Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.3%

Time: 10.8s

Alternatives: 17

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: 72.8% 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: 98.3% 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 75.7%

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

   1. associate-/l* 76.7%

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

   2. distribute-lft-neg-out 76.7%

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

   3. distribute-rgt-neg-in 76.7%

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

   4. associate-/r* 85.7%

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

   5. distribute-neg-frac2 85.7%

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

3. Simplified 85.7%

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

   1. associate-*r/ 97.3%

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

   2. +-commutative 97.3%

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

   3. distribute-neg-in 97.3%

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

   4. sub-neg 97.3%

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

   5. associate-*l/ 97.6%

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

   6. frac-2neg 97.6%

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

   7. associate-*r/ 98.6%

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

   8. remove-double-neg 98.6%

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

   9. sub-neg 98.6%

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

   10. distribute-neg-in 98.6%

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

   11. +-commutative 98.6%

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

   12. frac-2neg 98.6%

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

   13. add-sqr-sqrt 48.8%

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

   14. sqrt-unprod 44.0%

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

   15. sqr-neg 44.0%

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

   16. sqrt-unprod 17.5%

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

   17. add-sqr-sqrt 34.4%

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

   18. add-sqr-sqrt 16.6%

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

   19. sqrt-unprod 55.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 2: 90.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ t_2 := \frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{if}\;t1 \leq -5.4 \cdot 10^{+137}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -2.7 \cdot 10^{-195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{-164}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;t1 \leq 1.3 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (/ v (+ t1 u)) (- (- u) t1))))
        (t_2 (/ v (- (* u (- 2.0)) t1))))
   (if (<= t1 -5.4e+137)
     t_2
     (if (<= t1 -2.7e-195)
       t_1
       (if (<= t1 1.45e-164)
         (/ v (* (/ u t1) (- t1 u)))
         (if (<= t1 1.3e+83) 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 / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -5.4e+137) {
		tmp = t_2;
	} else if (t1 <= -2.7e-195) {
		tmp = t_1;
	} else if (t1 <= 1.45e-164) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (t1 <= 1.3e+83) {
		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 / ((u * -2.0d0) - t1)
    if (t1 <= (-5.4d+137)) then
        tmp = t_2
    else if (t1 <= (-2.7d-195)) then
        tmp = t_1
    else if (t1 <= 1.45d-164) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (t1 <= 1.3d+83) 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 / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -5.4e+137) {
		tmp = t_2;
	} else if (t1 <= -2.7e-195) {
		tmp = t_1;
	} else if (t1 <= 1.45e-164) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (t1 <= 1.3e+83) {
		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 / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -5.4e+137:
		tmp = t_2
	elif t1 <= -2.7e-195:
		tmp = t_1
	elif t1 <= 1.45e-164:
		tmp = v / ((u / t1) * (t1 - u))
	elif t1 <= 1.3e+83:
		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(v / Float64(Float64(u * Float64(-2.0)) - t1))
	tmp = 0.0
	if (t1 <= -5.4e+137)
		tmp = t_2;
	elseif (t1 <= -2.7e-195)
		tmp = t_1;
	elseif (t1 <= 1.45e-164)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (t1 <= 1.3e+83)
		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 / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -5.4e+137)
		tmp = t_2;
	elseif (t1 <= -2.7e-195)
		tmp = t_1;
	elseif (t1 <= 1.45e-164)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (t1 <= 1.3e+83)
		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[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.4e+137], t$95$2, If[LessEqual[t1, -2.7e-195], t$95$1, If[LessEqual[t1, 1.45e-164], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.3e+83], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.40000000000000034e137 or 1.3000000000000001e83 < t1

   1. Initial program 45.9%

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

      1. associate-/l* 47.4%

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

      2. distribute-lft-neg-out 47.4%

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

      3. distribute-rgt-neg-in 47.4%

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

      4. associate-/r* 69.6%

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

      5. distribute-neg-frac2 69.6%

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

   3. Simplified 69.6%

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

      1. associate-*r/ 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. frac-2neg 99.9%

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

      7. clear-num 100.0%

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

      8. frac-times 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. sub-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around 0 96.8%

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

      1. *-commutative 96.8%

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

   9. Simplified 96.8%

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

   if -5.40000000000000034e137 < t1 < -2.7e-195 or 1.45e-164 < t1 < 1.3000000000000001e83

   1. Initial program 90.6%

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

      1. associate-/l* 91.4%

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

      2. distribute-lft-neg-out 91.4%

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

      3. distribute-rgt-neg-in 91.4%

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

      4. associate-/r* 95.1%

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

      5. distribute-neg-frac2 95.1%

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

   3. Simplified 95.1%

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

   if -2.7e-195 < t1 < 1.45e-164

   1. Initial program 78.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 83.6%

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

      5. distribute-neg-frac2 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in t1 around 0 77.6%

