Rosa's DopplerBench

Percentage Accurate: 73.1% → 97.8%

Time: 9.7s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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: 97.8% 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 77.0%

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

   1. times-frac 96.6%

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

3. Simplified 96.6%

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

5. Final simplification 96.6%

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

Alternative 2: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.1 \cdot 10^{-109} \lor \neg \left(t1 \leq 5.5 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.1e-109) (not (<= t1 5.5e-40)))
   (/ v (- (* u -2.0) t1))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.1e-109) || !(t1 <= 5.5e-40)) {
		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 <= (-2.1d-109)) .or. (.not. (t1 <= 5.5d-40))) 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 <= -2.1e-109) || !(t1 <= 5.5e-40)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.1e-109) or not (t1 <= 5.5e-40):
		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 <= -2.1e-109) || !(t1 <= 5.5e-40))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.1e-109) || ~((t1 <= 5.5e-40)))
		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, -2.1e-109], N[Not[LessEqual[t1, 5.5e-40]], $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 < -2.09999999999999996e-109 or 5.50000000000000002e-40 < 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-/r* 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 97.6%

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

      5. +-commutative 97.6%

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

      6. remove-double-neg 97.6%

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

      7. unsub-neg 97.6%

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

      8. div-sub 97.6%

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

      9. sub-neg 97.6%

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

      10. *-inverses 97.6%

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

      11. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in t1 around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. unsub-neg 81.1%

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

      3. *-commutative 81.1%

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

   7. Simplified 81.1%

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

   if -2.09999999999999996e-109 < t1 < 5.50000000000000002e-40

   1. Initial program 84.3%

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

      1. times-frac 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t1 around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

   7. Simplified 80.3%

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

   8. Taylor expanded in t1 around 0 83.6%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.1 \cdot 10^{-109} \lor \neg \left(t1 \leq 5.5 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.3 \cdot 10^{-109} \lor \neg \left(t1 \leq 2.1 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.3e-109) (not (<= t1 2.1e-43)))
   (/ v (- (* u -2.0) t1))
   (/ (* v (/ t1 u)) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.3e-109) || !(t1 <= 2.1e-43)) {
		tmp = v / ((u * -2.0) - t1);
	} 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 <= (-2.3d-109)) .or. (.not. (t1 <= 2.1d-43))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    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 <= -2.3e-109) || !(t1 <= 2.1e-43)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.3e-109) or not (t1 <= 2.1e-43):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (v * (t1 / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.3e-109) || !(t1 <= 2.1e-43))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.3e-109) || ~((t1 <= 2.1e-43)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (v * (t1 / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.3e-109], N[Not[LessEqual[t1, 2.1e-43]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.3000000000000001e-109 or 2.1000000000000001e-43 < 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-/r* 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 97.6%

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

      5. +-commutative 97.6%

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

      6. remove-double-neg 97.6%

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

      7. unsub-neg 97.6%

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

      8. div-sub 97.6%

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

      9. sub-neg 97.6%

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

      10. *-inverses 97.6%

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

      11. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in t1 around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. unsub-neg 81.1%

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

      3. *-commutative 81.1%

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

   7. Simplified 81.1%

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

   if -2.3000000000000001e-109 < t1 < 2.1000000000000001e-43

   1. Initial program 84.3%

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

      1. times-frac 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t1 around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

