Rosa's DopplerBench

Percentage Accurate: 72.5% → 97.9%

Time: 9.8s

Alternatives: 11

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: 72.5% 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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. times-frac 98.9%

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

   2. distribute-frac-neg 98.9%

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

   3. distribute-neg-frac2 98.9%

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

   4. +-commutative 98.9%

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

   5. distribute-neg-in 98.9%

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

   6. unsub-neg 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{-44} \lor \neg \left(u \leq 1.1 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3e-44) (not (<= u 1.1e-33)))
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e-44) || !(u <= 1.1e-33)) {
		tmp = (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 <= (-3d-44)) .or. (.not. (u <= 1.1d-33))) then
        tmp = (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 <= -3e-44) || !(u <= 1.1e-33)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3e-44) or not (u <= 1.1e-33):
		tmp = (v / (t1 + u)) * (t1 / -u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3e-44) || !(u <= 1.1e-33))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3e-44) || ~((u <= 1.1e-33)))
		tmp = (v / (t1 + u)) * (t1 / -u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3e-44], N[Not[LessEqual[u, 1.1e-33]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.0000000000000002e-44 or 1.10000000000000003e-33 < u

   1. Initial program 81.6%

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

      1. times-frac 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-neg-frac2 99.4%

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

      4. +-commutative 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t1 around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. mul-1-neg 81.9%

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

   7. Simplified 81.9%

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

   if -3.0000000000000002e-44 < u < 1.10000000000000003e-33

   1. Initial program 63.0%

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

      1. times-frac 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around inf 80.7%

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

      1. associate-*r/ 80.7%

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

      2. neg-mul-1 80.7%

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

   7. Simplified 80.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{-44} \lor \neg \left(u \leq 1.1 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{-50} \lor \neg \left(u \leq 4 \cdot 10^{-34}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.3e-50) (not (<= u 4e-34)))
   (* t1 (/ (/ v (+ t1 u)) (- u)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.3e-50) || !(u <= 4e-34)) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.3d-50)) .or. (.not. (u <= 4d-34))) then
        tmp = t1 * ((v / (t1 + u)) / -u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.3e-50) || !(u <= 4e-34)) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.3e-50) or not (u <= 4e-34):
		tmp = t1 * ((v / (t1 + u)) / -u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.3e-50) || !(u <= 4e-34))
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(-u)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.3e-50) || ~((u <= 4e-34)))
		tmp = t1 * ((v / (t1 + u)) / -u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.3e-50], N[Not[LessEqual[u, 4e-34]], $MachinePrecision]], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.29999999999999997e-50 or 3.99999999999999971e-34 < u

   1. Initial program 81.6%

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

      1. associate-/l* 83.4%

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

      2. distribute-lft-neg-out 83.4%

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

      3. distribute-rgt-neg-in 83.4%

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

      4. associate-/r* 92.5%

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

      5. distribute-neg-frac2 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in t1 around 0 79.8%

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

   if -4.29999999999999997e-50 < u < 3.99999999999999971e-34

   1. Initial program 63.0%

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

      1. times-frac 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around inf 80.7%

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

      1. associate-*r/ 80.7%

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

      2. neg-mul-1 80.7%

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

   7. Simplified 80.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{-50} \lor \neg \left(u \leq 4 \cdot 10^{-34}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+18} \lor \neg \left(t1 \leq 1.95 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.6e+18) (not (<= t1 1.95e-119)))
   (/ v (- u t1))
   (* t1 (/ (/ v u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.6e+18) || !(t1 <= 1.95e-119)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.6d+18)) .or. (.not. (t1 <= 1.95d-119))) then
        tmp = v / (u - t1)
    else
        tmp = t1 * ((v / u) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.6e+18) || !(t1 <= 1.95e-119)) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.6e+18) or not (t1 <= 1.95e-119):
		tmp = v / (u - t1)
	else:
		tmp = t1 * ((v / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.6e+18) || !(t1 <= 1.95e-119))
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.6e+18) || ~((t1 <= 1.95e-119)))
		tmp = v / (u - t1);
	else
		tmp = t1 * ((v / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.6e+18], N[Not[LessEqual[t1, 1.95e-119]], $MachinePrecision]], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.6e18 or 1.94999999999999995e-119 < t1

