue}{\frac{t1}{\frac{u}{v}}}}{u + \left(-t1\right)} \]

       3. associate-/r/ 57.1%

          \[\leadsto \frac{\color{blue}{\frac{t1}{u} \cdot v}}{u + \left(-t1\right)} \]

       4. /-rgt-identity 57.1%

          \[\leadsto \frac{\frac{t1}{u} \cdot \color{blue}{\frac{v}{1}}}{u + \left(-t1\right)} \]

       5. times-frac 56.7%

          \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{u \cdot 1}}}{u + \left(-t1\right)} \]

       6. *-rgt-identity 56.7%

          \[\leadsto \frac{\frac{t1 \cdot v}{\color{blue}{u}}}{u + \left(-t1\right)} \]

       7. associate-/l* 57.1%

          \[\leadsto \frac{\color{blue}{t1 \cdot \frac{v}{u}}}{u + \left(-t1\right)} \]

       8. sub-neg 57.1%

          \[\leadsto \frac{t1 \cdot \frac{v}{u}}{\color{blue}{u - t1}} \]

   13. Simplified 57.1%

       \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{u}}{u - t1}} \]
   14. Step-by-step derivation

       1. frac-2neg 57.1%

          \[\leadsto \color{blue}{\frac{-t1 \cdot \frac{v}{u}}{-\left(u - t1\right)}} \]

       2. div-inv 57.1%

          \[\leadsto \color{blue}{\left(-t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{-\left(u - t1\right)}} \]

       3. distribute-lft-neg-in 57.1%

          \[\leadsto \color{blue}{\left(\left(-t1\right) \cdot \frac{v}{u}\right)} \cdot \frac{1}{-\left(u - t1\right)} \]

       4. add-sqr-sqrt 26.5%

          \[\leadsto \left(\color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{v}{u}\right) \cdot \frac{1}{-\left(u - t1\right)} \]

       5. sqrt-unprod 60.9%

          \[\leadsto \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{v}{u}\right) \cdot \frac{1}{-\left(u - t1\right)} \]

       6. sqr-neg 60.9%

          \[\leadsto \left(\sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{v}{u}\right) \cdot \frac{1}{-\left(u - t1\right)} \]

       7. sqrt-unprod 45.7%

          \[\leadsto \left(\color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{v}{u}\right) \cdot \frac{1}{-\left(u - t1\right)} \]

       8. add-sqr-sqrt 83.4%

          \[\leadsto \left(\color{blue}{t1} \cdot \frac{v}{u}\right) \cdot \frac{1}{-\left(u - t1\right)} \]

       9. sub-neg 83.4%

          \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{-\color{blue}{\left(u + \left(-t1\right)\right)}} \]

       10. distribute-neg-in 83.4%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\color{blue}{\left(-u\right) + \left(-\left(-t1\right)\right)}} \]

       11. add-sqr-sqrt 37.6%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\left(-u\right) + \left(-\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}\right)} \]

       12. sqrt-unprod 79.5%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\left(-u\right) + \left(-\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}\right)} \]

       13. sqr-neg 79.5%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\left(-u\right) + \left(-\sqrt{\color{blue}{t1 \cdot t1}}\right)} \]

       14. sqrt-unprod 46.1%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\left(-u\right) + \left(-\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}\right)} \]

       15. add-sqr-sqrt 83.6%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\left(-u\right) + \left(-\color{blue}{t1}\right)} \]

       16. distribute-neg-in 83.6%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\color{blue}{-\left(u + t1\right)}} \]

       17. +-commutative 83.6%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{-\color{blue}{\left(t1 + u\right)}} \]

       18. distribute-neg-in 83.6%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\color{blue}{\left(-t1\right) + \left(-u\right)}} \]

       19. add-sqr-sqrt 37.5%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)} \]

       20. sqrt-unprod 79.4%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)} \]

       21. sqr-neg 79.4%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)} \]

       22. sqrt-unprod 45.9%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)} \]

       23. add-sqr-sqrt 83.4%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\color{blue}{t1} + \left(-u\right)} \]

       24. sub-neg 83.4%

           \[\leadsto \left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{\color{blue}{t1 - u}} \]

   15. Applied egg-rr 83.4%

       \[\leadsto \color{blue}{\left(t1 \cdot \frac{v}{u}\right) \cdot \frac{1}{t1 - u}} \]
   16. Step-by-step derivation

       1. associate-*r/ 83.5%

          \[\leadsto \color{blue}{\frac{\left(t1 \cdot \frac{v}{u}\right) \cdot 1}{t1 - u}} \]

       2. *-rgt-identity 83.5%

          \[\leadsto \frac{\color{blue}{t1 \cdot \frac{v}{u}}}{t1 - u} \]

   17. Simplified 83.5%

       \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{u}}{t1 - u}} \]

   if -1.9000000000000001e79 < u < -2.2e6

   1. Initial program 67.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 76.7%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. clear-num 89.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(-u\right) - t1}{t1}}} \cdot \frac{v}{t1} \]

