Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.3%

Time: 11.2s

Alternatives: 16

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 74.4%

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

   1. associate-/l* 74.4%

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

   2. distribute-lft-neg-out 74.4%

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

   3. distribute-rgt-neg-in 74.4%

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

   4. associate-/r* 84.6%

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

   5. distribute-neg-frac2 84.6%

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

3. Simplified 84.6%

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

   1. associate-*r/ 98.0%

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

   2. +-commutative 98.0%

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

   3. distribute-neg-in 98.0%

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

   4. sub-neg 98.0%

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

   5. associate-*l/ 98.8%

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

   6. frac-2neg 98.8%

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

   7. associate-*r/ 99.0%

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

   8. remove-double-neg 99.0%

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

   9. sub-neg 99.0%

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

   10. distribute-neg-in 99.0%

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

   11. +-commutative 99.0%

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

   12. frac-2neg 99.0%

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

   13. add-sqr-sqrt 50.6%

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

   14. sqrt-unprod 42.6%

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

   15. sqr-neg 42.6%

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

   16. sqrt-unprod 17.8%

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

   17. add-sqr-sqrt 38.3%

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

   18. add-sqr-sqrt 21.1%

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

   19. sqrt-unprod 61.2%

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

6. Applied egg-rr 99.0%

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

Alternative 2: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+140} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3e+140) (not (<= t1 4.2e+130)))
   (/ v (- (- t1) (* u 2.0)))
   (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e+140) || !(t1 <= 4.2e+130)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3d+140)) .or. (.not. (t1 <= 4.2d+130))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e+140) || !(t1 <= 4.2e+130)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3e+140) or not (t1 <= 4.2e+130):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3e+140) || !(t1 <= 4.2e+130))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3e+140) || ~((t1 <= 4.2e+130)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3e+140], N[Not[LessEqual[t1, 4.2e+130]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.99999999999999997e140 or 4.19999999999999981e130 < t1

   1. Initial program 47.8%

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

      1. associate-/l* 45.0%

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

      2. distribute-lft-neg-out 45.0%

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

      3. distribute-rgt-neg-in 45.0%

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

      4. associate-/r* 67.9%

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

      5. distribute-neg-frac2 67.9%

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

   3. Simplified 67.9%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-*l/ 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 99.9%

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

      8. frac-times 98.8%

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

      9. *-un-lft-identity 98.8%

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

      10. +-commutative 98.8%

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

      11. distribute-neg-in 98.8%

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

      12. sub-neg 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in u around 0 96.2%

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

      1. *-commutative 96.2%

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

   9. Simplified 96.2%

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

   if -2.99999999999999997e140 < t1 < 4.19999999999999981e130

   1. Initial program 85.6%

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

      1. associate-/l* 86.8%

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

      2. distribute-lft-neg-out 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

      4. associate-/r* 91.6%

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

      5. distribute-neg-frac2 91.6%

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

   3. Simplified 91.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+140} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.3 \cdot 10^{+22} \lor \neg \left(t1 \leq 3.7 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.3e+22) (not (<= t1 3.7e-63)))
   (/ v (- (- t1) (* u 2.0)))
   (/ (* v (/ t1 (- u))) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.3e+22) || !(t1 <= 3.7e-63)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v * (t1 / -u)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.3d+22)) .or. (.not. (t1 <= 3.7d-63))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (v * (t1 / -u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.3e+22) || !(t1 <= 3.7e-63)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v * (t1 / -u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.3e+22) or not (t1 <= 3.7e-63):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (v * (t1 / -u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.3e+22) || !(t1 <= 3.7e-63))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v * Float64(t1 / Float64(-u))) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.3e+22) || ~((t1 <= 3.7e-63)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (v * (t1 / -u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.3e+22], N[Not[LessEqual[t1, 3.7e-63]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.2999999999999998e22 or 3.70000000000000012e-63 < t1

   1. Initial program 64.9%

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

      1. associate-/l* 64.3%

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

      2. distribute-lft-neg-out 64.3%

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

      3. distribute-rgt-neg-in 64.3%

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

      4. associate-/r* 80.6%

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

      5. distribute-neg-frac2 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. clear-num 99.9%

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

      8. frac-times 96.5%

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

      9. *-un-lft-identity 96.5%

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

      10. +-commutative 96.5%

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

      11. distribute-neg-in 96.5%

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

      12. sub-neg 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in u around 0 83.1%