   1. Initial program 68.4%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 83.3%

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

      1. add-sqr-sqrt 39.4%

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

      2. add-sqr-sqrt 22.8%

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

      3. difference-of-squares 22.8%

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

      4. add-sqr-sqrt 22.8%

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

      5. sqrt-unprod 25.5%

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

      6. sqr-neg 25.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 22.2%

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

      11. sqr-neg 22.2%

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

      12. sqrt-unprod 22.1%

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

      13. add-sqr-sqrt 22.1%

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

   7. Applied egg-rr 22.1%

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

      1. difference-of-squares 22.1%

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

      2. rem-square-sqrt 45.2%

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

      3. rem-square-sqrt 83.5%

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

   9. Simplified 83.5%

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

   10. Taylor expanded in v around 0 83.5%

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

   if -2.6e18 < t1 < 1.94999999999999995e-119

   1. Initial program 77.7%

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

      1. associate-/l* 83.0%

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

      2. distribute-lft-neg-out 83.0%

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

      3. distribute-rgt-neg-in 83.0%

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

      4. associate-/r* 89.9%

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

      5. distribute-neg-frac2 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in t1 around 0 72.8%

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

   6. Taylor expanded in t1 around 0 74.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+18} \lor \neg \left(t1 \leq 1.95 \cdot 10^{-119}\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+181} \lor \neg \left(u \leq 6.8 \cdot 10^{+202}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.35e+181) (not (<= u 6.8e+202)))
   (* t1 (/ (/ v u) u))
   (/ v (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.35e+181) || !(u <= 6.8e+202)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.35d+181)) .or. (.not. (u <= 6.8d+202))) then
        tmp = t1 * ((v / u) / u)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.35e+181) || !(u <= 6.8e+202)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.35e+181) or not (u <= 6.8e+202):
		tmp = t1 * ((v / u) / u)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.35e+181) || !(u <= 6.8e+202))
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.35e+181) || ~((u <= 6.8e+202)))
		tmp = t1 * ((v / u) / u);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.35e+181], N[Not[LessEqual[u, 6.8e+202]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.35000000000000004e181 or 6.8e202 < u

   1. Initial program 84.2%

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

      1. associate-*l/ 84.5%

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

      2. *-commutative 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in t1 around 0 84.5%

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

   6. Taylor expanded in t1 around 0 84.5%

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

      1. associate-*r/ 84.2%

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

      2. associate-/r* 94.2%

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

      3. *-commutative 94.2%

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

      4. add-sqr-sqrt 44.2%

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

      5. sqrt-unprod 76.9%

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

      6. sqr-neg 76.9%

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

      7. sqrt-unprod 49.3%

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

      8. add-sqr-sqrt 84.1%

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

   8. Applied egg-rr 84.1%

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

      1. associate-/l* 84.2%

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

      2. associate-/l* 84.4%

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

   10. Applied egg-rr 84.4%

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

   if -1.35000000000000004e181 < u < 6.8e202

   1. Initial program 70.1%

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

      1. times-frac 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t1 around inf 63.6%

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

      1. add-sqr-sqrt 29.3%

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

      2. add-sqr-sqrt 15.2%

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

      3. difference-of-squares 15.2%

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

      4. add-sqr-sqrt 15.2%

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

      5. sqrt-unprod 15.2%

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

      6. sqr-neg 15.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 14.1%

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

      11. sqr-neg 14.1%

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

      12. sqrt-unprod 13.7%

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

      13. add-sqr-sqrt 13.7%

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

   7. Applied egg-rr 13.7%

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

      1. difference-of-squares 13.7%

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

      2. rem-square-sqrt 29.3%

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

      3. rem-square-sqrt 64.8%

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

   9. Simplified 64.8%

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

   10. Taylor expanded in v around 0 64.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+181} \lor \neg \left(u \leq 6.8 \cdot 10^{+202}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e+198) (/ v (- u t1)) (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+198) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.2d+198)) then
        tmp = v / (u - t1)
    else
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+198) {
		tmp = v / (u - t1);
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e+198:
		tmp = v / (u - t1)
	else:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e+198)
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e+198)
		tmp = v / (u - t1);
	else
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e+198], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.2000000000000001e198

   1. Initial program 34.2%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 96.4%

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

      1. add-sqr-sqrt 30.8%

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

      2. add-sqr-sqrt 0.0%

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

      3. difference-of-squares 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. sqr-neg 0.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. difference-of-squares 0.0%