      2. frac-times 89.8%

         \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{\left(-u\right) - t1}{t1} \cdot t1}} \]

      3. *-un-lft-identity 89.8%

         \[\leadsto \frac{\color{blue}{v}}{\frac{\left(-u\right) - t1}{t1} \cdot t1} \]

      4. add-sqr-sqrt 89.8%

         \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} - t1}{t1} \cdot t1} \]

      5. sqrt-unprod 89.8%

         \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} - t1}{t1} \cdot t1} \]

      6. sqr-neg 89.8%

         \[\leadsto \frac{v}{\frac{\sqrt{\color{blue}{u \cdot u}} - t1}{t1} \cdot t1} \]

      7. sqrt-unprod 0.0%

         \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{u} \cdot \sqrt{u}} - t1}{t1} \cdot t1} \]

      8. add-sqr-sqrt 89.3%

         \[\leadsto \frac{v}{\frac{\color{blue}{u} - t1}{t1} \cdot t1} \]

   7. Applied egg-rr 89.3%

      \[\leadsto \color{blue}{\frac{v}{\frac{u - t1}{t1} \cdot t1}} \]

   if -2.2e6 < u < -2.64999999999999987e-51

   1. Initial program 94.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 99.8%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 99.7%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 99.7%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 81.5%

      \[\leadsto t1 \cdot \frac{\color{blue}{\frac{v}{u}}}{-\left(t1 + u\right)} \]

   if -2.64999999999999987e-51 < u < 2.2000000000000001e25

   1. Initial program 66.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 64.5%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 64.5%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 64.5%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 71.8%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 71.8%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 95.8%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. +-commutative 95.8%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{-\color{blue}{\left(u + t1\right)}} \]

      3. distribute-neg-in 95.8%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      4. sub-neg 95.8%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{\left(-u\right) - t1}} \]

      5. associate-*l/ 97.7%

         \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]

      6. clear-num 97.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(-u\right) - t1}{t1}}} \cdot \frac{v}{t1 + u} \]

      7. frac-2neg 97.6%

         \[\leadsto \frac{1}{\frac{\left(-u\right) - t1}{t1}} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

      8. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-v\right)}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)}} \]

      9. *-un-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{-v}}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      10. frac-2neg 99.8%

          \[\leadsto \frac{-v}{\color{blue}{\frac{-\left(\left(-u\right) - t1\right)}{-t1}} \cdot \left(-\left(t1 + u\right)\right)} \]

      11. sub-neg 99.8%

          \[\leadsto \frac{-v}{\frac{-\color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      12. distribute-neg-in 99.8%

          \[\leadsto \frac{-v}{\frac{-\color{blue}{\left(-\left(u + t1\right)\right)}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{-v}{\frac{-\left(-\color{blue}{\left(t1 + u\right)}\right)}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      14. remove-double-neg 99.8%

          \[\leadsto \frac{-v}{\frac{\color{blue}{t1 + u}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      15. add-sqr-sqrt 42.7%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

      16. sqrt-unprod 21.6%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}} \cdot \left(-\left(t1 + u\right)\right)} \]

      17. sqr-neg 21.6%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\sqrt{\color{blue}{t1 \cdot t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

      18. sqrt-unprod 5.7%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

      19. add-sqr-sqrt 14.4%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{t1}} \cdot \left(-\left(t1 + u\right)\right)} \]

      20. add-sqr-sqrt 6.9%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\left(\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}\right)}} \]

      21. sqrt-unprod 55.0%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\sqrt{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{-v}{\frac{t1 + u}{t1} \cdot \left(t1 + u\right)}} \]

   7. Taylor expanded in u around 0 78.8%

      \[\leadsto \frac{-v}{\color{blue}{t1 + 2 \cdot u}} \]
   8. Step-by-step derivation

      1. *-commutative 78.8%

         \[\leadsto \frac{-v}{t1 + \color{blue}{u \cdot 2}} \]

   9. Simplified 78.8%

      \[\leadsto \frac{-v}{\color{blue}{t1 + u \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+79}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq -2200000:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \mathbf{elif}\;u \leq -2.65 \cdot 10^{-51}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+122} \lor \neg \left(u \leq 3.9 \cdot 10^{+68}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.1e+122) (not (<= u 3.9e+68)))
   (* t1 (/ (/ v u) (+ t1 u)))
   (/ (- v) (+ t1 (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.1e+122) || !(u <= 3.9e+68)) {
		tmp = t1 * ((v / u) / (t1 + u));
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	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 <= (-2.1d+122)) .or. (.not. (u <= 3.9d+68))) then
        tmp = t1 * ((v / u) / (t1 + u))
    else
        tmp = -v / (t1 + (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.1e+122) || !(u <= 3.9e+68)) {
		tmp = t1 * ((v / u) / (t1 + u));
	} else {
		tmp = -v / (t1 + (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.1e+122) or not (u <= 3.9e+68):
		tmp = t1 * ((v / u) / (t1 + u))
	else:
		tmp = -v / (t1 + (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.1e+122) || !(u <= 3.9e+68))
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.1e+122) || ~((u <= 3.9e+68)))
		tmp = t1 * ((v / u) / (t1 + u));
	else
		tmp = -v / (t1 + (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.1e+122], N[Not[LessEqual[u, 3.9e+68]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.10000000000000016e122 or 3.90000000000000019e68 < u

