Rosa's DopplerBench

Percentage Accurate: 72.5% → 98.0%

Time: 9.7s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. times-frac 98.7%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 2: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{+70} \lor \neg \left(t1 \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.8e+70) (not (<= t1 9e-58)))
   (/ v (- (* u -2.0) t1))
   (* (/ (- t1) u) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.8e+70) || !(t1 <= 9e-58)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.8d+70)) .or. (.not. (t1 <= 9d-58))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (-t1 / u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.8e+70) || !(t1 <= 9e-58)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.8e+70) or not (t1 <= 9e-58):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-t1 / u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.8e+70) || !(t1 <= 9e-58))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.8e+70) || ~((t1 <= 9e-58)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-t1 / u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.8e+70], N[Not[LessEqual[t1, 9e-58]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.7999999999999997e70 or 9.0000000000000006e-58 < t1

   1. Initial program 49.8%

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

      1. associate-/r* 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-/l* 100.0%

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

      4. associate-/l/ 96.8%

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

      5. +-commutative 96.8%

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

      6. remove-double-neg 96.8%

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

      7. unsub-neg 96.8%

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

      8. div-sub 96.8%

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

      9. sub-neg 96.8%

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

      10. *-inverses 96.8%

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

      11. metadata-eval 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in t1 around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. *-commutative 90.6%

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

   7. Simplified 90.6%

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

   if -5.7999999999999997e70 < t1 < 9.0000000000000006e-58

   1. Initial program 84.1%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in t1 around 0 78.7%

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

      1. associate-*r/ 78.7%

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

      2. mul-1-neg 78.7%

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

   7. Simplified 78.7%

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

      1. associate-*r/ 77.8%

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

      2. frac-2neg 77.8%

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

      3. div-inv 77.8%

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

      4. div-inv 77.8%

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

      5. add-sqr-sqrt 47.5%

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

      6. sqrt-unprod 56.1%

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

      7. sqr-neg 56.1%

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

      8. sqrt-unprod 19.1%

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

      9. add-sqr-sqrt 46.5%

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

      10. distribute-lft-neg-out 46.5%

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

      11. distribute-frac-neg 46.5%

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

      12. *-commutative 46.5%

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

      13. add-sqr-sqrt 27.3%

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

      14. sqrt-unprod 50.9%

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

      15. sqr-neg 50.9%

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

      16. sqrt-unprod 30.2%

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

      17. add-sqr-sqrt 77.8%

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

      18. distribute-neg-in 77.8%

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

   9. Applied egg-rr 78.3%

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

       1. associate-/l* 77.3%

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

       2. associate-/r/ 79.2%

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

   11. Applied egg-rr 79.2%

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

   12. Taylor expanded in t1 around 0 79.7%

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

       1. associate-*r/ 79.7%

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

       2. neg-mul-1 79.7%

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

   14. Simplified 79.7%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{+70} \lor \neg \left(t1 \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.26 \cdot 10^{+69} \lor \neg \left(t1 \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.26e+69) (not (<= t1 9e-58)))
   (/ v (- (* u -2.0) t1))
   (/ (/ (- t1) u) (/ u v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.26e+69) || !(t1 <= 9e-58)) {
		tmp = v / ((u * -2.0) - 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 <= (-1.26d+69)) .or. (.not. (t1 <= 9d-58))) then
        tmp = v / ((u * (-2.0d0)) - 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 <= -1.26e+69) || !(t1 <= 9e-58)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.26e+69) or not (t1 <= 9e-58):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-t1 / u) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.26e+69) || !(t1 <= 9e-58))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.26e+69) || ~((t1 <= 9e-58)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-t1 / u) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.26e+69], N[Not[LessEqual[t1, 9e-58]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.26000000000000005e69 or 9.0000000000000006e-58 < t1

   1. Initial program 49.8%

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

      1. associate-/r* 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-/l* 100.0%