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

      1. associate-/l/ 73.3%

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

      2. associate-*r/ 72.1%

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

      3. *-commutative 72.1%

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

      4. add-sqr-sqrt 40.2%

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

      5. sqrt-unprod 58.2%

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

      6. sqr-neg 58.2%

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

      7. sqrt-unprod 18.0%

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

      8. add-sqr-sqrt 48.9%

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

   7. Applied egg-rr 48.9%

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

      1. associate-/l* 48.9%

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

   9. Simplified 48.9%

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

       1. add-sqr-sqrt 23.5%

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

       2. sqrt-unprod 43.3%

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

       3. sqr-neg 43.3%

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

       4. sqrt-unprod 29.1%

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

       5. add-sqr-sqrt 73.3%

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

       6. associate-/l* 72.1%

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

       7. frac-times 81.5%

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

       8. clear-num 80.7%

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

       9. frac-2neg 80.7%

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

       10. frac-times 86.3%

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

       11. *-un-lft-identity 86.3%

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

       12. add-sqr-sqrt 48.0%

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

       13. sqrt-unprod 56.7%

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

       14. sqr-neg 56.7%

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

       15. sqrt-unprod 26.8%

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

       16. add-sqr-sqrt 49.1%

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

       17. add-sqr-sqrt 25.4%

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

       18. sqrt-unprod 43.9%

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

       19. sqr-neg 43.9%

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

       20. sqrt-unprod 49.7%

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

       21. add-sqr-sqrt 86.3%

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

       22. distribute-neg-in 86.3%

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

   11. Applied egg-rr 92.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;t1 \leq -2.7 \cdot 10^{-195}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{-164}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;t1 \leq 1.3 \cdot 10^{+83}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := \frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{if}\;t1 \leq -2.2 \cdot 10^{+124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -4.8 \cdot 10^{-195}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{t\_1}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{+133}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (/ v (- (* u (- 2.0)) t1))))
   (if (<= t1 -2.2e+124)
     t_2
     (if (<= t1 -4.8e-195)
       (* t1 (/ (/ v (+ t1 u)) t_1))
       (if (<= t1 2.5e-163)
         (/ v (* (/ u t1) (- t1 u)))
         (if (<= t1 1.5e+133) (* 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 / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.2e+124) {
		tmp = t_2;
	} else if (t1 <= -4.8e-195) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else if (t1 <= 2.5e-163) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (t1 <= 1.5e+133) {
		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 / ((u * -2.0d0) - t1)
    if (t1 <= (-2.2d+124)) then
        tmp = t_2
    else if (t1 <= (-4.8d-195)) then
        tmp = t1 * ((v / (t1 + u)) / t_1)
    else if (t1 <= 2.5d-163) then
        tmp = v / ((u / t1) * (t1 - u))
    else if (t1 <= 1.5d+133) 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 / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2.2e+124) {
		tmp = t_2;
	} else if (t1 <= -4.8e-195) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else if (t1 <= 2.5e-163) {
		tmp = v / ((u / t1) * (t1 - u));
	} else if (t1 <= 1.5e+133) {
		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 / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -2.2e+124:
		tmp = t_2
	elif t1 <= -4.8e-195:
		tmp = t1 * ((v / (t1 + u)) / t_1)
	elif t1 <= 2.5e-163:
		tmp = v / ((u / t1) * (t1 - u))
	elif t1 <= 1.5e+133:
		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(v / Float64(Float64(u * Float64(-2.0)) - t1))
	tmp = 0.0
	if (t1 <= -2.2e+124)
		tmp = t_2;
	elseif (t1 <= -4.8e-195)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / t_1));
	elseif (t1 <= 2.5e-163)
		tmp = Float64(v / Float64(Float64(u / t1) * Float64(t1 - u)));
	elseif (t1 <= 1.5e+133)
		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 / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -2.2e+124)
		tmp = t_2;
	elseif (t1 <= -4.8e-195)
		tmp = t1 * ((v / (t1 + u)) / t_1);
	elseif (t1 <= 2.5e-163)
		tmp = v / ((u / t1) * (t1 - u));
	elseif (t1 <= 1.5e+133)
		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[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.2e+124], t$95$2, If[LessEqual[t1, -4.8e-195], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.5e-163], N[(v / N[(N[(u / t1), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.5e+133], 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 < -2.2000000000000001e124 or 1.50000000000000003e133 < t1

   1. Initial program 43.0%

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

      1. associate-/l* 43.1%

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

      2. distribute-lft-neg-out 43.1%

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

      3. distribute-rgt-neg-in 43.1%

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

      4. associate-/r* 67.6%

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

      5. distribute-neg-frac2 67.6%

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

   3. Simplified 67.6%

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

      1. associate-*r/ 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/ 100.0%

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

      6. frac-2neg 100.0%

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

      7. clear-num 100.0%

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

      8. frac-times 100.0%

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

      9. *-un-lft-identity 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-neg-in 100.0%

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

      12. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 97.9%

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

      1. *-commutative 97.9%

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

   9. Simplified 97.9%

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

   if -2.2000000000000001e124 < t1 < -4.8e-195

   1. Initial program 89.3%

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

      1. associate-/l* 89.6%

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

      2. distribute-lft-neg-out 89.6%

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

      3. distribute-rgt-neg-in 89.6%

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

      4. associate-/r* 95.0%

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

      5. distribute-neg-frac2 95.0%

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

   3. Simplified 95.0%

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

   if -4.8e-195 < t1 < 2.49999999999999989e-163

   1. Initial program 78.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 83.6%

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

      5. distribute-neg-frac2 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in t1 around 0 77.6%