   1. Initial program 69.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 71.3%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 71.3%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 71.3%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 86.1%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 86.1%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 86.1%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-frac-neg2 86.1%

         \[\leadsto t1 \cdot \color{blue}{\left(-\frac{\frac{v}{t1 + u}}{t1 + u}\right)} \]

      2. distribute-rgt-neg-out 86.1%

         \[\leadsto \color{blue}{-t1 \cdot \frac{\frac{v}{t1 + u}}{t1 + u}} \]

      3. associate-/r* 71.3%

         \[\leadsto -t1 \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      4. distribute-lft-neg-out 71.3%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      5. associate-/l* 69.4%

         \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      6. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      7. frac-2neg 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

      8. associate-*r/ 98.7%

         \[\leadsto \color{blue}{\frac{\frac{-t1}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)}} \]

      9. add-sqr-sqrt 44.4%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

      10. sqrt-unprod 61.2%

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

      11. sqr-neg 61.2%

          \[\leadsto \frac{\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

      12. sqrt-unprod 33.8%

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

      13. add-sqr-sqrt 63.9%

          \[\leadsto \frac{\frac{\color{blue}{t1}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

      14. add-sqr-sqrt 27.2%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}}} \]

      15. sqrt-unprod 69.0%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

      16. sqr-neg 69.0%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}} \]

      17. sqrt-prod 55.2%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{t1 + u} \cdot \sqrt{t1 + u}}} \]

      18. add-sqr-sqrt 98.7%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{t1 + u}} \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{t1 + u}} \]

   7. Taylor expanded in t1 around 0 76.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{t1 \cdot v}{u}}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 76.0%

         \[\leadsto \frac{\color{blue}{-\frac{t1 \cdot v}{u}}}{t1 + u} \]

      2. associate-/l* 85.2%

         \[\leadsto \frac{-\color{blue}{t1 \cdot \frac{v}{u}}}{t1 + u} \]

      3. distribute-rgt-neg-in 85.2%

         \[\leadsto \frac{\color{blue}{t1 \cdot \left(-\frac{v}{u}\right)}}{t1 + u} \]

      4. distribute-neg-frac2 85.2%

         \[\leadsto \frac{t1 \cdot \color{blue}{\frac{v}{-u}}}{t1 + u} \]

   9. Simplified 85.2%

      \[\leadsto \frac{\color{blue}{t1 \cdot \frac{v}{-u}}}{t1 + u} \]
   10. Step-by-step derivation

       1. associate-/l* 80.1%

          \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{-u}}{t1 + u}} \]

       2. *-commutative 80.1%

          \[\leadsto \color{blue}{\frac{\frac{v}{-u}}{t1 + u} \cdot t1} \]

       3. add-sqr-sqrt 36.9%

          \[\leadsto \frac{\frac{v}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}}}}{t1 + u} \cdot t1 \]

       4. sqrt-unprod 66.7%

          \[\leadsto \frac{\frac{v}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}}}{t1 + u} \cdot t1 \]

       5. sqr-neg 66.7%

          \[\leadsto \frac{\frac{v}{\sqrt{\color{blue}{u \cdot u}}}}{t1 + u} \cdot t1 \]

       6. sqrt-unprod 35.4%

          \[\leadsto \frac{\frac{v}{\color{blue}{\sqrt{u} \cdot \sqrt{u}}}}{t1 + u} \cdot t1 \]

       7. add-sqr-sqrt 64.3%

          \[\leadsto \frac{\frac{v}{\color{blue}{u}}}{t1 + u} \cdot t1 \]

   11. Applied egg-rr 64.3%

       \[\leadsto \color{blue}{\frac{\frac{v}{u}}{t1 + u} \cdot t1} \]

   if -2.10000000000000016e122 < u < 3.90000000000000019e68

   1. Initial program 71.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 68.9%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. distribute-lft-neg-out 68.9%

         \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 68.9%

         \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

      4. associate-/r* 76.0%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 76.0%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 76.0%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 96.7%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

      2. +-commutative 96.7%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{-\color{blue}{\left(u + t1\right)}} \]

      3. distribute-neg-in 96.7%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      4. sub-neg 96.7%

         \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{\left(-u\right) - t1}} \]

      5. associate-*l/ 98.2%

         \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]

      6. clear-num 98.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(-u\right) - t1}{t1}}} \cdot \frac{v}{t1 + u} \]

      7. frac-2neg 98.1%

         \[\leadsto \frac{1}{\frac{\left(-u\right) - t1}{t1}} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

      8. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-v\right)}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)}} \]