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

      4. associate-/l/ 96.8%

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

      5. +-commutative 96.8%

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

      6. remove-double-neg 96.8%

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

      7. unsub-neg 96.8%

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

      8. div-sub 96.8%

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

      9. sub-neg 96.8%

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

      10. *-inverses 96.8%

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

      11. metadata-eval 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in t1 around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. *-commutative 90.6%

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

   7. Simplified 90.6%

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

   if -1.26000000000000005e69 < t1 < 9.0000000000000006e-58

   1. Initial program 84.1%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

      1. *-commutative 97.6%

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

      2. clear-num 97.6%

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

      3. frac-2neg 97.6%

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

      4. frac-times 89.2%

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

      5. *-un-lft-identity 89.2%

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

      6. remove-double-neg 89.2%

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

      7. distribute-neg-in 89.2%

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

      8. add-sqr-sqrt 55.2%

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

      9. sqrt-unprod 86.7%

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

      10. sqr-neg 86.7%

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

      11. sqrt-unprod 31.6%

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

      12. add-sqr-sqrt 76.1%

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

      13. sub-neg 76.1%

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

   6. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/r* 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in t1 around 0 80.1%

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

   10. Taylor expanded in t1 around 0 80.6%

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

       1. associate-*r/ 80.6%

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

       2. mul-1-neg 80.6%

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

   12. Simplified 80.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.26 \cdot 10^{+69} \lor \neg \left(t1 \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+86} \lor \neg \left(u \leq 8.5 \cdot 10^{+192}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9.2e+86) (not (<= u 8.5e+192))) (/ 1.0 (/ u v)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.2e+86) || !(u <= 8.5e+192)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-9.2d+86)) .or. (.not. (u <= 8.5d+192))) then
        tmp = 1.0d0 / (u / v)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.2e+86) || !(u <= 8.5e+192)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9.2e+86) or not (u <= 8.5e+192):
		tmp = 1.0 / (u / v)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9.2e+86) || !(u <= 8.5e+192))
		tmp = Float64(1.0 / Float64(u / v));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9.2e+86) || ~((u <= 8.5e+192)))
		tmp = 1.0 / (u / v);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9.2e+86], N[Not[LessEqual[u, 8.5e+192]], $MachinePrecision]], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -9.19999999999999958e86 or 8.49999999999999965e192 < u

   1. Initial program 80.0%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. frac-2neg 99.9%

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

      4. frac-times 90.2%

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

      5. *-un-lft-identity 90.2%

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

      6. remove-double-neg 90.2%

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

      7. distribute-neg-in 90.2%

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

      8. add-sqr-sqrt 46.4%

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

      9. sqrt-unprod 86.5%

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

      10. sqr-neg 86.5%

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

      11. sqrt-unprod 43.8%

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

      12. add-sqr-sqrt 88.9%

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

      13. sub-neg 88.9%

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

   6. Applied egg-rr 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-/r* 91.2%

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

   8. Simplified 91.2%

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

   9. Taylor expanded in t1 around 0 90.2%

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

   10. Taylor expanded in t1 around inf 49.0%

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

   if -9.19999999999999958e86 < u < 8.49999999999999965e192

   1. Initial program 62.8%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in t1 around inf 66.0%

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

      1. associate-*r/ 66.0%

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

      2. neg-mul-1 66.0%

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

   7. Simplified 66.0%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+86} \lor \neg \left(u \leq 8.5 \cdot 10^{+192}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+192}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.5e+83)
   (/ v (+ t1 u))
   (if (<= u 2.5e+192) (/ (- v) t1) (/ 1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.5e+83) {
		tmp = v / (t1 + u);
	} else if (u <= 2.5e+192) {
		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 <= (-3.5d+83)) then
        tmp = v / (t1 + u)
    else if (u <= 2.5d+192) 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 <= -3.5e+83) {
		tmp = v / (t1 + u);
	} else if (u <= 2.5e+192) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.5e+83:
		tmp = v / (t1 + u)
	elif u <= 2.5e+192:
		tmp = -v / t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.5e+83)
		tmp = Float64(v / Float64(t1 + u));
	elseif (u <= 2.5e+192)
		tmp = Float64(Float64(-v) / 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 <= -3.5e+83)
		tmp = v / (t1 + u);
	elseif (u <= 2.5e+192)
		tmp = -v / t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.5e+83], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.5e+192], N[((-v) / t1), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.49999999999999977e83