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

      1. associate-/l/ 73.3%

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

      2. associate-*r/ 72.1%

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

      3. *-commutative 72.1%

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

      4. add-sqr-sqrt 40.2%

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

      5. sqrt-unprod 58.2%

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

      6. sqr-neg 58.2%

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

      7. sqrt-unprod 18.0%

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

      8. add-sqr-sqrt 48.9%

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

   7. Applied egg-rr 48.9%

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

      1. associate-/l* 48.9%

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

   9. Simplified 48.9%

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

       1. add-sqr-sqrt 23.5%

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

       2. sqrt-unprod 43.3%

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

       3. sqr-neg 43.3%

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

       4. sqrt-unprod 29.1%

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

       5. add-sqr-sqrt 73.3%

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

       6. associate-/l* 72.1%

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

       7. frac-times 81.5%

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

       8. clear-num 80.7%

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

       9. frac-2neg 80.7%

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

       10. frac-times 86.3%

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

       11. *-un-lft-identity 86.3%

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

       12. add-sqr-sqrt 48.0%

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

       13. sqrt-unprod 56.7%

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

       14. sqr-neg 56.7%

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

       15. sqrt-unprod 26.8%

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

       16. add-sqr-sqrt 49.1%

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

       17. add-sqr-sqrt 25.4%

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

       18. sqrt-unprod 43.9%

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

       19. sqr-neg 43.9%

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

       20. sqrt-unprod 49.7%

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

       21. add-sqr-sqrt 86.3%

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

       22. distribute-neg-in 86.3%

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

   11. Applied egg-rr 92.3%

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

   if 2.49999999999999989e-163 < t1 < 1.50000000000000003e133

   1. Initial program 90.2%

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

      1. associate-*l/ 96.5%

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

      2. *-commutative 96.5%

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

   3. Simplified 96.5%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;t1 \leq -4.8 \cdot 10^{-195}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{+133}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{-17} \lor \neg \left(u \leq 1.15 \cdot 10^{-54}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e-17) (not (<= u 1.15e-54)))
   (* t1 (/ (/ v u) (- (- u) t1)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e-17) || !(u <= 1.15e-54)) {
		tmp = t1 * ((v / u) / (-u - t1));
	} 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 <= (-3.4d-17)) .or. (.not. (u <= 1.15d-54))) then
        tmp = t1 * ((v / u) / (-u - t1))
    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 <= -3.4e-17) || !(u <= 1.15e-54)) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e-17) or not (u <= 1.15e-54):
		tmp = t1 * ((v / u) / (-u - t1))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e-17) || !(u <= 1.15e-54))
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.4e-17) || ~((u <= 1.15e-54)))
		tmp = t1 * ((v / u) / (-u - t1));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e-17], N[Not[LessEqual[u, 1.15e-54]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.3999999999999998e-17 or 1.1499999999999999e-54 < u

   1. Initial program 82.6%

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

      1. associate-/l* 83.0%

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

      2. distribute-lft-neg-out 83.0%

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

      3. distribute-rgt-neg-in 83.0%

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

      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

   5. Taylor expanded in t1 around 0 80.7%

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

   if -3.3999999999999998e-17 < u < 1.1499999999999999e-54

   1. Initial program 66.6%

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

      1. associate-/l* 68.5%

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

      2. distribute-lft-neg-out 68.5%

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

      3. distribute-rgt-neg-in 68.5%

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

      4. associate-/r* 77.4%

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

      5. distribute-neg-frac2 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in t1 around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. neg-mul-1 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{-17} \lor \neg \left(u \leq 1.15 \cdot 10^{-54}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.62 \cdot 10^{-18}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{-u}}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.62e-18)
   (* t1 (/ (/ v u) (- (- u) t1)))
   (if (<= u 4.8e-67) (/ v (- t1)) (/ (/ t1 (- u)) (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.62e-18) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (u <= 4.8e-67) {
		tmp = v / -t1;
	} else {
		tmp = (t1 / -u) / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.62d-18)) then
        tmp = t1 * ((v / u) / (-u - t1))
    else if (u <= 4.8d-67) then
        tmp = v / -t1
    else
        tmp = (t1 / -u) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.62e-18) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else if (u <= 4.8e-67) {
		tmp = v / -t1;
	} else {
		tmp = (t1 / -u) / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.62e-18:
		tmp = t1 * ((v / u) / (-u - t1))
	elif u <= 4.8e-67:
		tmp = v / -t1
	else:
		tmp = (t1 / -u) / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.62e-18)
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(Float64(-u) - t1)));
	elseif (u <= 4.8e-67)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(t1 / Float64(-u)) / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.62e-18)
		tmp = t1 * ((v / u) / (-u - t1));
	elseif (u <= 4.8e-67)
		tmp = v / -t1;
	else
		tmp = (t1 / -u) / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.62e-18], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.8e-67], N[(v / (-t1)), $MachinePrecision], N[(N[(t1 / (-u)), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.62000000000000005e-18