      9. *-un-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{-v}}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      10. frac-2neg 99.8%

          \[\leadsto \frac{-v}{\color{blue}{\frac{-\left(\left(-u\right) - t1\right)}{-t1}} \cdot \left(-\left(t1 + u\right)\right)} \]

      11. sub-neg 99.8%

          \[\leadsto \frac{-v}{\frac{-\color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      12. distribute-neg-in 99.8%

          \[\leadsto \frac{-v}{\frac{-\color{blue}{\left(-\left(u + t1\right)\right)}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{-v}{\frac{-\left(-\color{blue}{\left(t1 + u\right)}\right)}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      14. remove-double-neg 99.8%

          \[\leadsto \frac{-v}{\frac{\color{blue}{t1 + u}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      15. add-sqr-sqrt 40.8%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

      16. sqrt-unprod 27.3%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}} \cdot \left(-\left(t1 + u\right)\right)} \]

      17. sqr-neg 27.3%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\sqrt{\color{blue}{t1 \cdot t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

      18. sqrt-unprod 8.7%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

      19. add-sqr-sqrt 17.7%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{t1}} \cdot \left(-\left(t1 + u\right)\right)} \]

      20. add-sqr-sqrt 11.3%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\left(\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}\right)}} \]

      21. sqrt-unprod 52.9%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\sqrt{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{-v}{\frac{t1 + u}{t1} \cdot \left(t1 + u\right)}} \]

   7. Taylor expanded in u around 0 70.2%

      \[\leadsto \frac{-v}{\color{blue}{t1 + 2 \cdot u}} \]
   8. Step-by-step derivation

      1. *-commutative 70.2%

         \[\leadsto \frac{-v}{t1 + \color{blue}{u \cdot 2}} \]

   9. Simplified 70.2%

      \[\leadsto \frac{-v}{\color{blue}{t1 + u \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+122} \lor \neg \left(u \leq 3.9 \cdot 10^{+68}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+157} \lor \neg \left(t1 \leq 4.3 \cdot 10^{+72}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.5e+157) (not (<= t1 4.3e+72))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.5e+157) || !(t1 <= 4.3e+72)) {
		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 <= (-9.5d+157)) .or. (.not. (t1 <= 4.3d+72))) 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 <= -9.5e+157) || !(t1 <= 4.3e+72)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.5e+157) or not (t1 <= 4.3e+72):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.5e+157) || !(t1 <= 4.3e+72))
		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 <= -9.5e+157) || ~((t1 <= 4.3e+72)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.5e+157], N[Not[LessEqual[t1, 4.3e+72]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.4999999999999996e157 or 4.3000000000000001e72 < t1

   1. Initial program 47.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 87.5%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. remove-double-neg 87.5%

         \[\leadsto \frac{\color{blue}{-\left(-t1\right)}}{\left(-u\right) - t1} \cdot \frac{v}{t1} \]

      2. sub-neg 87.5%

         \[\leadsto \frac{-\left(-t1\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1} \]

      3. distribute-neg-in 87.5%

         \[\leadsto \frac{-\left(-t1\right)}{\color{blue}{-\left(u + t1\right)}} \cdot \frac{v}{t1} \]

      4. +-commutative 87.5%

         \[\leadsto \frac{-\left(-t1\right)}{-\color{blue}{\left(t1 + u\right)}} \cdot \frac{v}{t1} \]

      5. frac-2neg 87.5%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u}} \cdot \frac{v}{t1} \]

      6. div-inv 87.3%

         \[\leadsto \color{blue}{\left(\left(-t1\right) \cdot \frac{1}{t1 + u}\right)} \cdot \frac{v}{t1} \]

      7. add-sqr-sqrt 31.1%

         \[\leadsto \left(\color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      8. sqrt-unprod 5.1%

         \[\leadsto \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      9. sqr-neg 5.1%

         \[\leadsto \left(\sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      10. sqrt-unprod 13.8%

          \[\leadsto \left(\color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      11. add-sqr-sqrt 27.1%

          \[\leadsto \left(\color{blue}{t1} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

   7. Applied egg-rr 27.1%

      \[\leadsto \color{blue}{\left(t1 \cdot \frac{1}{t1 + u}\right)} \cdot \frac{v}{t1} \]
   8. Step-by-step derivation

      1. associate-*r/ 27.1%

         \[\leadsto \color{blue}{\frac{t1 \cdot 1}{t1 + u}} \cdot \frac{v}{t1} \]

      2. *-rgt-identity 27.1%

         \[\leadsto \frac{\color{blue}{t1}}{t1 + u} \cdot \frac{v}{t1} \]

   9. Simplified 27.1%

      \[\leadsto \color{blue}{\frac{t1}{t1 + u}} \cdot \frac{v}{t1} \]

   10. Taylor expanded in t1 around inf 26.8%

       \[\leadsto \color{blue}{\frac{v}{t1}} \]

   if -9.4999999999999996e157 < t1 < 4.3000000000000001e72

   1. Initial program 82.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 97.6%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 97.6%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 97.6%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 97.6%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 97.6%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 97.6%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 53.0%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. remove-double-neg 53.0%

         \[\leadsto \frac{\color{blue}{-\left(-t1\right)}}{\left(-u\right) - t1} \cdot \frac{v}{t1} \]