   1. Initial program 75.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. frac-2neg 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. add-sqr-sqrt 56.6%

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

      8. sqrt-unprod 82.5%

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

      9. sqr-neg 82.5%

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

      10. sqrt-unprod 38.8%

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

      11. add-sqr-sqrt 88.8%

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

      12. sub-neg 88.8%

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

      13. remove-double-neg 88.8%

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

   6. Applied egg-rr 88.8%

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

   7. Taylor expanded in t1 around inf 45.6%

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

   if -3.49999999999999977e83 < u < 2.50000000000000017e192

   1. Initial program 62.8%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in t1 around inf 66.0%

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

      1. associate-*r/ 66.0%

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

      2. neg-mul-1 66.0%

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

   7. Simplified 66.0%

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

   if 2.50000000000000017e192 < u

   1. Initial program 85.4%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 100.0%

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

      3. frac-2neg 100.0%

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

      4. frac-times 91.6%

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

      5. *-un-lft-identity 91.6%

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

      6. remove-double-neg 91.6%

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

      7. distribute-neg-in 91.6%

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

      8. add-sqr-sqrt 41.4%

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

      9. sqrt-unprod 91.6%

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

      10. sqr-neg 91.6%

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

      11. sqrt-unprod 50.2%

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

      12. add-sqr-sqrt 91.6%

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

      13. sub-neg 91.6%

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

   6. Applied egg-rr 91.6%

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

      1. *-commutative 91.6%

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

      2. associate-/r* 94.4%

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

   8. Simplified 94.4%

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

   9. Taylor expanded in t1 around 0 94.4%

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

   10. Taylor expanded in t1 around inf 55.3%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+192}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+86} \lor \neg \left(u \leq 4.6 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9e+86) (not (<= u 4.6e+217))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9e+86) || !(u <= 4.6e+217)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-9d+86)) .or. (.not. (u <= 4.6d+217))) then
        tmp = v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9e+86) || !(u <= 4.6e+217)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9e+86) or not (u <= 4.6e+217):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9e+86) || !(u <= 4.6e+217))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9e+86) || ~((u <= 4.6e+217)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9e+86], N[Not[LessEqual[u, 4.6e+217]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -8.99999999999999986e86 or 4.5999999999999998e217 < u