   1. Initial program 79.7%

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

      1. associate-/l* 81.4%

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

      2. distribute-lft-neg-out 81.4%

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

      3. distribute-rgt-neg-in 81.4%

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

      4. associate-/r* 92.3%

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

      5. distribute-neg-frac2 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in t1 around 0 77.8%

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

   if -1.62000000000000005e-18 < u < 4.8e-67

   1. Initial program 65.6%

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

      1. associate-/l* 67.7%

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

      2. distribute-lft-neg-out 67.7%

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

      3. distribute-rgt-neg-in 67.7%

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

      4. associate-/r* 76.8%

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

      5. distribute-neg-frac2 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   7. Simplified 85.4%

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

   if 4.8e-67 < u

   1. Initial program 86.2%

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

      1. associate-/l* 85.3%

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

      2. distribute-lft-neg-out 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

      4. associate-/r* 92.1%

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

      5. distribute-neg-frac2 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in t1 around 0 81.6%

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

      1. associate-/l/ 79.0%

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

      2. associate-*r/ 78.1%

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

      3. *-commutative 78.1%

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

      4. add-sqr-sqrt 0.2%

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

      5. sqrt-unprod 55.5%

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

      6. sqr-neg 55.5%

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

      7. sqrt-unprod 52.1%

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

      8. add-sqr-sqrt 52.7%

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

   7. Applied egg-rr 52.7%

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

      1. associate-/l* 54.9%

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

   9. Simplified 54.9%

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

       1. add-sqr-sqrt 28.8%

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

       2. sqrt-unprod 47.5%

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

       3. sqr-neg 47.5%

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

       4. sqrt-unprod 35.5%

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

       5. add-sqr-sqrt 79.0%

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

       6. associate-/l* 78.1%

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

       7. frac-times 82.1%

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

       8. frac-2neg 82.1%

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

       9. clear-num 82.7%

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

       10. frac-times 82.9%

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

       11. remove-double-neg 82.9%

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

       12. *-commutative 82.9%

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

       13. *-un-lft-identity 82.9%

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

   11. Applied egg-rr 82.9%

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

       1. associate-/r* 82.8%

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

   13. Simplified 82.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.62 \cdot 10^{-18}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{-u}}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{-36} \lor \neg \left(u \leq 1.15 \cdot 10^{-54}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(t1 - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.02e-36) (not (<= u 1.15e-54)))
   (* t1 (/ v (* u (- t1 u))))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.02e-36) || !(u <= 1.15e-54)) {
		tmp = t1 * (v / (u * (t1 - u)));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.02d-36)) .or. (.not. (u <= 1.15d-54))) then
        tmp = t1 * (v / (u * (t1 - u)))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.02e-36) || !(u <= 1.15e-54)) {
		tmp = t1 * (v / (u * (t1 - u)));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.02e-36) or not (u <= 1.15e-54):
		tmp = t1 * (v / (u * (t1 - u)))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.02e-36) || !(u <= 1.15e-54))
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(t1 - u))));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.02e-36) || ~((u <= 1.15e-54)))
		tmp = t1 * (v / (u * (t1 - u)));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.02e-36], N[Not[LessEqual[u, 1.15e-54]], $MachinePrecision]], N[(t1 * N[(v / N[(u * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.02e-36 or 1.1499999999999999e-54 < u

   1. Initial program 82.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 92.1%

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

      5. distribute-neg-frac2 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in t1 around 0 80.3%