      2. sub-neg 53.0%

         \[\leadsto \frac{-\left(-t1\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1} \]

      3. distribute-neg-in 53.0%

         \[\leadsto \frac{-\left(-t1\right)}{\color{blue}{-\left(u + t1\right)}} \cdot \frac{v}{t1} \]

      4. +-commutative 53.0%

         \[\leadsto \frac{-\left(-t1\right)}{-\color{blue}{\left(t1 + u\right)}} \cdot \frac{v}{t1} \]

      5. frac-2neg 53.0%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u}} \cdot \frac{v}{t1} \]

      6. div-inv 53.0%

         \[\leadsto \color{blue}{\left(\left(-t1\right) \cdot \frac{1}{t1 + u}\right)} \cdot \frac{v}{t1} \]

      7. add-sqr-sqrt 22.5%

         \[\leadsto \left(\color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      8. sqrt-unprod 35.5%

         \[\leadsto \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      9. sqr-neg 35.5%

         \[\leadsto \left(\sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      10. sqrt-unprod 11.2%

          \[\leadsto \left(\color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      11. add-sqr-sqrt 20.3%

          \[\leadsto \left(\color{blue}{t1} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

   7. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{\left(t1 \cdot \frac{1}{t1 + u}\right)} \cdot \frac{v}{t1} \]
   8. Step-by-step derivation

      1. associate-*r/ 20.3%

         \[\leadsto \color{blue}{\frac{t1 \cdot 1}{t1 + u}} \cdot \frac{v}{t1} \]

      2. *-rgt-identity 20.3%

         \[\leadsto \frac{\color{blue}{t1}}{t1 + u} \cdot \frac{v}{t1} \]

   9. Simplified 20.3%

      \[\leadsto \color{blue}{\frac{t1}{t1 + u}} \cdot \frac{v}{t1} \]

   10. Taylor expanded in t1 around 0 17.1%

       \[\leadsto \color{blue}{\frac{v}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{+157} \lor \neg \left(t1 \leq 4.3 \cdot 10^{+72}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\frac{t1}{t1 + u} \cdot \frac{v}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (- (* (/ t1 (+ t1 u)) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	return -((t1 / (t1 + u)) * (v / (t1 + u)));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = -((t1 / (t1 + u)) * (v / (t1 + u)))
end function

Java

public static double code(double u, double v, double t1) {
	return -((t1 / (t1 + u)) * (v / (t1 + u)));
}

Python

def code(u, v, t1):
	return -((t1 / (t1 + u)) * (v / (t1 + u)))

Julia

function code(u, v, t1)
	return Float64(-Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(t1 + u))))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = -((t1 / (t1 + u)) * (v / (t1 + u)));
end

Wolfram

code[u_, v_, t1_] := (-N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 70.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 98.4%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 98.4%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 98.4%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 98.4%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 98.4%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Final simplification 98.4%

   \[\leadsto -\frac{t1}{t1 + u} \cdot \frac{v}{t1 + u} \]
6. Add Preprocessing

Alternative 13: 57.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+170}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.2e+170) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.2e+170) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.2d+170)) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.2e+170) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.2e+170:
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.2e+170)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.2e+170)
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.2e+170], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.19999999999999996e170

   1. Initial program 64.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 50.0%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. remove-double-neg 50.0%

         \[\leadsto \frac{\color{blue}{-\left(-t1\right)}}{\left(-u\right) - t1} \cdot \frac{v}{t1} \]

      2. sub-neg 50.0%

         \[\leadsto \frac{-\left(-t1\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1} \]

      3. distribute-neg-in 50.0%

         \[\leadsto \frac{-\left(-t1\right)}{\color{blue}{-\left(u + t1\right)}} \cdot \frac{v}{t1} \]

      4. +-commutative 50.0%

         \[\leadsto \frac{-\left(-t1\right)}{-\color{blue}{\left(t1 + u\right)}} \cdot \frac{v}{t1} \]

      5. frac-2neg 50.0%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u}} \cdot \frac{v}{t1} \]

      6. div-inv 50.0%

         \[\leadsto \color{blue}{\left(\left(-t1\right) \cdot \frac{1}{t1 + u}\right)} \cdot \frac{v}{t1} \]

      7. add-sqr-sqrt 25.3%

         \[\leadsto \left(\color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      8. sqrt-unprod 41.8%

         \[\leadsto \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      9. sqr-neg 41.8%

         \[\leadsto \left(\sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      10. sqrt-unprod 21.0%

          \[\leadsto \left(\color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

      11. add-sqr-sqrt 45.7%

          \[\leadsto \left(\color{blue}{t1} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

   7. Applied egg-rr 45.7%

      \[\leadsto \color{blue}{\left(t1 \cdot \frac{1}{t1 + u}\right)} \cdot \frac{v}{t1} \]
   8. Step-by-step derivation