   1. Initial program 82.9%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 93.0%

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

      1. associate-*r/ 93.0%

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

      2. mul-1-neg 93.0%

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

   7. Simplified 93.0%

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

      1. associate-*r/ 91.6%

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

      2. frac-2neg 91.6%

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

      3. div-inv 91.5%

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

      4. div-inv 91.6%

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

      5. add-sqr-sqrt 48.8%

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

      6. sqrt-unprod 71.2%

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

      7. sqr-neg 71.2%

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

      8. sqrt-unprod 37.2%

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

      9. add-sqr-sqrt 75.7%

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

      10. distribute-lft-neg-out 75.7%

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

      11. distribute-frac-neg 75.7%

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

      12. *-commutative 75.7%

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

      13. add-sqr-sqrt 38.5%

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

      14. sqrt-unprod 69.4%

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

      15. sqr-neg 69.4%

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

      16. sqrt-unprod 42.7%

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

      17. add-sqr-sqrt 91.6%

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

      18. distribute-neg-in 91.6%

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

   9. Applied egg-rr 91.6%

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

   10. Taylor expanded in t1 around inf 51.0%

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

   if -8.99999999999999986e86 < u < 4.5999999999999998e217

   1. Initial program 62.7%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. neg-mul-1 64.2%

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

   7. Simplified 64.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+86} \lor \neg \left(u \leq 4.6 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.4e+81) (/ (- v) u) (if (<= u 5e+217) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+81) {
		tmp = -v / u;
	} else if (u <= 5e+217) {
		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 <= (-1.4d+81)) then
        tmp = -v / u
    else if (u <= 5d+217) 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 <= -1.4e+81) {
		tmp = -v / u;
	} else if (u <= 5e+217) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.4e+81:
		tmp = -v / u
	elif u <= 5e+217:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.4e+81)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 5e+217)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.4e+81)
		tmp = -v / u;
	elseif (u <= 5e+217)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.4e+81], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 5e+217], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.39999999999999997e81

   1. Initial program 76.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 89.2%

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

      1. associate-*r/ 89.2%

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

      2. mul-1-neg 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in t1 around inf 42.2%

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

      1. associate-*r/ 42.2%

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

      2. neg-mul-1 42.2%

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

   10. Simplified 42.2%

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

   if -1.39999999999999997e81 < u < 5.00000000000000041e217

   1. Initial program 62.3%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   7. Simplified 64.8%

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

   if 5.00000000000000041e217 < u

   1. Initial program 95.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r/ 100.0%

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

      2. frac-2neg 100.0%

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

      3. div-inv 100.0%

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

      4. div-inv 100.0%

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

      5. add-sqr-sqrt 50.0%

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

      6. sqrt-unprod 83.6%

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

      7. sqr-neg 83.6%

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

      8. sqrt-unprod 50.0%

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

      9. add-sqr-sqrt 96.0%

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

      10. distribute-lft-neg-out 96.0%

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

      11. distribute-frac-neg 96.0%

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

      12. *-commutative 96.0%

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

      13. add-sqr-sqrt 46.0%

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

      14. sqrt-unprod 83.6%

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

      15. sqr-neg 83.6%

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

      16. sqrt-unprod 50.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. distribute-neg-in 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in t1 around inf 64.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{+80}:\\ \;\;\;\;-0.5 \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+219}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.6e+80)
   (* -0.5 (/ v u))
   (if (<= u 3.8e+219) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.6e+80) {
		tmp = -0.5 * (v / u);
	} else if (u <= 3.8e+219) {
		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 <= (-7.6d+80)) then
        tmp = (-0.5d0) * (v / u)
    else if (u <= 3.8d+219) 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 <= -7.6e+80) {
		tmp = -0.5 * (v / u);
	} else if (u <= 3.8e+219) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.6e+80:
		tmp = -0.5 * (v / u)
	elif u <= 3.8e+219:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.6e+80)
		tmp = Float64(-0.5 * Float64(v / u));
	elseif (u <= 3.8e+219)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.6e+80)
		tmp = -0.5 * (v / u);
	elseif (u <= 3.8e+219)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.6e+80], N[(-0.5 * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.8e+219], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.59999999999999995e80

   1. Initial program 76.8%

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

      1. associate-/r* 87.2%

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

      2. *-commutative 87.2%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 91.7%

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

      5. +-commutative 91.7%

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

      6. remove-double-neg 91.7%

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

      7. unsub-neg 91.7%

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

      8. div-sub 91.7%

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

      9. sub-neg 91.7%

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

      10. *-inverses 91.7%

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

      11. metadata-eval 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in t1 around inf 54.6%