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

      2. rem-square-sqrt 0.0%

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

      3. rem-square-sqrt 96.6%

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

   9. Simplified 96.6%

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

   10. Taylor expanded in v around 0 96.6%

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

   if -1.2000000000000001e198 < t1

   1. Initial program 76.9%

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

      1. associate-/l* 81.9%

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

      2. distribute-lft-neg-out 81.9%

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

      3. distribute-rgt-neg-in 81.9%

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

      4. associate-/r* 92.6%

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

      5. distribute-neg-frac2 92.6%

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

   3. Simplified 92.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+181} \lor \neg \left(u \leq 4.8 \cdot 10^{+199}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4e+181) (not (<= u 4.8e+199))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4e+181) || !(u <= 4.8e+199)) {
		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 <= (-4d+181)) .or. (.not. (u <= 4.8d+199))) 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 <= -4e+181) || !(u <= 4.8e+199)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4e+181) or not (u <= 4.8e+199):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4e+181) || !(u <= 4.8e+199))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4e+181) || ~((u <= 4.8e+199)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4e+181], N[Not[LessEqual[u, 4.8e+199]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.9999999999999997e181 or 4.8000000000000003e199 < u

   1. Initial program 84.6%

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

      1. times-frac 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-neg-frac2 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t1 around inf 55.2%

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

      1. add-sqr-sqrt 36.8%

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

      2. add-sqr-sqrt 20.4%

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

      3. difference-of-squares 20.4%

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

      4. add-sqr-sqrt 20.4%

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

      5. sqrt-unprod 28.9%

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

      6. sqr-neg 28.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 11.3%

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

      11. sqr-neg 11.3%

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

      12. sqrt-unprod 11.2%

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

      13. add-sqr-sqrt 11.2%

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

   7. Applied egg-rr 11.2%

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

      1. difference-of-squares 11.2%

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

      2. rem-square-sqrt 31.6%

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

      3. rem-square-sqrt 55.3%

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

   9. Simplified 55.3%

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

   10. Taylor expanded in t1 around 0 43.2%

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

   if -3.9999999999999997e181 < u < 4.8000000000000003e199

   1. Initial program 70.0%

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

      1. times-frac 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t1 around inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+181} \lor \neg \left(u \leq 4.8 \cdot 10^{+199}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+181}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 9.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.45e+181) (/ v (- u)) (if (<= u 9.2e+198) (/ v (- t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.45e+181) {
		tmp = v / -u;
	} else if (u <= 9.2e+198) {
		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 <= (-1.45d+181)) then
        tmp = v / -u
    else if (u <= 9.2d+198) 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 <= -1.45e+181) {
		tmp = v / -u;
	} else if (u <= 9.2e+198) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.45e+181:
		tmp = v / -u
	elif u <= 9.2e+198:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.45e+181)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 9.2e+198)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.45e+181)
		tmp = v / -u;
	elseif (u <= 9.2e+198)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.45e+181], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 9.2e+198], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.45e181

   1. Initial program 86.9%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 41.3%

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

   6. Taylor expanded in t1 around 0 38.2%

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

      1. associate-*r/ 38.2%

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

      2. mul-1-neg 38.2%

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

   8. Simplified 38.2%

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

   if -1.45e181 < u < 9.2000000000000002e198

   1. Initial program 70.3%

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

      1. times-frac 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t1 around inf 62.9%

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

      1. associate-*r/ 62.9%

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

      2. neg-mul-1 62.9%

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

   7. Simplified 62.9%

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

   if 9.2000000000000002e198 < u

   1. Initial program 75.9%

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

      1. times-frac 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. distribute-neg-frac2 98.1%

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

      4. +-commutative 98.1%

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

      5. distribute-neg-in 98.1%

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

      6. unsub-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in t1 around inf 48.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. difference-of-squares 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 29.8%

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

      11. sqr-neg 29.8%

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

      12. sqrt-unprod 29.6%

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

      13. add-sqr-sqrt 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. difference-of-squares 29.6%

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

      2. rem-square-sqrt 29.6%

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

      3. rem-square-sqrt 49.3%

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

   9. Simplified 49.3%

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

   10. Taylor expanded in t1 around 0 50.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+181}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 9.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 22.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+205} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.4e+205) (not (<= t1 7.5e+76))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.4e+205) || !(t1 <= 7.5e+76)) {
		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 <= (-7.4d+205)) .or. (.not. (t1 <= 7.5d+76))) 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 <= -7.4e+205) || !(t1 <= 7.5e+76)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7.4e+205) or not (t1 <= 7.5e+76):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.4e+205) || !(t1 <= 7.5e+76))
		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 <= -7.4e+205) || ~((t1 <= 7.5e+76)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7.4e+205], N[Not[LessEqual[t1, 7.5e+76]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.39999999999999961e205 or 7.4999999999999995e76 < t1