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

      1. *-commutative 83.1%

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

   9. Simplified 83.1%

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

   if -5.2999999999999998e22 < t1 < 3.70000000000000012e-63

   1. Initial program 86.0%

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

      1. associate-/l* 86.8%

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

      2. distribute-lft-neg-out 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

      4. associate-/r* 89.3%

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

      5. distribute-neg-frac2 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in t1 around 0 76.2%

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

   6. Taylor expanded in t1 around 0 79.9%

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

      1. associate-*r/ 81.4%

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

      2. clear-num 81.4%

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

      3. un-div-inv 81.6%

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

      4. add-sqr-sqrt 30.6%

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

      5. sqrt-unprod 54.4%

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

      6. sqr-neg 54.4%

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

      7. sqrt-unprod 23.9%

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

      8. add-sqr-sqrt 43.0%

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

   8. Applied egg-rr 43.0%

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

      1. frac-2neg 43.0%

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

      2. associate-/r/ 43.0%

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

      3. add-sqr-sqrt 26.9%

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

      4. sqrt-unprod 55.1%

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

      5. sqr-neg 55.1%

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

      6. sqrt-unprod 37.4%

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

      7. add-sqr-sqrt 83.7%

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

   10. Applied egg-rr 83.7%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.3 \cdot 10^{+22} \lor \neg \left(t1 \leq 3.7 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+22} \lor \neg \left(t1 \leq 3.45 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.5e+22) (not (<= t1 3.45e-63)))
   (/ v (- (- t1) (* u 2.0)))
   (/ (* t1 (/ v (- u))) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.5e+22) || !(t1 <= 3.45e-63)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (t1 * (v / -u)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.5d+22)) .or. (.not. (t1 <= 3.45d-63))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (t1 * (v / -u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.5e+22) || !(t1 <= 3.45e-63)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (t1 * (v / -u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.5e+22) or not (t1 <= 3.45e-63):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (t1 * (v / -u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.5e+22) || !(t1 <= 3.45e-63))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.5e+22) || ~((t1 <= 3.45e-63)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (t1 * (v / -u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.5e+22], N[Not[LessEqual[t1, 3.45e-63]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.4999999999999998e22 or 3.45e-63 < t1

   1. Initial program 64.9%

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

      1. associate-/l* 64.3%

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

      2. distribute-lft-neg-out 64.3%

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

      3. distribute-rgt-neg-in 64.3%

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

      4. associate-/r* 80.6%

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

      5. distribute-neg-frac2 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. clear-num 99.9%

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

      8. frac-times 96.5%

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

      9. *-un-lft-identity 96.5%

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

      10. +-commutative 96.5%

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

      11. distribute-neg-in 96.5%

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

      12. sub-neg 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in u around 0 83.1%

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

      1. *-commutative 83.1%

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

   9. Simplified 83.1%

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

   if -2.4999999999999998e22 < t1 < 3.45e-63

   1. Initial program 86.0%

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

      1. associate-/l* 86.8%

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

      2. distribute-lft-neg-out 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

      4. associate-/r* 89.3%

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

      5. distribute-neg-frac2 89.3%

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

   3. Simplified 89.3%

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

      1. associate-*r/ 95.6%

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

      2. +-commutative 95.6%

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

      3. distribute-neg-in 95.6%

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

      4. sub-neg 95.6%

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

      5. associate-*l/ 97.5%

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

      6. frac-2neg 97.5%

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

      7. associate-*r/ 98.0%

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

      8. remove-double-neg 98.0%

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

      9. sub-neg 98.0%

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

      10. distribute-neg-in 98.0%

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

      11. +-commutative 98.0%

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

      12. frac-2neg 98.0%

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

      13. add-sqr-sqrt 51.3%

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

      14. sqrt-unprod 62.1%

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

      15. sqr-neg 62.1%

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

      16. sqrt-unprod 23.9%

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

      17. add-sqr-sqrt 43.0%

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

      18. add-sqr-sqrt 19.1%

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

      19. sqrt-unprod 69.6%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in t1 around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. associate-/l* 77.7%