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

      1. associate-/l/ 77.5%

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

      2. associate-*r/ 76.3%

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

      3. *-commutative 76.3%

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

      4. add-sqr-sqrt 37.2%

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

      5. sqrt-unprod 65.1%

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

      6. sqr-neg 65.1%

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

      7. sqrt-unprod 26.8%

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

      8. add-sqr-sqrt 50.6%

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

   7. Applied egg-rr 50.6%

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

      1. associate-/l* 52.1%

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

   9. Simplified 52.1%

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

       1. frac-2neg 52.1%

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

       2. div-inv 52.1%

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

       3. add-sqr-sqrt 25.9%

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

       4. sqrt-unprod 51.6%

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

       5. sqr-neg 51.6%

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

       6. sqrt-unprod 38.8%

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

       7. add-sqr-sqrt 77.4%

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

       8. distribute-rgt-neg-in 77.4%

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

       9. distribute-neg-in 77.4%

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

       10. add-sqr-sqrt 36.0%

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

       11. sqrt-unprod 77.6%

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

       12. sqr-neg 77.6%

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

       13. sqrt-unprod 41.5%

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

       14. add-sqr-sqrt 77.6%

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

       15. sub-neg 77.6%

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

   11. Applied egg-rr 77.6%

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

       1. associate-*r/ 77.6%

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

       2. *-rgt-identity 77.6%

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

   13. Simplified 77.6%

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

   if -1.02e-36 < u < 1.1499999999999999e-54

   1. Initial program 66.0%

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

      1. associate-/l* 68.0%

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

      2. distribute-lft-neg-out 68.0%

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

      3. distribute-rgt-neg-in 68.0%

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

      4. associate-/r* 77.0%

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

      5. distribute-neg-frac2 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{-36} \lor \neg \left(u \leq 1.15 \cdot 10^{-54}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(t1 - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.06 \cdot 10^{-36} \lor \neg \left(u \leq 1.7 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 - u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.06e-36) (not (<= u 1.7e-67)))
   (/ (* t1 (/ v (- t1 u))) u)
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.06e-36) || !(u <= 1.7e-67)) {
		tmp = (t1 * (v / (t1 - u))) / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.06d-36)) .or. (.not. (u <= 1.7d-67))) then
        tmp = (t1 * (v / (t1 - u))) / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.06e-36) || !(u <= 1.7e-67)) {
		tmp = (t1 * (v / (t1 - u))) / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.06e-36) or not (u <= 1.7e-67):
		tmp = (t1 * (v / (t1 - u))) / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.06e-36) || !(u <= 1.7e-67))
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 - u))) / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.06e-36) || ~((u <= 1.7e-67)))
		tmp = (t1 * (v / (t1 - u))) / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.06e-36], N[Not[LessEqual[u, 1.7e-67]], $MachinePrecision]], N[(N[(t1 * N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.05999999999999999e-36 or 1.70000000000000005e-67 < u

   1. Initial program 83.2%

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

      1. associate-/l* 83.6%

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

      2. distribute-lft-neg-out 83.6%

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

      3. distribute-rgt-neg-in 83.6%

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

      4. associate-/r* 92.3%

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

      5. distribute-neg-frac2 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in t1 around 0 79.3%

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

      1. associate-/l/ 76.6%

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

      2. associate-*r/ 75.5%

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

      3. *-commutative 75.5%

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

      4. add-sqr-sqrt 36.5%

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

      5. sqrt-unprod 64.4%

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

      6. sqr-neg 64.4%

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

      7. sqrt-unprod 26.3%

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

      8. add-sqr-sqrt 49.7%

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

   7. Applied egg-rr 49.7%

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

      1. associate-/l* 51.2%

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

   9. Simplified 51.2%

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

       1. add-sqr-sqrt 26.8%

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

       2. sqrt-unprod 48.6%

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

       3. sqr-neg 48.6%

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

       4. sqrt-unprod 35.2%

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

       5. add-sqr-sqrt 76.6%

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

       6. associate-/l* 75.5%

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

       7. frac-times 78.6%

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

       8. associate-*l/ 79.5%

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

   11. Applied egg-rr 79.6%

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

   if -1.05999999999999999e-36 < u < 1.70000000000000005e-67

   1. Initial program 65.0%

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

      1. associate-/l* 67.1%

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

      2. distribute-lft-neg-out 67.1%

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

      3. distribute-rgt-neg-in 67.1%

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

      4. associate-/r* 76.3%

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

      5. distribute-neg-frac2 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in t1 around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. neg-mul-1 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.06 \cdot 10^{-36} \lor \neg \left(u \leq 1.7 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 - u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-18} \lor \neg \left(t1 \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.6e-18) (not (<= t1 5.4e-9)))
   (/ v (- (- u) t1))
   (* (/ v u) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-18) || !(t1 <= 5.4e-9)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (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 ((t1 <= (-1.6d-18)) .or. (.not. (t1 <= 5.4d-9))) then
        tmp = v / (-u - t1)
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-18) || !(t1 <= 5.4e-9)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.6e-18) or not (t1 <= 5.4e-9):
		tmp = v / (-u - t1)
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.6e-18) || !(t1 <= 5.4e-9))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.6e-18) || ~((t1 <= 5.4e-9)))
		tmp = v / (-u - t1);
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.6e-18], N[Not[LessEqual[t1, 5.4e-9]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6e-18 or 5.4000000000000004e-9 < t1

   1. Initial program 66.3%

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

      1. associate-/l* 65.7%

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

      2. distribute-lft-neg-out 65.7%

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

      3. distribute-rgt-neg-in 65.7%

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

      4. associate-/r* 80.2%

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

      5. distribute-neg-frac2 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. remove-double-neg 99.9%

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

      9. sub-neg 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. +-commutative 99.9%

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

      12. frac-2neg 99.9%

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

      13. add-sqr-sqrt 52.5%

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

      14. sqrt-unprod 34.6%

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

      15. sqr-neg 34.6%

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

      16. sqrt-unprod 15.6%

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

      17. add-sqr-sqrt 31.9%

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

      18. add-sqr-sqrt 14.5%

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

      19. sqrt-unprod 51.2%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 86.5%

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

      1. mul-1-neg 86.5%

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

   9. Simplified 86.5%

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

   if -1.6e-18 < t1 < 5.4000000000000004e-9

   1. Initial program 84.3%

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

      1. times-frac 95.4%

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

      2. distribute-frac-neg 95.4%

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

      3. distribute-neg-frac2 95.4%

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

      4. +-commutative 95.4%

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

      5. distribute-neg-in 95.4%

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

      6. unsub-neg 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in t1 around 0 72.1%