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

      1. mul-1-neg 54.6%

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

      2. unsub-neg 54.6%

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

      3. *-commutative 54.6%

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

   7. Simplified 54.6%

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

   8. Taylor expanded in u around inf 42.3%

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

   if -7.59999999999999995e80 < u < 3.79999999999999996e219

   1. Initial program 62.3%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   7. Simplified 64.8%

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

   if 3.79999999999999996e219 < u

   1. Initial program 95.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r/ 100.0%

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

      2. frac-2neg 100.0%

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

      3. div-inv 100.0%

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

      4. div-inv 100.0%

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

      5. add-sqr-sqrt 50.0%

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

      6. sqrt-unprod 83.6%

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

      7. sqr-neg 83.6%

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

      8. sqrt-unprod 50.0%

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

      9. add-sqr-sqrt 96.0%

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

      10. distribute-lft-neg-out 96.0%

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

      11. distribute-frac-neg 96.0%

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

      12. *-commutative 96.0%

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

      13. add-sqr-sqrt 46.0%

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

      14. sqrt-unprod 83.6%

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

      15. sqr-neg 83.6%

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

      16. sqrt-unprod 50.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. distribute-neg-in 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in t1 around inf 64.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{+80}:\\ \;\;\;\;-0.5 \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+219}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{+87} \lor \neg \left(t1 \leq 4.5 \cdot 10^{+52}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.2e+87) (not (<= t1 4.5e+52))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.2e+87) || !(t1 <= 4.5e+52)) {
		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 <= (-2.2d+87)) .or. (.not. (t1 <= 4.5d+52))) 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 <= -2.2e+87) || !(t1 <= 4.5e+52)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.2e+87) or not (t1 <= 4.5e+52):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.2e+87) || !(t1 <= 4.5e+52))
		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 <= -2.2e+87) || ~((t1 <= 4.5e+52)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.2e+87], N[Not[LessEqual[t1, 4.5e+52]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.2000000000000001e87 or 4.5e52 < t1

   1. Initial program 43.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 97.7%

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

      3. frac-2neg 97.7%

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

      4. frac-times 65.4%

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

      5. *-un-lft-identity 65.4%

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

      6. remove-double-neg 65.4%

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

      7. distribute-neg-in 65.4%

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

      8. add-sqr-sqrt 29.0%

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

      9. sqrt-unprod 39.3%

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

      10. sqr-neg 39.3%

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

      11. sqrt-unprod 22.2%

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

      12. add-sqr-sqrt 36.2%

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

      13. sub-neg 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. *-commutative 36.2%

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

      2. associate-/r* 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in t1 around inf 27.3%

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

   if -2.2000000000000001e87 < t1 < 4.5e52

   1. Initial program 85.1%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 75.1%

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

      1. associate-*r/ 75.1%

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

      2. mul-1-neg 75.1%

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

   7. Simplified 75.1%

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

      1. associate-*r/ 74.3%

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

      2. frac-2neg 74.3%

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

      3. div-inv 74.3%

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

      4. div-inv 74.3%

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

      5. add-sqr-sqrt 42.5%

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

      6. sqrt-unprod 53.8%

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

      7. sqr-neg 53.8%

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

      8. sqrt-unprod 20.7%

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

      9. add-sqr-sqrt 45.2%

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

      10. distribute-lft-neg-out 45.2%

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

      11. distribute-frac-neg 45.2%

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

      12. *-commutative 45.2%

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

      13. add-sqr-sqrt 24.5%

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

      14. sqrt-unprod 50.3%

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

      15. sqr-neg 50.3%

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

      16. sqrt-unprod 31.7%

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

      17. add-sqr-sqrt 74.3%

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

      18. distribute-neg-in 74.3%

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

   9. Applied egg-rr 75.0%

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

   10. Taylor expanded in t1 around inf 23.2%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{+87} \lor \neg \left(t1 \leq 4.5 \cdot 10^{+52}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / ((-1.0d0) - (u / t1))) / (t1 + u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. times-frac 98.7%