   7. Simplified 80.3%

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

   8. Taylor expanded in t1 around 0 83.6%

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

      1. distribute-frac-neg 83.6%

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

      2. frac-2neg 83.6%

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

      3. distribute-frac-neg 83.6%

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

      4. remove-double-neg 83.6%

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

      5. associate-*l/ 83.7%

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

      6. associate-*r/ 89.6%

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

      7. associate-*l/ 88.7%

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

      8. *-commutative 88.7%

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

   10. Applied egg-rr 88.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.3 \cdot 10^{-109} \lor \neg \left(t1 \leq 2.1 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{-109} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{\frac{u}{t1}}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.4e-109) (not (<= t1 3.1e-30)))
   (/ v (- (* u -2.0) t1))
   (/ (/ (- v) (/ u t1)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.4e-109) || !(t1 <= 3.1e-30)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-v / (u / t1)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.4d-109)) .or. (.not. (t1 <= 3.1d-30))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (-v / (u / t1)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.4e-109) || !(t1 <= 3.1e-30)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-v / (u / t1)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.4e-109) or not (t1 <= 3.1e-30):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-v / (u / t1)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.4e-109) || !(t1 <= 3.1e-30))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-v) / Float64(u / t1)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.4e-109) || ~((t1 <= 3.1e-30)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-v / (u / t1)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.4e-109], N[Not[LessEqual[t1, 3.1e-30]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-v) / N[(u / t1), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.39999999999999989e-109 or 3.09999999999999991e-30 < 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-/r* 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 97.6%

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

      5. +-commutative 97.6%

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

      6. remove-double-neg 97.6%

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

      7. unsub-neg 97.6%

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

      8. div-sub 97.6%

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

      9. sub-neg 97.6%

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

      10. *-inverses 97.6%

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

      11. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in t1 around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. unsub-neg 81.1%

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

      3. *-commutative 81.1%

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

   7. Simplified 81.1%

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

   if -2.39999999999999989e-109 < t1 < 3.09999999999999991e-30

   1. Initial program 84.3%

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

      1. times-frac 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t1 around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. mul-1-neg 80.3%

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

   7. Simplified 80.3%

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

   8. Taylor expanded in t1 around 0 83.6%

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

      1. associate-*l/ 83.7%

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

      2. add-sqr-sqrt 41.8%

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

      3. sqrt-unprod 45.4%

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

      4. sqr-neg 45.4%

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

      5. sqrt-unprod 16.8%

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

      6. add-sqr-sqrt 37.7%

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

      7. associate-*r/ 37.8%

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

      8. associate-*l/ 37.8%

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

      9. *-commutative 37.8%

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

   10. Applied egg-rr 37.8%

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

       1. associate-*r/ 37.8%

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

       2. *-commutative 37.8%

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

       3. frac-2neg 37.8%

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

       4. add-sqr-sqrt 23.2%

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

       5. sqrt-unprod 50.8%

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

       6. sqr-neg 50.8%

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

       7. sqrt-unprod 33.3%

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

       8. add-sqr-sqrt 89.6%

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

       9. distribute-neg-frac 89.6%

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

       10. *-commutative 89.6%

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

       11. associate-*r/ 88.7%

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

       12. neg-sub0 88.7%

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

       13. clear-num 88.7%

           \[\leadsto \frac{0 - v \cdot \color{blue}{\frac{1}{\frac{u}{t1}}}}{u} \]

       14. un-div-inv 88.9%

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

   12. Applied egg-rr 88.9%

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

       1. neg-sub0 88.9%

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

       2. distribute-neg-frac 88.9%

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

   14. Simplified 88.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{-109} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{\frac{u}{t1}}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.9 \cdot 10^{+164} \lor \neg \left(u \leq 2.3 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.9e+164) (not (<= u 2.3e+43)))
   (/ t1 (/ u (/ v u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.9e+164) || !(u <= 2.3e+43)) {
		tmp = t1 / (u / (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 <= (-3.9d+164)) .or. (.not. (u <= 2.3d+43))) then
        tmp = t1 / (u / (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 <= -3.9e+164) || !(u <= 2.3e+43)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.9e+164) or not (u <= 2.3e+43):
		tmp = t1 / (u / (v / u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.9e+164) || !(u <= 2.3e+43))
		tmp = Float64(t1 / Float64(u / 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 <= -3.9e+164) || ~((u <= 2.3e+43)))
		tmp = t1 / (u / (v / u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.9e+164], N[Not[LessEqual[u, 2.3e+43]], $MachinePrecision]], N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.89999999999999985e164 or 2.3000000000000002e43 < u

