Rosa's DopplerBench

Percentage Accurate: 72.1% → 97.9%

Time: 7.0s

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: 72.1% 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: 97.9% 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 73.3%

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

   1. times-frac 96.9%

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

3. Simplified 96.9%

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

4. Final simplification 96.9%

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

Alternative 2: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-34} \lor \neg \left(u \leq 1.25 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.9e-34) (not (<= u 1.25e+33)))
   (/ (* t1 (/ v (+ t1 u))) (- t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.9e-34) || !(u <= 1.25e+33)) {
		tmp = (t1 * (v / (t1 + u))) / (t1 - u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.9d-34)) .or. (.not. (u <= 1.25d+33))) then
        tmp = (t1 * (v / (t1 + u))) / (t1 - u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.9e-34) || !(u <= 1.25e+33)) {
		tmp = (t1 * (v / (t1 + u))) / (t1 - u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.9e-34) or not (u <= 1.25e+33):
		tmp = (t1 * (v / (t1 + u))) / (t1 - u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.9e-34) || !(u <= 1.25e+33))
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 + u))) / Float64(t1 - u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.9e-34) || ~((u <= 1.25e+33)))
		tmp = (t1 * (v / (t1 + u))) / (t1 - u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.9e-34], N[Not[LessEqual[u, 1.25e+33]], $MachinePrecision]], N[(N[(t1 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.9000000000000002e-34 or 1.24999999999999993e33 < u

   1. Initial program 79.5%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

      1. frac-2neg 96.4%

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

      2. remove-double-neg 96.4%

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

      3. associate-*l/ 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. add-sqr-sqrt 45.0%

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

      6. sqrt-unprod 87.2%

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

      7. sqr-neg 87.2%

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

      8. sqrt-unprod 48.0%

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

      9. add-sqr-sqrt 88.3%

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

      10. sub-neg 88.3%

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

   5. Applied egg-rr 88.3%

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

   if -2.9000000000000002e-34 < u < 1.24999999999999993e33

   1. Initial program 68.3%

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

      1. times-frac 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in t1 around inf 77.9%

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

      1. associate-*r/ 77.9%

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

      2. neg-mul-1 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-34} \lor \neg \left(u \leq 1.25 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{-35} \lor \neg \left(u \leq 2.9 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.5e-35) (not (<= u 2.9e+28)))
   (* (/ (- t1) u) (/ v u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.5e-35) || !(u <= 2.9e+28)) {
		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 <= (-4.5d-35)) .or. (.not. (u <= 2.9d+28))) 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 <= -4.5e-35) || !(u <= 2.9e+28)) {
		tmp = (-t1 / u) * (v / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.5e-35) or not (u <= 2.9e+28):
		tmp = (-t1 / u) * (v / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.5e-35) || !(u <= 2.9e+28))
		tmp = Float64(Float64(Float64(-t1) / 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 <= -4.5e-35) || ~((u <= 2.9e+28)))
		tmp = (-t1 / u) * (v / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.5e-35], N[Not[LessEqual[u, 2.9e+28]], $MachinePrecision]], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.5000000000000001e-35 or 2.9000000000000001e28 < u

   1. Initial program 79.5%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t1 around 0 80.2%

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

   5. Taylor expanded in t1 around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-neg-frac 79.5%

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

   7. Simplified 79.5%

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

   if -4.5000000000000001e-35 < u < 2.9000000000000001e28

   1. Initial program 68.3%

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

      1. times-frac 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in t1 around inf 77.9%