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

      1. associate-*r/ 72.1%

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

      2. mul-1-neg 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in t1 around 0 75.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-18} \lor \neg \left(t1 \leq 5.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{-15} \lor \neg \left(t1 \leq 2.8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3e-15) (not (<= t1 2.8e-11)))
   (/ v (- (* u (- 2.0)) t1))
   (* (/ v u) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e-15) || !(t1 <= 2.8e-11)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (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 ((t1 <= (-3d-15)) .or. (.not. (t1 <= 2.8d-11))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e-15) || !(t1 <= 2.8e-11)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3e-15) or not (t1 <= 2.8e-11):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3e-15) || !(t1 <= 2.8e-11))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3e-15) || ~((t1 <= 2.8e-11)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3e-15], N[Not[LessEqual[t1, 2.8e-11]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3e-15 or 2.8e-11 < t1

   1. Initial program 66.3%

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

      1. associate-/l* 65.7%

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

      2. distribute-lft-neg-out 65.7%

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

      3. distribute-rgt-neg-in 65.7%

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

      4. associate-/r* 80.2%

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

      5. distribute-neg-frac2 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. clear-num 99.9%

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

      8. frac-times 98.3%

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

      9. *-un-lft-identity 98.3%

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

      10. +-commutative 98.3%

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

      11. distribute-neg-in 98.3%

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

      12. sub-neg 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in u around 0 86.7%

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

      1. *-commutative 86.7%

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

   9. Simplified 86.7%

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

   if -3e-15 < t1 < 2.8e-11

   1. Initial program 84.3%

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

      1. times-frac 95.4%

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

      2. distribute-frac-neg 95.4%

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

      3. distribute-neg-frac2 95.4%

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

      4. +-commutative 95.4%

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

      5. distribute-neg-in 95.4%

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

      6. unsub-neg 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in t1 around 0 72.1%

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

      1. associate-*r/ 72.1%

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

      2. mul-1-neg 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in t1 around 0 75.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{-15} \lor \neg \left(t1 \leq 2.8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+177} \lor \neg \left(u \leq 8.5 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.2e+177) (not (<= u 8.5e+224)))
   (* (/ v u) -0.5)
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.2e+177) || !(u <= 8.5e+224)) {
		tmp = (v / u) * -0.5;
	} 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 <= (-5.2d+177)) .or. (.not. (u <= 8.5d+224))) then
        tmp = (v / u) * (-0.5d0)
    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 <= -5.2e+177) || !(u <= 8.5e+224)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.2e+177) or not (u <= 8.5e+224):
		tmp = (v / u) * -0.5
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.2e+177) || !(u <= 8.5e+224))
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.2e+177) || ~((u <= 8.5e+224)))
		tmp = (v / u) * -0.5;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.2e+177], N[Not[LessEqual[u, 8.5e+224]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.19999999999999959e177 or 8.50000000000000046e224 < u

   1. Initial program 91.2%

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

      1. associate-/l* 91.3%

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

      2. distribute-lft-neg-out 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

      4. associate-/r* 100.0%

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

      5. distribute-neg-frac2 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 98.6%

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

      6. frac-2neg 98.6%

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

      7. clear-num 97.5%

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

      8. frac-times 95.5%

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

      9. *-un-lft-identity 95.5%

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

      10. +-commutative 95.5%

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

      11. distribute-neg-in 95.5%

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

      12. sub-neg 95.5%

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

   6. Applied egg-rr 95.5%

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

   7. Taylor expanded in u around 0 46.1%

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

      1. *-commutative 46.1%

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

   9. Simplified 46.1%

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

   10. Taylor expanded in t1 around 0 42.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]

   if -5.19999999999999959e177 < u < 8.50000000000000046e224

   1. Initial program 72.3%

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

      1. associate-/l* 73.5%

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

      2. distribute-lft-neg-out 73.5%

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

      3. distribute-rgt-neg-in 73.5%

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

      4. associate-/r* 82.6%

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

      5. distribute-neg-frac2 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in t1 around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   7. Simplified 62.1%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+177} \lor \neg \left(u \leq 8.5 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.05 \cdot 10^{+169} \lor \neg \left(u \leq 5 \cdot 10^{+176}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.05e+169) (not (<= u 5e+176))) (/ 1.0 (/ u v)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.05e+169) || !(u <= 5e+176)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.05d+169)) .or. (.not. (u <= 5d+176))) then
        tmp = 1.0d0 / (u / v)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.05e+169) || !(u <= 5e+176)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.05e+169) or not (u <= 5e+176):
		tmp = 1.0 / (u / v)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.05e+169) || !(u <= 5e+176))
		tmp = Float64(1.0 / Float64(u / v));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.05e+169) || ~((u <= 5e+176)))
		tmp = 1.0 / (u / v);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.05e+169], N[Not[LessEqual[u, 5e+176]], $MachinePrecision]], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.0500000000000002e169 or 5e176 < u

   1. Initial program 90.8%

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

      1. associate-/l* 91.0%

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

      2. distribute-lft-neg-out 91.0%

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

      3. distribute-rgt-neg-in 91.0%

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

      4. associate-/r* 100.0%

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

      5. distribute-neg-frac2 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around 0 96.7%