   1. Initial program 54.9%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 97.1%

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

      1. associate-*r/ 97.1%

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

      2. neg-mul-1 97.1%

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

   7. Simplified 97.1%

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

      1. distribute-frac-neg 97.1%

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

      2. div-inv 96.8%

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

      3. distribute-rgt-neg-in 96.8%

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

      4. frac-2neg 96.8%

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

      5. metadata-eval 96.8%

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

      6. add-sqr-sqrt 37.0%

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

      7. sqrt-unprod 40.5%

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

      8. sqr-neg 40.5%

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

      9. sqrt-unprod 25.0%

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

      10. add-sqr-sqrt 39.7%

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

   9. Applied egg-rr 39.7%

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

       1. distribute-rgt-neg-out 39.7%

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

       2. *-commutative 39.7%

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

       3. associate-*l/ 39.7%

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

       4. mul-1-neg 39.7%

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

       5. distribute-neg-frac 39.7%

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

       6. remove-double-neg 39.7%

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

   11. Simplified 39.7%

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

   if -7.39999999999999961e205 < t1 < 7.4999999999999995e76

   1. Initial program 78.2%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around inf 50.7%

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

      1. add-sqr-sqrt 26.6%

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

      2. add-sqr-sqrt 11.1%

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

      3. difference-of-squares 11.1%

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

      4. add-sqr-sqrt 11.1%

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

      5. sqrt-unprod 13.0%

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

      6. sqr-neg 13.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 9.2%

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

      11. sqr-neg 9.2%

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

      12. sqrt-unprod 8.8%

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

      13. add-sqr-sqrt 8.8%

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

   7. Applied egg-rr 8.8%

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

      1. difference-of-squares 8.8%

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

      2. rem-square-sqrt 20.3%

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

      3. rem-square-sqrt 52.1%

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

   9. Simplified 52.1%

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

   10. Taylor expanded in t1 around 0 15.4%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+205} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. times-frac 98.9%

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

   2. distribute-frac-neg 98.9%

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

   3. distribute-neg-frac2 98.9%

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

   4. +-commutative 98.9%

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

   5. distribute-neg-in 98.9%

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

   6. unsub-neg 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in t1 around inf 61.9%

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

   1. add-sqr-sqrt 30.7%

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

   2. add-sqr-sqrt 16.2%

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

   3. difference-of-squares 16.2%

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

   4. add-sqr-sqrt 16.2%

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

   5. sqrt-unprod 17.7%

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

   6. sqr-neg 17.7%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 0.0%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 13.6%

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

   11. sqr-neg 13.6%

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

   12. sqrt-unprod 13.3%

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

   13. add-sqr-sqrt 13.3%

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

7. Applied egg-rr 13.3%

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

   1. difference-of-squares 13.3%

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

   2. rem-square-sqrt 29.8%

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

   3. rem-square-sqrt 63.0%

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

9. Simplified 63.0%

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

10. Taylor expanded in v around 0 60.9%

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

Alternative 11: 14.1% 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 72.5%

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

   1. times-frac 98.9%

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

   2. distribute-frac-neg 98.9%

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

   3. distribute-neg-frac2 98.9%

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

   4. +-commutative 98.9%

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

   5. distribute-neg-in 98.9%

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

   6. unsub-neg 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in t1 around inf 54.2%

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

   1. associate-*r/ 54.2%

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

   2. neg-mul-1 54.2%

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

7. Simplified 54.2%

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

   1. distribute-frac-neg 54.2%

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

   2. div-inv 54.0%

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

   3. distribute-rgt-neg-in 54.0%

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

   4. frac-2neg 54.0%

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

   5. metadata-eval 54.0%

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

   6. add-sqr-sqrt 28.7%

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

   7. sqrt-unprod 27.6%

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

   8. sqr-neg 27.6%

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

   9. sqrt-unprod 7.5%

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

   10. add-sqr-sqrt 12.6%

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

9. Applied egg-rr 12.6%

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

    1. distribute-rgt-neg-out 12.6%

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

    2. *-commutative 12.6%

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

    3. associate-*l/ 12.6%

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

    4. mul-1-neg 12.6%

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

    5. distribute-neg-frac 12.6%

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

    6. remove-double-neg 12.6%

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

11. Simplified 12.6%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))