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

      1. associate-*r/ 77.9%

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

      2. neg-mul-1 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{-35} \lor \neg \left(u \leq 2.9 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 4: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{-31} \lor \neg \left(u \leq 1.26 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2e-31) (not (<= u 1.26e+28)))
   (/ t1 (* u (/ (- u) v)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e-31) || !(u <= 1.26e+28)) {
		tmp = t1 / (u * (-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 <= (-2d-31)) .or. (.not. (u <= 1.26d+28))) then
        tmp = t1 / (u * (-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 <= -2e-31) || !(u <= 1.26e+28)) {
		tmp = t1 / (u * (-u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2e-31) or not (u <= 1.26e+28):
		tmp = t1 / (u * (-u / v))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2e-31) || !(u <= 1.26e+28))
		tmp = Float64(t1 / Float64(u * Float64(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 <= -2e-31) || ~((u <= 1.26e+28)))
		tmp = t1 / (u * (-u / v));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2e-31], N[Not[LessEqual[u, 1.26e+28]], $MachinePrecision]], N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2e-31 or 1.26e28 < u

   1. Initial program 79.5%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t1 around 0 80.2%

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

   5. Taylor expanded in t1 around 0 79.5%

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

      1. mul-1-neg 79.5%

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

      2. distribute-neg-frac 79.5%

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

   7. Simplified 79.5%

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

      1. *-commutative 79.5%

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

      2. clear-num 80.3%

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

      3. frac-2neg 80.3%

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

      4. frac-times 85.1%

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

      5. *-un-lft-identity 85.1%

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

      6. remove-double-neg 85.1%

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

   9. Applied egg-rr 85.1%

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

   if -2e-31 < u < 1.26e28

   1. Initial program 68.3%

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

      1. times-frac 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in t1 around inf 77.9%

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

      1. associate-*r/ 77.9%

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

      2. neg-mul-1 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{-31} \lor \neg \left(u \leq 1.26 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 5: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.5e-33)
   (/ (/ t1 (/ u v)) (- u))
   (if (<= u 7.6e+30) (/ (- v) t1) (/ t1 (* u (/ (- u) v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.5e-33) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= 7.6e+30) {
		tmp = -v / 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 (u <= (-3.5d-33)) then
        tmp = (t1 / (u / v)) / -u
    else if (u <= 7.6d+30) then
        tmp = -v / 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 (u <= -3.5e-33) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= 7.6e+30) {
		tmp = -v / t1;
	} else {
		tmp = t1 / (u * (-u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.5e-33:
		tmp = (t1 / (u / v)) / -u
	elif u <= 7.6e+30:
		tmp = -v / t1
	else:
		tmp = t1 / (u * (-u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.5e-33)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(-u));
	elseif (u <= 7.6e+30)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(-u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.5e-33)
		tmp = (t1 / (u / v)) / -u;
	elseif (u <= 7.6e+30)
		tmp = -v / t1;
	else
		tmp = t1 / (u * (-u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.5e-33], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, 7.6e+30], N[((-v) / t1), $MachinePrecision], N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.4999999999999999e-33

   1. Initial program 77.5%

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

      1. times-frac 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in t1 around 0 78.6%

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

   5. Taylor expanded in t1 around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. distribute-neg-frac 77.7%

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

   7. Simplified 77.7%

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

      1. associate-*l/ 82.2%

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

      2. frac-2neg 82.2%

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

      3. add-sqr-sqrt 39.8%

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

      4. sqrt-unprod 57.5%

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

      5. sqr-neg 57.5%

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

      6. sqrt-unprod 27.9%

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

      7. add-sqr-sqrt 45.2%

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

      8. distribute-lft-neg-in 45.2%

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

      9. clear-num 46.5%

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

      10. un-div-inv 46.5%

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

      11. add-sqr-sqrt 17.3%

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

      12. sqrt-unprod 47.9%

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

      13. sqr-neg 47.9%

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

      14. sqrt-unprod 43.6%

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

      15. add-sqr-sqrt 83.6%

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

   9. Applied egg-rr 83.6%

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

   if -3.4999999999999999e-33 < u < 7.6000000000000003e30

   1. Initial program 68.3%

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

      1. times-frac 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in t1 around inf 77.9%