      1. associate-*r/ 45.7%

         \[\leadsto \color{blue}{\frac{t1 \cdot 1}{t1 + u}} \cdot \frac{v}{t1} \]

      2. *-rgt-identity 45.7%

         \[\leadsto \frac{\color{blue}{t1}}{t1 + u} \cdot \frac{v}{t1} \]

   9. Simplified 45.7%

      \[\leadsto \color{blue}{\frac{t1}{t1 + u}} \cdot \frac{v}{t1} \]

   10. Taylor expanded in t1 around 0 35.2%

       \[\leadsto \color{blue}{\frac{v}{u}} \]

   if -4.19999999999999996e170 < u

   1. Initial program 71.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 78.6%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 78.6%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 58.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 58.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 58.9%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+170}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.8e+170) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.8e+170) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.8d+170)) then
        tmp = v / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.8e+170) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.8e+170:
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.8e+170)
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.8e+170)
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.8e+170], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.8000000000000001e170

   1. Initial program 64.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 50.0%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 35.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 35.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 35.2%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 35.2%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]

   if -5.8000000000000001e170 < u

   1. Initial program 71.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 78.6%

         \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

      5. distribute-neg-frac2 78.6%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 58.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 58.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 58.9%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ v \cdot \frac{\frac{t1}{u - t1}}{t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (* v (/ (/ t1 (- u t1)) t1)))

C

double code(double u, double v, double t1) {
	return v * ((t1 / (u - t1)) / 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)) / t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v * ((t1 / (u - t1)) / t1);
}

Python

def code(u, v, t1):
	return v * ((t1 / (u - t1)) / t1)

Julia

function code(u, v, t1)
	return Float64(v * Float64(Float64(t1 / Float64(u - t1)) / t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v * ((t1 / (u - t1)) / t1);
end

Wolfram

code[u_, v_, t1_] := N[(v * N[(N[(t1 / N[(u - t1), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 98.4%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 98.4%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 98.4%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 98.4%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 98.4%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Taylor expanded in t1 around inf 64.6%

   \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
6. Step-by-step derivation

   1. clear-num 63.7%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{1}{\frac{t1}{v}}} \]

   2. frac-times 47.1%

      \[\leadsto \color{blue}{\frac{t1 \cdot 1}{\left(\left(-u\right) - t1\right) \cdot \frac{t1}{v}}} \]

   3. *-commutative 47.1%

      \[\leadsto \frac{\color{blue}{1 \cdot t1}}{\left(\left(-u\right) - t1\right) \cdot \frac{t1}{v}} \]

   4. *-un-lft-identity 47.1%

      \[\leadsto \frac{\color{blue}{t1}}{\left(\left(-u\right) - t1\right) \cdot \frac{t1}{v}} \]

   5. add-sqr-sqrt 25.3%

      \[\leadsto \frac{t1}{\left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} - t1\right) \cdot \frac{t1}{v}} \]

   6. sqrt-unprod 51.3%

      \[\leadsto \frac{t1}{\left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} - t1\right) \cdot \frac{t1}{v}} \]

   7. sqr-neg 51.3%

      \[\leadsto \frac{t1}{\left(\sqrt{\color{blue}{u \cdot u}} - t1\right) \cdot \frac{t1}{v}} \]

   8. sqrt-unprod 21.7%

      \[\leadsto \frac{t1}{\left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} - t1\right) \cdot \frac{t1}{v}} \]

   9. add-sqr-sqrt 46.0%

      \[\leadsto \frac{t1}{\left(\color{blue}{u} - t1\right) \cdot \frac{t1}{v}} \]

7. Applied egg-rr 46.0%

   \[\leadsto \color{blue}{\frac{t1}{\left(u - t1\right) \cdot \frac{t1}{v}}} \]
8. Step-by-step derivation

   1. associate-/r* 62.5%

      \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1}}{\frac{t1}{v}}} \]

   2. associate-/r/ 64.5%

      \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1}}{t1} \cdot v} \]

9. Simplified 64.5%

   \[\leadsto \color{blue}{\frac{\frac{t1}{u - t1}}{t1} \cdot v} \]

10. Final simplification 64.5%

    \[\leadsto v \cdot \frac{\frac{t1}{u - t1}}{t1} \]
11. Add Preprocessing

Alternative 16: 69.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{v}{t1 \cdot \frac{u - t1}{t1}} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (* t1 (/ (- u t1) t1))))

C

double code(double u, double v, double t1) {
	return v / (t1 * ((u - t1) / 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) / t1))
end function

Java

public static double code(double u, double v, double t1) {
	return v / (t1 * ((u - t1) / t1));
}

Python

def code(u, v, t1):
	return v / (t1 * ((u - t1) / t1))

Julia

function code(u, v, t1)
	return Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (t1 * ((u - t1) / t1));
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 98.4%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 98.4%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 98.4%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 98.4%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 98.4%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Taylor expanded in t1 around inf 64.6%

   \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
6. Step-by-step derivation

   1. clear-num 66.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(-u\right) - t1}{t1}}} \cdot \frac{v}{t1} \]