   1. Initial program 81.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   7. Simplified 91.4%

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

   8. Taylor expanded in t1 around 0 90.4%

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

      1. associate-*l/ 90.3%

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

      2. associate-/l* 86.9%

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

      3. add-sqr-sqrt 55.8%

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

      4. sqrt-unprod 63.9%

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

      5. sqr-neg 63.9%

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

      6. sqrt-unprod 25.5%

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

      7. add-sqr-sqrt 74.8%

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

   10. Applied egg-rr 74.8%

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

   if -3.89999999999999985e164 < u < 2.3000000000000002e43

   1. Initial program 74.9%

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

      1. times-frac 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in t1 around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. neg-mul-1 68.1%

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

   7. Simplified 68.1%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.9 \cdot 10^{+164} \lor \neg \left(u \leq 2.3 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+163} \lor \neg \left(u \leq 2.3 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.5e+163) (not (<= u 2.3e+43)))
   (/ t1 (/ u (/ v u)))
   (/ v (- (* u -2.0) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.5e+163) || !(u <= 2.3e+43)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.5d+163)) .or. (.not. (u <= 2.3d+43))) then
        tmp = t1 / (u / (v / u))
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.5e+163) || !(u <= 2.3e+43)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.5e+163) or not (u <= 2.3e+43):
		tmp = t1 / (u / (v / u))
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.5e+163) || !(u <= 2.3e+43))
		tmp = Float64(t1 / Float64(u / Float64(v / u)));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.5e+163) || ~((u <= 2.3e+43)))
		tmp = t1 / (u / (v / u));
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.5e+163], N[Not[LessEqual[u, 2.3e+43]], $MachinePrecision]], N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.5000000000000003e163 or 2.3000000000000002e43 < u

   1. Initial program 81.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   7. Simplified 91.4%

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

   8. Taylor expanded in t1 around 0 90.4%

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

      1. associate-*l/ 90.3%

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

      2. associate-/l* 86.9%

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

      3. add-sqr-sqrt 55.8%

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

      4. sqrt-unprod 63.9%

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

      5. sqr-neg 63.9%

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

      6. sqrt-unprod 25.5%

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

      7. add-sqr-sqrt 74.8%

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

   10. Applied egg-rr 74.8%

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

   if -3.5000000000000003e163 < u < 2.3000000000000002e43

   1. Initial program 74.9%

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

      1. associate-/r* 85.7%

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

      2. *-commutative 85.7%

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

      3. associate-/l* 97.8%

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

      4. associate-/l/ 95.6%

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

      5. +-commutative 95.6%

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

      6. remove-double-neg 95.6%

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

      7. unsub-neg 95.6%

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

      8. div-sub 95.6%

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

      9. sub-neg 95.6%

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

      10. *-inverses 95.6%

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

      11. metadata-eval 95.6%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in t1 around inf 68.6%

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

      1. mul-1-neg 68.6%

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

      2. unsub-neg 68.6%

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

      3. *-commutative 68.6%

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

   7. Simplified 68.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+163} \lor \neg \left(u \leq 2.3 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+185} \lor \neg \left(u \leq 4.2 \cdot 10^{+204}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.5e+185) (not (<= u 4.2e+204))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.5e+185) || !(u <= 4.2e+204)) {
		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 <= (-6.5d+185)) .or. (.not. (u <= 4.2d+204))) 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 <= -6.5e+185) || !(u <= 4.2e+204)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.5e+185) or not (u <= 4.2e+204):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.5e+185) || !(u <= 4.2e+204))
		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 <= -6.5e+185) || ~((u <= 4.2e+204)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.5e+185], N[Not[LessEqual[u, 4.2e+204]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.5000000000000002e185 or 4.2000000000000001e204 < u

   1. Initial program 86.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 96.2%

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

      1. associate-*r/ 96.2%

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

      2. mul-1-neg 96.2%

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

   7. Simplified 96.2%

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

      1. expm1-log1p-u 96.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)\right)} \]