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

      3. distribute-lft-neg-in 77.7%

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

   9. Simplified 77.7%

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

   10. Taylor expanded in t1 around 0 81.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+22} \lor \neg \left(t1 \leq 3.45 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{+22} \lor \neg \left(t1 \leq 3.8 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.85e+22) (not (<= t1 3.8e-63)))
   (/ v (- (- t1) (* u 2.0)))
   (* t1 (/ (/ v u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.85e+22) || !(t1 <= 3.8e-63)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.85d+22)) .or. (.not. (t1 <= 3.8d-63))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t1 * ((v / u) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.85e+22) || !(t1 <= 3.8e-63)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.85e+22) or not (t1 <= 3.8e-63):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t1 * ((v / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.85e+22) || !(t1 <= 3.8e-63))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.85e+22) || ~((t1 <= 3.8e-63)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t1 * ((v / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.85e+22], N[Not[LessEqual[t1, 3.8e-63]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.8499999999999999e22 or 3.80000000000000017e-63 < t1

   1. Initial program 64.9%

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

      1. associate-/l* 64.3%

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

      2. distribute-lft-neg-out 64.3%

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

      3. distribute-rgt-neg-in 64.3%

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

      4. associate-/r* 80.6%

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

      5. distribute-neg-frac2 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. clear-num 99.9%

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

      8. frac-times 96.5%

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

      9. *-un-lft-identity 96.5%

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

      10. +-commutative 96.5%

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

      11. distribute-neg-in 96.5%

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

      12. sub-neg 96.5%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in u around 0 83.1%

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

      1. *-commutative 83.1%

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

   9. Simplified 83.1%

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

   if -1.8499999999999999e22 < t1 < 3.80000000000000017e-63

   1. Initial program 86.0%

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

      1. associate-/l* 86.8%

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

      2. distribute-lft-neg-out 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

      4. associate-/r* 89.3%

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

      5. distribute-neg-frac2 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in t1 around 0 76.2%

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

   6. Taylor expanded in t1 around 0 79.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{+22} \lor \neg \left(t1 \leq 3.8 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{+25} \lor \neg \left(t1 \leq 4.05 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.4e+25) (not (<= t1 4.05e-63)))
   (/ v (- (- u) t1))
   (* t1 (/ (/ v u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.4e+25) || !(t1 <= 4.05e-63)) {
		tmp = v / (-u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.4d+25)) .or. (.not. (t1 <= 4.05d-63))) then
        tmp = v / (-u - t1)
    else
        tmp = t1 * ((v / u) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.4e+25) || !(t1 <= 4.05e-63)) {
		tmp = v / (-u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.4e+25) or not (t1 <= 4.05e-63):
		tmp = v / (-u - t1)
	else:
		tmp = t1 * ((v / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.4e+25) || !(t1 <= 4.05e-63))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.4e+25) || ~((t1 <= 4.05e-63)))
		tmp = v / (-u - t1);
	else
		tmp = t1 * ((v / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.4e+25], N[Not[LessEqual[t1, 4.05e-63]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.4e25 or 4.04999999999999975e-63 < t1

   1. Initial program 64.9%

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

      1. associate-/l* 64.3%

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

      2. distribute-lft-neg-out 64.3%

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

      3. distribute-rgt-neg-in 64.3%

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

      4. associate-/r* 80.6%

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

      5. distribute-neg-frac2 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. 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 50.1%

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

      14. sqrt-unprod 26.8%

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

      15. sqr-neg 26.8%

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

      16. sqrt-unprod 12.9%

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

      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 22.8%

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

      19. sqrt-unprod 54.4%

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

   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 82.8%

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

      1. mul-1-neg 82.8%

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

   9. Simplified 82.8%

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

   if -5.4e25 < t1 < 4.04999999999999975e-63

   1. Initial program 86.0%

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

      1. associate-/l* 86.8%

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

      2. distribute-lft-neg-out 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

      4. associate-/r* 89.3%

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

      5. distribute-neg-frac2 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in t1 around 0 76.2%