   2. frac-times 68.0%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{\left(-u\right) - t1}{t1} \cdot t1}} \]

   3. *-un-lft-identity 68.0%

      \[\leadsto \frac{\color{blue}{v}}{\frac{\left(-u\right) - t1}{t1} \cdot t1} \]

   4. add-sqr-sqrt 34.6%

      \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} - t1}{t1} \cdot t1} \]

   5. sqrt-unprod 67.7%

      \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} - t1}{t1} \cdot t1} \]

   6. sqr-neg 67.7%

      \[\leadsto \frac{v}{\frac{\sqrt{\color{blue}{u \cdot u}} - t1}{t1} \cdot t1} \]

   7. sqrt-unprod 33.2%

      \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{u} \cdot \sqrt{u}} - t1}{t1} \cdot t1} \]

   8. add-sqr-sqrt 66.9%

      \[\leadsto \frac{v}{\frac{\color{blue}{u} - t1}{t1} \cdot t1} \]

7. Applied egg-rr 66.9%

   \[\leadsto \color{blue}{\frac{v}{\frac{u - t1}{t1} \cdot t1}} \]

8. Final simplification 66.9%

   \[\leadsto \frac{v}{t1 \cdot \frac{u - t1}{t1}} \]
9. Add Preprocessing

Alternative 17: 63.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-v}{t1 + u \cdot 2} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (- v) (+ t1 (* u 2.0))))

C

double code(double u, double v, double t1) {
	return -v / (t1 + (u * 2.0));
}

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 * 2.0d0))
end function

Java

public static double code(double u, double v, double t1) {
	return -v / (t1 + (u * 2.0));
}

Python

def code(u, v, t1):
	return -v / (t1 + (u * 2.0))

Julia

function code(u, v, t1)
	return Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = -v / (t1 + (u * 2.0));
end

Wolfram

code[u_, v_, t1_] := N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 69.7%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. distribute-lft-neg-out 69.7%

      \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. distribute-rgt-neg-in 69.7%

      \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

   4. associate-/r* 79.3%

      \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

   5. distribute-neg-frac2 79.3%

      \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

3. Simplified 79.3%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 97.7%

      \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

   2. +-commutative 97.7%

      \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{-\color{blue}{\left(u + t1\right)}} \]

   3. distribute-neg-in 97.7%

      \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   4. sub-neg 97.7%

      \[\leadsto \frac{t1 \cdot \frac{v}{t1 + u}}{\color{blue}{\left(-u\right) - t1}} \]

   5. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]

   6. clear-num 98.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(-u\right) - t1}{t1}}} \cdot \frac{v}{t1 + u} \]

   7. frac-2neg 98.3%

      \[\leadsto \frac{1}{\frac{\left(-u\right) - t1}{t1}} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

   8. frac-times 95.7%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(-v\right)}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)}} \]

   9. *-un-lft-identity 95.7%

      \[\leadsto \frac{\color{blue}{-v}}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)} \]

   10. frac-2neg 95.7%

       \[\leadsto \frac{-v}{\color{blue}{\frac{-\left(\left(-u\right) - t1\right)}{-t1}} \cdot \left(-\left(t1 + u\right)\right)} \]

   11. sub-neg 95.7%

       \[\leadsto \frac{-v}{\frac{-\color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

   12. distribute-neg-in 95.7%

       \[\leadsto \frac{-v}{\frac{-\color{blue}{\left(-\left(u + t1\right)\right)}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

   13. +-commutative 95.7%

       \[\leadsto \frac{-v}{\frac{-\left(-\color{blue}{\left(t1 + u\right)}\right)}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

   14. remove-double-neg 95.7%

       \[\leadsto \frac{-v}{\frac{\color{blue}{t1 + u}}{-t1} \cdot \left(-\left(t1 + u\right)\right)} \]

   15. add-sqr-sqrt 40.4%

       \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

   16. sqrt-unprod 36.9%

       \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}} \cdot \left(-\left(t1 + u\right)\right)} \]

   17. sqr-neg 36.9%

       \[\leadsto \frac{-v}{\frac{t1 + u}{\sqrt{\color{blue}{t1 \cdot t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

   18. sqrt-unprod 16.9%

       \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}} \cdot \left(-\left(t1 + u\right)\right)} \]

   19. add-sqr-sqrt 32.8%

       \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{t1}} \cdot \left(-\left(t1 + u\right)\right)} \]

   20. add-sqr-sqrt 16.5%

       \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\left(\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}\right)}} \]

   21. sqrt-unprod 57.7%

       \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\sqrt{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

6. Applied egg-rr 95.7%

   \[\leadsto \color{blue}{\frac{-v}{\frac{t1 + u}{t1} \cdot \left(t1 + u\right)}} \]

7. Taylor expanded in u around 0 62.4%

   \[\leadsto \frac{-v}{\color{blue}{t1 + 2 \cdot u}} \]
8. Step-by-step derivation

   1. *-commutative 62.4%

      \[\leadsto \frac{-v}{t1 + \color{blue}{u \cdot 2}} \]