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

   6. Taylor expanded in u around 0 45.1%

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

      1. associate-*r/ 45.1%

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

      2. mul-1-neg 45.1%

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

   8. Simplified 45.1%

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

      1. clear-num 45.1%

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

      2. un-div-inv 45.1%

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

      3. associate-/l* 44.8%

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

      4. add-sqr-sqrt 19.9%

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

      5. sqrt-unprod 44.2%

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

      6. sqr-neg 44.2%

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

      7. sqrt-unprod 24.7%

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

      8. add-sqr-sqrt 44.5%

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

   10. Applied egg-rr 44.5%

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

       1. associate-/r* 42.9%

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

       2. *-inverses 42.9%

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

   12. Simplified 42.9%

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

   if -2.0500000000000002e169 < u < 5e176

   1. Initial program 71.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 81.8%

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

      5. distribute-neg-frac2 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in t1 around inf 63.4%

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

      1. associate-*r/ 63.4%

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

      2. neg-mul-1 63.4%

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

   7. Simplified 63.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.05 \cdot 10^{+169} \lor \neg \left(u \leq 5 \cdot 10^{+176}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+169}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+224}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6e+169)
   (* (/ v u) -0.5)
   (if (<= u 9e+224) (/ v (- t1)) (* v (/ -0.5 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e+169) {
		tmp = (v / u) * -0.5;
	} else if (u <= 9e+224) {
		tmp = v / -t1;
	} else {
		tmp = v * (-0.5 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-6d+169)) then
        tmp = (v / u) * (-0.5d0)
    else if (u <= 9d+224) then
        tmp = v / -t1
    else
        tmp = v * ((-0.5d0) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e+169) {
		tmp = (v / u) * -0.5;
	} else if (u <= 9e+224) {
		tmp = v / -t1;
	} else {
		tmp = v * (-0.5 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6e+169:
		tmp = (v / u) * -0.5
	elif u <= 9e+224:
		tmp = v / -t1
	else:
		tmp = v * (-0.5 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6e+169)
		tmp = Float64(Float64(v / u) * -0.5);
	elseif (u <= 9e+224)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v * Float64(-0.5 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6e+169)
		tmp = (v / u) * -0.5;
	elseif (u <= 9e+224)
		tmp = v / -t1;
	else
		tmp = v * (-0.5 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6e+169], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[u, 9e+224], N[(v / (-t1)), $MachinePrecision], N[(v * N[(-0.5 / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.9999999999999999e169

   1. Initial program 91.9%

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

      1. associate-/l* 92.0%

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

      2. distribute-lft-neg-out 92.0%

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

      3. distribute-rgt-neg-in 92.0%

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

      4. associate-/r* 100.0%

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

      5. distribute-neg-frac2 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 97.6%

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

      6. frac-2neg 97.6%

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

      7. clear-num 95.7%

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

      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. +-commutative 95.7%

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

      11. distribute-neg-in 95.7%

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

      12. sub-neg 95.7%

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

   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 39.3%

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

      1. *-commutative 39.3%

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

   9. Simplified 39.3%

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

   10. Taylor expanded in t1 around 0 35.9%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]

   if -5.9999999999999999e169 < u < 8.9999999999999995e224

   1. Initial program 72.3%

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

      1. associate-/l* 73.5%

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

      2. distribute-lft-neg-out 73.5%

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

      3. distribute-rgt-neg-in 73.5%

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

      4. associate-/r* 82.6%

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

      5. distribute-neg-frac2 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in t1 around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   7. Simplified 62.1%

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

   if 8.9999999999999995e224 < u

   1. Initial program 90.3%

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

      1. associate-/l* 90.5%

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

      2. distribute-lft-neg-out 90.5%

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

      3. distribute-rgt-neg-in 90.5%

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

      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. Step-by-step derivation

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. clear-num 99.9%

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

      8. frac-times 95.2%

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

      9. *-un-lft-identity 95.2%

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

      10. +-commutative 95.2%

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

      11. distribute-neg-in 95.2%

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

      12. sub-neg 95.2%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in u around 0 55.1%

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

      1. *-commutative 55.1%

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

   9. Simplified 55.1%

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

   10. Taylor expanded in t1 around 0 50.6%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   11. Step-by-step derivation

       1. associate-*r/ 50.6%

          \[\leadsto \color{blue}{\frac{-0.5 \cdot v}{u}} \]

       2. *-commutative 50.6%

          \[\leadsto \frac{\color{blue}{v \cdot -0.5}}{u} \]

       3. associate-/l* 50.6%

          \[\leadsto \color{blue}{v \cdot \frac{-0.5}{u}} \]

   12. Simplified 50.6%

       \[\leadsto \color{blue}{v \cdot \frac{-0.5}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+169}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+224}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+170} \lor \neg \left(u \leq 1.05 \cdot 10^{+225}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.2e+170) (not (<= u 1.05e+225))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.2e+170) || !(u <= 1.05e+225)) {
		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)) .or. (.not. (u <= 1.05d+225))) 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) || !(u <= 1.05e+225)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.2e+170) or not (u <= 1.05e+225):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.2e+170) || !(u <= 1.05e+225))
		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) || ~((u <= 1.05e+225)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.2e+170], N[Not[LessEqual[u, 1.05e+225]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.19999999999999996e170 or 1.05e225 < u