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

   6. Taylor expanded in t1 around 0 79.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{+25} \lor \neg \left(t1 \leq 4.05 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{-211} \lor \neg \left(t1 \leq 2.15 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.2e-211) (not (<= t1 2.15e-165)))
   (/ v (- (- u) t1))
   (/ (/ (* t1 v) u) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.2e-211) || !(t1 <= 2.15e-165)) {
		tmp = v / (-u - t1);
	} else {
		tmp = ((t1 * v) / u) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.2d-211)) .or. (.not. (t1 <= 2.15d-165))) then
        tmp = v / (-u - t1)
    else
        tmp = ((t1 * v) / u) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.2e-211) || !(t1 <= 2.15e-165)) {
		tmp = v / (-u - t1);
	} else {
		tmp = ((t1 * v) / u) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.2e-211) or not (t1 <= 2.15e-165):
		tmp = v / (-u - t1)
	else:
		tmp = ((t1 * v) / u) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.2e-211) || !(t1 <= 2.15e-165))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(Float64(t1 * v) / u) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.2e-211) || ~((t1 <= 2.15e-165)))
		tmp = v / (-u - t1);
	else
		tmp = ((t1 * v) / u) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.2e-211], N[Not[LessEqual[t1, 2.15e-165]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.20000000000000015e-211 or 2.15000000000000003e-165 < t1

   1. Initial program 73.2%

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

      1. associate-/l* 71.0%

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

      2. distribute-lft-neg-out 71.0%

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

      3. distribute-rgt-neg-in 71.0%

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

      4. associate-/r* 83.8%

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

      5. distribute-neg-frac2 83.8%

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

   3. Simplified 83.8%

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

      1. associate-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. sub-neg 99.4%

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

      5. associate-*l/ 99.4%

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

      6. frac-2neg 99.4%

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

      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 53.9%

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

      14. sqrt-unprod 39.0%

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

      15. sqr-neg 39.0%

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

      16. sqrt-unprod 12.0%

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

      17. add-sqr-sqrt 33.6%

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

      18. add-sqr-sqrt 18.9%

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

      19. sqrt-unprod 56.7%

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

   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 71.4%

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

      1. mul-1-neg 71.4%

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

   9. Simplified 71.4%

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

   if -4.20000000000000015e-211 < t1 < 2.15000000000000003e-165

   1. Initial program 78.9%

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

      1. associate-/l* 87.5%

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

      2. distribute-lft-neg-out 87.5%

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

      3. distribute-rgt-neg-in 87.5%

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

      4. associate-/r* 87.6%

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

      5. distribute-neg-frac2 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in t1 around 0 85.7%

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

   6. Taylor expanded in t1 around 0 87.6%

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

      1. associate-*r/ 89.2%

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

      2. clear-num 89.2%

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

      3. un-div-inv 89.2%

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

      4. add-sqr-sqrt 36.1%

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

      5. sqrt-unprod 62.5%

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

      6. sqr-neg 62.5%

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

      7. sqrt-unprod 26.4%

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

      8. add-sqr-sqrt 56.0%

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

   8. Applied egg-rr 56.0%

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

   9. Taylor expanded in t1 around 0 56.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{-211} \lor \neg \left(t1 \leq 2.15 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{-211} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.3e-211) (not (<= t1 2.4e-165)))
   (/ v (- (- u) t1))
   (/ (* t1 (/ v u)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.3e-211) || !(t1 <= 2.4e-165)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 * (v / u)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.3d-211)) .or. (.not. (t1 <= 2.4d-165))) then
        tmp = v / (-u - t1)
    else
        tmp = (t1 * (v / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.3e-211) || !(t1 <= 2.4e-165)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 * (v / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.3e-211) or not (t1 <= 2.4e-165):
		tmp = v / (-u - t1)
	else:
		tmp = (t1 * (v / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.3e-211) || !(t1 <= 2.4e-165))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.3e-211) || ~((t1 <= 2.4e-165)))
		tmp = v / (-u - t1);
	else
		tmp = (t1 * (v / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.3e-211], N[Not[LessEqual[t1, 2.4e-165]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.3000000000000002e-211 or 2.4000000000000002e-165 < t1

   1. Initial program 73.2%

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

      1. associate-/l* 71.0%

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

      2. distribute-lft-neg-out 71.0%

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

      3. distribute-rgt-neg-in 71.0%

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

      4. associate-/r* 83.8%

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

      5. distribute-neg-frac2 83.8%

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

   3. Simplified 83.8%

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

      1. associate-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. sub-neg 99.4%