9. Simplified 62.4%

   \[\leadsto \frac{-v}{\color{blue}{t1 + u \cdot 2}} \]

10. Final simplification 62.4%

    \[\leadsto \frac{-v}{t1 + u \cdot 2} \]
11. Add Preprocessing

Alternative 18: 62.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(-u\right) - t1} \end{array} \]

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(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 70.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 69.7%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. distribute-lft-neg-out 69.7%

      \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. distribute-rgt-neg-in 69.7%

      \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

   4. associate-/r* 79.3%

      \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

   5. distribute-neg-frac2 79.3%

      \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

3. Simplified 79.3%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-frac-neg2 79.3%

      \[\leadsto t1 \cdot \color{blue}{\left(-\frac{\frac{v}{t1 + u}}{t1 + u}\right)} \]

   2. distribute-rgt-neg-out 79.3%

      \[\leadsto \color{blue}{-t1 \cdot \frac{\frac{v}{t1 + u}}{t1 + u}} \]

   3. associate-/r* 69.7%

      \[\leadsto -t1 \cdot \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   4. distribute-lft-neg-out 69.7%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   5. associate-/l* 70.5%

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   6. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   7. frac-2neg 98.4%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

   8. associate-*r/ 98.4%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)}} \]

   9. add-sqr-sqrt 41.2%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

   10. sqrt-unprod 38.0%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

   11. sqr-neg 38.0%

       \[\leadsto \frac{\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

   12. sqrt-unprod 16.9%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

   13. add-sqr-sqrt 32.7%

       \[\leadsto \frac{\frac{\color{blue}{t1}}{t1 + u} \cdot \left(-v\right)}{-\left(t1 + u\right)} \]

   14. add-sqr-sqrt 16.5%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}}} \]

   15. sqrt-unprod 57.5%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

   16. sqr-neg 57.5%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}} \]

   17. sqrt-prod 53.8%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{t1 + u} \cdot \sqrt{t1 + u}}} \]

   18. add-sqr-sqrt 98.4%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{t1 + u}} \]

6. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{t1 + u}} \]

7. Taylor expanded in t1 around inf 61.8%

   \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
8. Step-by-step derivation

   1. mul-1-neg 61.8%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

9. Simplified 61.8%

   \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

10. Final simplification 61.8%

    \[\leadsto \frac{v}{\left(-u\right) - t1} \]
11. Add Preprocessing

Alternative 19: 15.2% 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 70.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 98.4%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 98.4%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 98.4%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 98.4%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 98.4%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Taylor expanded in t1 around inf 64.6%

   \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
6. Step-by-step derivation

   1. remove-double-neg 64.6%

      \[\leadsto \frac{\color{blue}{-\left(-t1\right)}}{\left(-u\right) - t1} \cdot \frac{v}{t1} \]

   2. sub-neg 64.6%

      \[\leadsto \frac{-\left(-t1\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1} \]

   3. distribute-neg-in 64.6%

      \[\leadsto \frac{-\left(-t1\right)}{\color{blue}{-\left(u + t1\right)}} \cdot \frac{v}{t1} \]

   4. +-commutative 64.6%

      \[\leadsto \frac{-\left(-t1\right)}{-\color{blue}{\left(t1 + u\right)}} \cdot \frac{v}{t1} \]

   5. frac-2neg 64.6%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u}} \cdot \frac{v}{t1} \]

   6. div-inv 64.5%

      \[\leadsto \color{blue}{\left(\left(-t1\right) \cdot \frac{1}{t1 + u}\right)} \cdot \frac{v}{t1} \]

   7. add-sqr-sqrt 25.3%

      \[\leadsto \left(\color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

   8. sqrt-unprod 25.3%

      \[\leadsto \left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

   9. sqr-neg 25.3%

      \[\leadsto \left(\sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

   10. sqrt-unprod 12.1%

       \[\leadsto \left(\color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

   11. add-sqr-sqrt 22.6%

       \[\leadsto \left(\color{blue}{t1} \cdot \frac{1}{t1 + u}\right) \cdot \frac{v}{t1} \]

7. Applied egg-rr 22.6%

   \[\leadsto \color{blue}{\left(t1 \cdot \frac{1}{t1 + u}\right)} \cdot \frac{v}{t1} \]
8. Step-by-step derivation

   1. associate-*r/ 22.6%

      \[\leadsto \color{blue}{\frac{t1 \cdot 1}{t1 + u}} \cdot \frac{v}{t1} \]

   2. *-rgt-identity 22.6%

      \[\leadsto \frac{\color{blue}{t1}}{t1 + u} \cdot \frac{v}{t1} \]

9. Simplified 22.6%

   \[\leadsto \color{blue}{\frac{t1}{t1 + u}} \cdot \frac{v}{t1} \]

10. Taylor expanded in t1 around inf 11.3%

    \[\leadsto \color{blue}{\frac{v}{t1}} \]

11. Final simplification 11.3%

    \[\leadsto \frac{v}{t1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))