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

3. Simplified 98.7%

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

   1. associate-*r/ 98.2%

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

   2. clear-num 97.9%

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

   3. associate-*l/ 98.0%

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

   4. *-un-lft-identity 98.0%

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

   5. frac-2neg 98.0%

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

   6. distribute-neg-in 98.0%

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

   7. add-sqr-sqrt 51.3%

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

   8. sqrt-unprod 66.9%

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

   9. sqr-neg 66.9%

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

   10. sqrt-unprod 27.6%

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

   11. add-sqr-sqrt 59.0%

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

   12. sub-neg 59.0%

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

   13. remove-double-neg 59.0%

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

6. Applied egg-rr 59.0%

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

   1. frac-2neg 59.0%

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

   2. add-sqr-sqrt 31.4%

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

   3. sqrt-unprod 44.4%

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

   4. sqr-neg 44.4%

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

   5. sqrt-unprod 37.6%

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

   6. add-sqr-sqrt 76.6%

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

   7. distribute-neg-frac 76.6%

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

   8. neg-sub0 76.6%

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

   9. div-sub 76.6%

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

   10. *-inverses 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(\color{blue}{1} - \frac{u}{t1}\right)}}{t1 + u} \]

   11. sub-neg 76.6%

       \[\leadsto \frac{\frac{v}{0 - \color{blue}{\left(1 + \left(-\frac{u}{t1}\right)\right)}}}{t1 + u} \]

   12. mul-1-neg 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{-1 \cdot \frac{u}{t1}}\right)}}{t1 + u} \]

   13. associate-*r/ 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{\frac{-1 \cdot u}{t1}}\right)}}{t1 + u} \]

   14. neg-mul-1 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{\color{blue}{-u}}{t1}\right)}}{t1 + u} \]

   15. add-sqr-sqrt 37.7%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}\right)}}{t1 + u} \]

   16. sqrt-unprod 79.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\color{blue}{\sqrt{t1 \cdot t1}}}\right)}}{t1 + u} \]

   17. sqr-neg 79.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}}\right)}}{t1 + u} \]

   18. sqrt-unprod 51.4%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}\right)}}{t1 + u} \]

   19. add-sqr-sqrt 98.0%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\color{blue}{-t1}}\right)}}{t1 + u} \]

   20. frac-2neg 98.0%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{\frac{u}{t1}}\right)}}{t1 + u} \]

8. Applied egg-rr 98.0%

   \[\leadsto \frac{\frac{v}{\color{blue}{0 - \left(1 + \frac{u}{t1}\right)}}}{t1 + u} \]
9. Step-by-step derivation

   1. associate--r+ 98.0%

      \[\leadsto \frac{\frac{v}{\color{blue}{\left(0 - 1\right) - \frac{u}{t1}}}}{t1 + u} \]

   2. metadata-eval 98.0%

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

10. Simplified 98.0%

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

11. Final simplification 98.0%

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

Alternative 11: 61.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u \cdot -2 - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (* u -2.0) t1)))

C

double code(double u, double v, double t1) {
	return v / ((u * -2.0) - 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 * (-2.0d0)) - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Python

def code(u, v, t1):
	return v / ((u * -2.0) - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(u * -2.0) - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((u * -2.0) - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

   1. associate-/r* 79.0%

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

   2. *-commutative 79.0%

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

   3. associate-/l* 98.0%

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

   4. associate-/l/ 96.2%

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

   5. +-commutative 96.2%

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

   6. remove-double-neg 96.2%

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

   7. unsub-neg 96.2%

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

   8. div-sub 96.2%

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

   9. sub-neg 96.2%

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

   10. *-inverses 96.2%

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

   11. metadata-eval 96.2%

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

3. Simplified 96.2%

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

5. Taylor expanded in t1 around inf 63.9%

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

   1. mul-1-neg 63.9%

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

   2. unsub-neg 63.9%

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

   3. *-commutative 63.9%

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

7. Simplified 63.9%

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

8. Final simplification 63.9%

   \[\leadsto \frac{v}{u \cdot -2 - t1} \]
9. Add Preprocessing

Alternative 12: 61.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. times-frac 98.7%