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

      5. associate-*l/ 99.4%

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

      6. frac-2neg 99.4%

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

      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 53.9%

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

      14. sqrt-unprod 39.0%

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

      15. sqr-neg 39.0%

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

      16. sqrt-unprod 12.0%

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

      17. add-sqr-sqrt 33.6%

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

      18. add-sqr-sqrt 18.9%

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

      19. sqrt-unprod 56.7%

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

   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 71.4%

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

      1. mul-1-neg 71.4%

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

   9. Simplified 71.4%

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

   if -3.3000000000000002e-211 < t1 < 2.4000000000000002e-165

   1. Initial program 78.9%

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

      1. associate-/l* 87.5%

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

      2. distribute-lft-neg-out 87.5%

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

      3. distribute-rgt-neg-in 87.5%

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

      4. associate-/r* 87.6%

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

      5. distribute-neg-frac2 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in t1 around 0 85.7%

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

   6. Taylor expanded in t1 around 0 87.6%

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

      1. associate-*r/ 89.2%

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

      2. clear-num 89.2%

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

      3. un-div-inv 89.2%

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

      4. add-sqr-sqrt 36.1%

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

      5. sqrt-unprod 62.5%

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

      6. sqr-neg 62.5%

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

      7. sqrt-unprod 26.4%

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

      8. add-sqr-sqrt 56.0%

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

   8. Applied egg-rr 56.0%

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

      1. clear-num 56.1%

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

      2. associate-/r/ 56.0%

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

      3. clear-num 56.0%

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

   10. Applied egg-rr 56.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{-211} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-208} \lor \neg \left(t1 \leq 2.15 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.35e-208) (not (<= t1 2.15e-165)))
   (/ v (- (- u) t1))
   (/ t1 (* u (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.35e-208) || !(t1 <= 2.15e-165)) {
		tmp = v / (-u - t1);
	} else {
		tmp = t1 / (u * (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 ((t1 <= (-1.35d-208)) .or. (.not. (t1 <= 2.15d-165))) then
        tmp = v / (-u - t1)
    else
        tmp = t1 / (u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.35e-208) || !(t1 <= 2.15e-165)) {
		tmp = v / (-u - t1);
	} else {
		tmp = t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.35e-208) or not (t1 <= 2.15e-165):
		tmp = v / (-u - t1)
	else:
		tmp = t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.35e-208) || !(t1 <= 2.15e-165))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.35e-208) || ~((t1 <= 2.15e-165)))
		tmp = v / (-u - t1);
	else
		tmp = t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.35e-208], N[Not[LessEqual[t1, 2.15e-165]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.35e-208 or 2.15000000000000003e-165 < t1

   1. Initial program 73.2%

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

      1. associate-/l* 71.0%

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

      2. distribute-lft-neg-out 71.0%

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

      3. distribute-rgt-neg-in 71.0%

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

      4. associate-/r* 83.8%

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

      5. distribute-neg-frac2 83.8%

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

   3. Simplified 83.8%

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

      1. associate-*r/ 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-neg-in 99.4%

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

      4. sub-neg 99.4%

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

      5. associate-*l/ 99.4%

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

      6. frac-2neg 99.4%

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

      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 53.9%

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

      14. sqrt-unprod 39.0%

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

      15. sqr-neg 39.0%

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

      16. sqrt-unprod 12.0%

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

      17. add-sqr-sqrt 33.6%

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

      18. add-sqr-sqrt 18.9%

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

      19. sqrt-unprod 56.7%

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

   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 71.4%

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

      1. mul-1-neg 71.4%

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

   9. Simplified 71.4%

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

   if -1.35e-208 < t1 < 2.15000000000000003e-165

   1. Initial program 78.9%

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

      1. associate-/l* 87.5%

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

      2. distribute-lft-neg-out 87.5%

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

      3. distribute-rgt-neg-in 87.5%

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

      4. associate-/r* 87.6%

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

      5. distribute-neg-frac2 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in t1 around 0 85.7%