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

      1. associate-*r/ 77.9%

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

      2. neg-mul-1 77.9%

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

   6. Simplified 77.9%

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

   if 7.6000000000000003e30 < u

   1. Initial program 82.2%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around 0 82.4%

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

   5. Taylor expanded in t1 around 0 82.0%

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

      1. mul-1-neg 82.0%

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

      2. distribute-neg-frac 82.0%

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

   7. Simplified 82.0%

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

      1. *-commutative 82.0%

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

      2. clear-num 82.1%

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

      3. frac-2neg 82.1%

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

      4. frac-times 90.0%

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

      5. *-un-lft-identity 90.0%

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

      6. remove-double-neg 90.0%

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

   9. Applied egg-rr 90.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \end{array} \]

Alternative 6: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+51} \lor \neg \left(u \leq 5.1 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.75e+51) (not (<= u 5.1e+120)))
   (/ v (* u (/ u t1)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+51) || !(u <= 5.1e+120)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.75d+51)) .or. (.not. (u <= 5.1d+120))) then
        tmp = v / (u * (u / t1))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+51) || !(u <= 5.1e+120)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.75e+51) or not (u <= 5.1e+120):
		tmp = v / (u * (u / t1))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.75e+51) || !(u <= 5.1e+120))
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.75e+51) || ~((u <= 5.1e+120)))
		tmp = v / (u * (u / t1));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.75e+51], N[Not[LessEqual[u, 5.1e+120]], $MachinePrecision]], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.75e51 or 5.10000000000000027e120 < u

   1. Initial program 79.4%

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

      1. times-frac 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in t1 around 0 84.7%

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

   5. Taylor expanded in t1 around 0 83.8%

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

      1. mul-1-neg 83.8%

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

      2. distribute-neg-frac 83.8%

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

   7. Simplified 83.8%

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

      1. clear-num 83.1%

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

      2. frac-times 70.9%

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

      3. *-un-lft-identity 70.9%

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

      4. add-sqr-sqrt 31.9%

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

      5. sqrt-unprod 64.9%

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

      6. sqr-neg 64.9%

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

      7. sqrt-unprod 36.8%

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

      8. add-sqr-sqrt 59.5%

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

   9. Applied egg-rr 59.5%

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

   if -1.75e51 < u < 5.10000000000000027e120

   1. Initial program 69.9%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in t1 around inf 72.8%

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

      1. associate-*r/ 72.8%

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

      2. neg-mul-1 72.8%

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

   6. Simplified 72.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+51} \lor \neg \left(u \leq 5.1 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 7: 61.7% 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 73.3%

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

   1. times-frac 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in t1 around inf 59.5%

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

5. Final simplification 59.5%

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

Alternative 8: 54.3% accurate, 3.0× 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(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 73.3%

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

   1. times-frac 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in t1 around inf 55.0%

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

   1. associate-*r/ 55.0%

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

   2. neg-mul-1 55.0%

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

6. Simplified 55.0%

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

7. Final simplification 55.0%

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

Alternative 9: 16.9% 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 73.3%

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

   1. times-frac 96.9%

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

3. Simplified 96.9%

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

   1. frac-2neg 96.9%

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

   2. remove-double-neg 96.9%

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

   3. associate-*l/ 97.6%

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

   4. distribute-neg-in 97.6%

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

   5. add-sqr-sqrt 48.7%

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

   6. sqrt-unprod 69.2%

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

   7. sqr-neg 69.2%

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

   8. sqrt-unprod 28.0%

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

   9. add-sqr-sqrt 55.0%

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

   10. sub-neg 55.0%

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

5. Applied egg-rr 55.0%

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

   1. associate-*r/ 52.6%

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

7. Simplified 52.6%

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

8. Taylor expanded in t1 around 0 47.4%

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

9. Taylor expanded in t1 around inf 13.6%

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

10. Final simplification 13.6%

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

Reproduce

herbie shell --seed 2023306 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))