   1. Initial program 91.2%

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

      1. associate-/l* 91.3%

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

      2. distribute-lft-neg-out 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

      4. associate-/r* 100.0%

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

      5. distribute-neg-frac2 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around 0 96.1%

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

      1. associate-/l/ 91.3%

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

      2. associate-*r/ 91.2%

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

      3. *-commutative 91.2%

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

      4. add-sqr-sqrt 51.9%

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

      5. sqrt-unprod 91.2%

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

      6. sqr-neg 91.2%

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

      7. sqrt-unprod 39.3%

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

      8. add-sqr-sqrt 91.2%

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

   7. Applied egg-rr 91.2%

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

      1. associate-/l* 91.3%

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

   9. Simplified 91.3%

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

   10. Taylor expanded in t1 around inf 41.8%

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

   if -4.19999999999999996e170 < u < 1.05e225

   1. Initial program 72.3%

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

      1. associate-/l* 73.5%

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

      2. distribute-lft-neg-out 73.5%

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

      3. distribute-rgt-neg-in 73.5%

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

      4. associate-/r* 82.6%

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

      5. distribute-neg-frac2 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in t1 around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   7. Simplified 62.1%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+170} \lor \neg \left(u \leq 1.05 \cdot 10^{+225}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{+169} \lor \neg \left(u \leq 8.5 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.05e+169) (not (<= u 8.5e+224))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.05e+169) || !(u <= 8.5e+224)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.05d+169)) .or. (.not. (u <= 8.5d+224))) then
        tmp = v / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.05e+169) || !(u <= 8.5e+224)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.05e+169) or not (u <= 8.5e+224):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.05e+169) || !(u <= 8.5e+224))
		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 <= -1.05e+169) || ~((u <= 8.5e+224)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.05e+169], N[Not[LessEqual[u, 8.5e+224]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.0500000000000001e169 or 8.50000000000000046e224 < u

   1. Initial program 91.2%

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

      1. associate-/l* 91.3%

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

      2. distribute-lft-neg-out 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

      4. associate-/r* 100.0%

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

      5. distribute-neg-frac2 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around 0 96.1%

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

   6. Taylor expanded in t1 around inf 42.2%

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

      1. associate-*r/ 42.2%

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

      2. mul-1-neg 42.2%

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

   8. Simplified 42.2%

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

   if -1.0500000000000001e169 < u < 8.50000000000000046e224

   1. Initial program 72.3%

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

      1. associate-/l* 73.5%

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

      2. distribute-lft-neg-out 73.5%

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

      3. distribute-rgt-neg-in 73.5%

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

      4. associate-/r* 82.6%

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

      5. distribute-neg-frac2 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in t1 around inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   7. Simplified 62.1%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{+169} \lor \neg \left(u \leq 8.5 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. Final simplification 97.6%

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

Alternative 16: 61.3% 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 75.7%

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

   1. associate-/l* 76.7%

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

   2. distribute-lft-neg-out 76.7%

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

   3. distribute-rgt-neg-in 76.7%

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

   4. associate-/r* 85.7%

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

   5. distribute-neg-frac2 85.7%

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

3. Simplified 85.7%

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

   1. associate-*r/ 97.3%

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

   2. +-commutative 97.3%

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

   3. distribute-neg-in 97.3%

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

   4. sub-neg 97.3%

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

   5. associate-*l/ 97.6%

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

   6. frac-2neg 97.6%

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

   7. associate-*r/ 98.6%

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

   8. remove-double-neg 98.6%

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

   9. sub-neg 98.6%

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

   10. distribute-neg-in 98.6%

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

   11. +-commutative 98.6%

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

   12. frac-2neg 98.6%

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

   13. add-sqr-sqrt 48.8%

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

   14. sqrt-unprod 44.0%

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

   15. sqr-neg 44.0%

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

   16. sqrt-unprod 17.5%

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

   17. add-sqr-sqrt 34.4%

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

   18. add-sqr-sqrt 16.6%

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

   19. sqrt-unprod 55.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in t1 around inf 60.6%

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

   1. mul-1-neg 60.6%

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

9. Simplified 60.6%

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

10. Final simplification 60.6%

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

Alternative 17: 17.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. associate-/l* 76.7%

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

   2. distribute-lft-neg-out 76.7%

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

   3. distribute-rgt-neg-in 76.7%

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

   4. associate-/r* 85.7%

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

   5. distribute-neg-frac2 85.7%

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

3. Simplified 85.7%

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

5. Taylor expanded in t1 around 0 52.7%

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

   1. associate-/l/ 50.3%

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

   2. associate-*r/ 48.6%

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

   3. *-commutative 48.6%

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

   4. add-sqr-sqrt 24.7%

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

   5. sqrt-unprod 43.9%

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

   6. sqr-neg 43.9%

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

   7. sqrt-unprod 15.9%

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

   8. add-sqr-sqrt 31.8%

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

7. Applied egg-rr 31.8%

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

   1. associate-/l* 33.0%

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

9. Simplified 33.0%

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

10. Taylor expanded in t1 around inf 16.6%

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

11. Final simplification 16.6%

    \[\leadsto \frac{v}{u} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))