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

   6. Taylor expanded in t1 around 0 87.6%

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

      1. clear-num 87.5%

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

      2. un-div-inv 87.6%

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

      3. div-inv 87.6%

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

      4. clear-num 87.6%

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

      5. add-sqr-sqrt 39.6%

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

      6. sqrt-unprod 66.1%

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

      7. sqr-neg 66.1%

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

      8. sqrt-unprod 26.4%

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

      9. add-sqr-sqrt 55.9%

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

   8. Applied egg-rr 55.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-208} \lor \neg \left(t1 \leq 2.15 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 23.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{+86} \lor \neg \left(t1 \leq 9 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.35e+86) (not (<= t1 9e+113))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.35e+86) || !(t1 <= 9e+113)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.35d+86)) .or. (.not. (t1 <= 9d+113))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.35e+86) || !(t1 <= 9e+113)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.35e+86) or not (t1 <= 9e+113):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.35e+86) || !(t1 <= 9e+113))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.35e+86) || ~((t1 <= 9e+113)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.35e+86], N[Not[LessEqual[t1, 9e+113]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.35000000000000009e86 or 9.0000000000000001e113 < t1

   1. Initial program 54.2%

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

      1. associate-/l* 54.0%

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

      2. distribute-lft-neg-out 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

      4. associate-/r* 73.8%

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

      5. distribute-neg-frac2 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in t1 around inf 91.9%

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

      1. associate-*r/ 91.9%

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

      2. neg-mul-1 91.9%

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

   7. Simplified 91.9%

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

      1. div-inv 91.6%

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

      2. add-sqr-sqrt 44.2%

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

      3. sqrt-unprod 56.0%

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

      4. sqr-neg 56.0%

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

      5. sqrt-unprod 21.7%

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

      6. add-sqr-sqrt 43.0%

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

   9. Applied egg-rr 43.0%

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

       1. associate-*r/ 43.0%

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

       2. *-rgt-identity 43.0%

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

   11. Simplified 43.0%

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

   if -1.35000000000000009e86 < t1 < 9.0000000000000001e113

   1. Initial program 86.7%

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

      1. associate-/l* 86.8%

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

      2. distribute-lft-neg-out 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

      4. associate-/r* 91.1%

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

      5. distribute-neg-frac2 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in t1 around 0 66.9%

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

   6. Taylor expanded in t1 around inf 17.1%

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

      1. associate-*r/ 17.1%

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

      2. mul-1-neg 17.1%

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

   8. Simplified 17.1%

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

      1. add-sqr-sqrt 7.5%

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

      2. sqrt-unprod 21.9%

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

      3. sqr-neg 21.9%

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

      4. sqrt-unprod 7.7%

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

      5. add-sqr-sqrt 16.9%

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

      6. div-inv 16.9%

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

   10. Applied egg-rr 16.9%

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

       1. associate-*r/ 16.9%

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

       2. *-rgt-identity 16.9%

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

   12. Simplified 16.9%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{+86} \lor \neg \left(t1 \leq 9 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% 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 74.4%

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

   1. times-frac 98.8%

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

   2. distribute-frac-neg 98.8%

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

   3. distribute-neg-frac2 98.8%

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

   4. +-commutative 98.8%

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

   5. distribute-neg-in 98.8%

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

   6. unsub-neg 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 12: 56.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 5e+189) (/ v (- t1)) (/ 1.0 (/ u v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 5e+189) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 5d+189) then
        tmp = v / -t1
    else
        tmp = 1.0d0 / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 5e+189) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 5e+189:
		tmp = v / -t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 5e+189)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 5e+189)
		tmp = v / -t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 5e+189], N[(v / (-t1)), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 5.0000000000000004e189

   1. Initial program 74.5%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 83.2%

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

      5. distribute-neg-frac2 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in t1 around inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. neg-mul-1 61.0%

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

   7. Simplified 61.0%

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

   if 5.0000000000000004e189 < u

   1. Initial program 73.7%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 96.2%

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

      5. distribute-neg-frac2 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t1 around 0 96.2%

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

   6. Taylor expanded in t1 around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. mul-1-neg 43.5%

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

   8. Simplified 43.5%

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

      1. add-sqr-sqrt 19.6%

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

      2. sqrt-unprod 43.0%

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

      3. sqr-neg 43.0%

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

      4. sqrt-unprod 23.8%

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

      5. add-sqr-sqrt 43.3%

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

      6. clear-num 45.1%

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

      7. inv-pow 45.1%

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

   10. Applied egg-rr 45.1%

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

       1. unpow-1 45.1%

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

   12. Simplified 45.1%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 5 \cdot 10^{+189}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 1.1 \cdot 10^{+190}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 1.1e+190) (/ v (- t1)) (/ v (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.1e+190) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 1.1d+190) then
        tmp = v / -t1
    else
        tmp = v / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.1e+190) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1.1e+190:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1.1e+190)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1.1e+190)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1.1e+190], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 1.1e190