      2. expm1-udef 87.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)} - 1} \]

   9. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v \cdot \frac{t1}{t1 - u}}{u}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 96.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v \cdot \frac{t1}{t1 - u}}{u}\right)\right)} \]

       2. expm1-log1p 96.3%

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

       3. associate-/l* 90.8%

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

   11. Simplified 90.8%

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

   12. Taylor expanded in u around 0 46.0%

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

   if -6.5000000000000002e185 < u < 4.2000000000000001e204

   1. Initial program 74.7%

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

      1. times-frac 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in t1 around inf 63.6%

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

      1. associate-*r/ 63.6%

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

      2. neg-mul-1 63.6%

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

   7. Simplified 63.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+185} \lor \neg \left(u \leq 4.2 \cdot 10^{+204}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+204}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.7e+168)
   (* (/ v u) -0.5)
   (if (<= u 3.4e+204) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.7e+168) {
		tmp = (v / u) * -0.5;
	} else if (u <= 3.4e+204) {
		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 <= (-1.7d+168)) then
        tmp = (v / u) * (-0.5d0)
    else if (u <= 3.4d+204) 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 <= -1.7e+168) {
		tmp = (v / u) * -0.5;
	} else if (u <= 3.4e+204) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.7e+168:
		tmp = (v / u) * -0.5
	elif u <= 3.4e+204:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.7e+168)
		tmp = Float64(Float64(v / u) * -0.5);
	elseif (u <= 3.4e+204)
		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 <= -1.7e+168)
		tmp = (v / u) * -0.5;
	elseif (u <= 3.4e+204)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.7e+168], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[u, 3.4e+204], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.70000000000000001e168

   1. Initial program 85.2%

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

      1. associate-/r* 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 96.9%

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

      5. +-commutative 96.9%

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

      6. remove-double-neg 96.9%

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

      7. unsub-neg 96.9%

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

      8. div-sub 96.9%

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

      9. sub-neg 96.9%

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

      10. *-inverses 96.9%

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

      11. metadata-eval 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t1 around inf 52.7%

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

      1. mul-1-neg 52.7%

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

      2. unsub-neg 52.7%

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

      3. *-commutative 52.7%

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

   7. Simplified 52.7%

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

   8. Taylor expanded in u around inf 45.1%

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

   if -1.70000000000000001e168 < u < 3.4000000000000001e204

   1. Initial program 74.6%

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

      1. times-frac 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

   if 3.4000000000000001e204 < u

   1. Initial program 87.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. expm1-log1p-u 99.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)\right)} \]

      2. expm1-udef 87.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)} - 1} \]

   9. Applied egg-rr 87.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v \cdot \frac{t1}{t1 - u}}{u}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 99.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v \cdot \frac{t1}{t1 - u}}{u}\right)\right)} \]

       2. expm1-log1p 99.9%

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

       3. associate-/l* 91.6%

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

   11. Simplified 91.6%

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

   12. Taylor expanded in u around 0 44.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+204}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 17.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. times-frac 96.6%

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

3. Simplified 96.6%

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

5. Taylor expanded in t1 around 0 51.7%

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

   1. associate-*r/ 51.7%

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

   2. mul-1-neg 51.7%

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

7. Simplified 51.7%

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

   1. expm1-log1p-u 47.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)\right)} \]

   2. expm1-udef 36.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{u} \cdot \frac{v}{t1 + u}\right)} - 1} \]

9. Applied egg-rr 41.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{v \cdot \frac{t1}{t1 - u}}{u}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 46.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v \cdot \frac{t1}{t1 - u}}{u}\right)\right)} \]

    2. expm1-log1p 51.3%

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

    3. associate-/l* 47.7%

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

11. Simplified 47.7%

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

12. Taylor expanded in u around 0 16.2%

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

13. Final simplification 16.2%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))