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

3. Simplified 98.7%

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

   1. associate-*r/ 98.2%

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

   2. clear-num 97.9%

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

   3. associate-*l/ 98.0%

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

   4. *-un-lft-identity 98.0%

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

   5. frac-2neg 98.0%

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

   6. distribute-neg-in 98.0%

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

   7. add-sqr-sqrt 51.3%

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

   8. sqrt-unprod 66.9%

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

   9. sqr-neg 66.9%

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

   10. sqrt-unprod 27.6%

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

   11. add-sqr-sqrt 59.0%

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

   12. sub-neg 59.0%

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

   13. remove-double-neg 59.0%

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

6. Applied egg-rr 59.0%

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

   1. frac-2neg 59.0%

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

   2. add-sqr-sqrt 31.4%

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

   3. sqrt-unprod 44.4%

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

   4. sqr-neg 44.4%

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

   5. sqrt-unprod 37.6%

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

   6. add-sqr-sqrt 76.6%

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

   7. distribute-neg-frac 76.6%

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

   8. neg-sub0 76.6%

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

   9. div-sub 76.6%

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

   10. *-inverses 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(\color{blue}{1} - \frac{u}{t1}\right)}}{t1 + u} \]

   11. sub-neg 76.6%

       \[\leadsto \frac{\frac{v}{0 - \color{blue}{\left(1 + \left(-\frac{u}{t1}\right)\right)}}}{t1 + u} \]

   12. mul-1-neg 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{-1 \cdot \frac{u}{t1}}\right)}}{t1 + u} \]

   13. associate-*r/ 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{\frac{-1 \cdot u}{t1}}\right)}}{t1 + u} \]

   14. neg-mul-1 76.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{\color{blue}{-u}}{t1}\right)}}{t1 + u} \]

   15. add-sqr-sqrt 37.7%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}\right)}}{t1 + u} \]

   16. sqrt-unprod 79.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\color{blue}{\sqrt{t1 \cdot t1}}}\right)}}{t1 + u} \]

   17. sqr-neg 79.6%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}}\right)}}{t1 + u} \]

   18. sqrt-unprod 51.4%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}\right)}}{t1 + u} \]

   19. add-sqr-sqrt 98.0%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \frac{-u}{\color{blue}{-t1}}\right)}}{t1 + u} \]

   20. frac-2neg 98.0%

       \[\leadsto \frac{\frac{v}{0 - \left(1 + \color{blue}{\frac{u}{t1}}\right)}}{t1 + u} \]

8. Applied egg-rr 98.0%

   \[\leadsto \frac{\frac{v}{\color{blue}{0 - \left(1 + \frac{u}{t1}\right)}}}{t1 + u} \]
9. Step-by-step derivation

   1. associate--r+ 98.0%

      \[\leadsto \frac{\frac{v}{\color{blue}{\left(0 - 1\right) - \frac{u}{t1}}}}{t1 + u} \]

   2. metadata-eval 98.0%

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

10. Simplified 98.0%

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

11. Taylor expanded in u around 0 63.7%

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

    1. neg-mul-1 63.7%

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

13. Simplified 63.7%

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

14. Final simplification 63.7%

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

Alternative 13: 13.9% 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 68.0%

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

   1. times-frac 98.7%

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

3. Simplified 98.7%

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

   1. *-commutative 98.7%

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

   2. clear-num 97.8%

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

   3. frac-2neg 97.8%

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

   4. frac-times 80.2%

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

   5. *-un-lft-identity 80.2%

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

   6. remove-double-neg 80.2%

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

   7. distribute-neg-in 80.2%

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

   8. add-sqr-sqrt 41.5%

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

   9. sqrt-unprod 65.2%

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

   10. sqr-neg 65.2%

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

   11. sqrt-unprod 28.6%

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

   12. add-sqr-sqrt 57.9%

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

   13. sub-neg 57.9%

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

6. Applied egg-rr 57.9%

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

   1. *-commutative 57.9%

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

   2. associate-/r* 60.2%

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

8. Simplified 60.2%

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

9. Taylor expanded in t1 around inf 12.4%

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

10. Final simplification 12.4%

    \[\leadsto \frac{v}{t1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))