   1. Initial program 74.5%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 83.2%

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

      5. distribute-neg-frac2 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in t1 around inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. neg-mul-1 61.0%

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

   7. Simplified 61.0%

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

   if 1.1e190 < u

   1. Initial program 73.7%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 96.2%

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

      5. distribute-neg-frac2 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t1 around 0 96.2%

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

   6. Taylor expanded in t1 around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. mul-1-neg 43.5%

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

   8. Simplified 43.5%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.1 \cdot 10^{+190}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 1.1 \cdot 10^{+190}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u 1.1e+190) (/ v (- t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.1e+190) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 1.1d+190) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.1e+190) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1.1e+190:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1.1e+190)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1.1e+190)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1.1e+190], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 1.1e190

   1. Initial program 74.5%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 83.2%

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

      5. distribute-neg-frac2 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in t1 around inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. neg-mul-1 61.0%

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

   7. Simplified 61.0%

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

   if 1.1e190 < u

   1. Initial program 73.7%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 96.2%

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

      5. distribute-neg-frac2 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t1 around 0 96.2%

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

   6. Taylor expanded in t1 around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. mul-1-neg 43.5%

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

   8. Simplified 43.5%

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

      1. add-sqr-sqrt 19.6%

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

      2. sqrt-unprod 43.0%

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

      3. sqr-neg 43.0%

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

      4. sqrt-unprod 23.8%

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

      5. add-sqr-sqrt 43.3%

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

      6. div-inv 43.3%

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

   10. Applied egg-rr 43.3%

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

       1. associate-*r/ 43.3%

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

       2. *-rgt-identity 43.3%

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

   12. Simplified 43.3%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.1 \cdot 10^{+190}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.6% 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 74.4%

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

   1. associate-/l* 74.4%

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

   2. distribute-lft-neg-out 74.4%

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

   3. distribute-rgt-neg-in 74.4%

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

   4. associate-/r* 84.6%

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

   5. distribute-neg-frac2 84.6%

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

3. Simplified 84.6%

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

   1. associate-*r/ 98.0%

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

   2. +-commutative 98.0%

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

   3. distribute-neg-in 98.0%

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

   4. sub-neg 98.0%

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

   5. associate-*l/ 98.8%

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

   6. frac-2neg 98.8%

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

   7. associate-*r/ 99.0%

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

   8. remove-double-neg 99.0%

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

   9. sub-neg 99.0%

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

   10. distribute-neg-in 99.0%

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

   11. +-commutative 99.0%

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

   12. frac-2neg 99.0%

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

   13. add-sqr-sqrt 50.6%

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

   14. sqrt-unprod 42.6%

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

   15. sqr-neg 42.6%

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

   16. sqrt-unprod 17.8%

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

   17. add-sqr-sqrt 38.3%

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

   18. add-sqr-sqrt 21.1%

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

   19. sqrt-unprod 61.2%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in t1 around inf 62.4%

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

   1. mul-1-neg 62.4%

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

9. Simplified 62.4%

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

10. Final simplification 62.4%

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

Alternative 16: 14.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. associate-/l* 74.4%

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

   2. distribute-lft-neg-out 74.4%

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

   3. distribute-rgt-neg-in 74.4%

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

   4. associate-/r* 84.6%

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

   5. distribute-neg-frac2 84.6%

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

3. Simplified 84.6%

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

5. Taylor expanded in t1 around inf 56.9%

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

   1. associate-*r/ 56.9%

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

   2. neg-mul-1 56.9%

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

7. Simplified 56.9%

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

   1. div-inv 56.8%

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

   2. add-sqr-sqrt 27.9%

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

   3. sqrt-unprod 37.7%

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

   4. sqr-neg 37.7%

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

   5. sqrt-unprod 8.7%

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

   6. add-sqr-sqrt 17.7%

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

9. Applied egg-rr 17.7%

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

    1. associate-*r/ 17.7%

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

    2. *-rgt-identity 17.7%

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

11. Simplified 17.7%

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

Reproduce

herbie shell --seed 2024136 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))