Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.1%

Time: 12.3s

Alternatives: 15

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: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. times-frac 98.4%

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

   2. distribute-frac-neg 98.4%

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

   3. distribute-neg-frac2 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-neg-in 98.4%

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

   6. unsub-neg 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -4.1 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq -8 \cdot 10^{+40}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq -3800000000000 \lor \neg \left(u \leq 1.3 \cdot 10^{+23}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ t1 (- u)) (/ v u))))
   (if (<= u -4.1e+126)
     t_1
     (if (<= u -8e+40)
       (/ v (- (- u) t1))
       (if (or (<= u -3800000000000.0) (not (<= u 1.3e+23)))
         t_1
         (/ v (- t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 / -u) * (v / u);
	double tmp;
	if (u <= -4.1e+126) {
		tmp = t_1;
	} else if (u <= -8e+40) {
		tmp = v / (-u - t1);
	} else if ((u <= -3800000000000.0) || !(u <= 1.3e+23)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (t1 / -u) * (v / u)
    if (u <= (-4.1d+126)) then
        tmp = t_1
    else if (u <= (-8d+40)) then
        tmp = v / (-u - t1)
    else if ((u <= (-3800000000000.0d0)) .or. (.not. (u <= 1.3d+23))) then
        tmp = t_1
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 / -u) * (v / u);
	double tmp;
	if (u <= -4.1e+126) {
		tmp = t_1;
	} else if (u <= -8e+40) {
		tmp = v / (-u - t1);
	} else if ((u <= -3800000000000.0) || !(u <= 1.3e+23)) {
		tmp = t_1;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 / -u) * (v / u)
	tmp = 0
	if u <= -4.1e+126:
		tmp = t_1
	elif u <= -8e+40:
		tmp = v / (-u - t1)
	elif (u <= -3800000000000.0) or not (u <= 1.3e+23):
		tmp = t_1
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 / Float64(-u)) * Float64(v / u))
	tmp = 0.0
	if (u <= -4.1e+126)
		tmp = t_1;
	elseif (u <= -8e+40)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif ((u <= -3800000000000.0) || !(u <= 1.3e+23))
		tmp = t_1;
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 / -u) * (v / u);
	tmp = 0.0;
	if (u <= -4.1e+126)
		tmp = t_1;
	elseif (u <= -8e+40)
		tmp = v / (-u - t1);
	elseif ((u <= -3800000000000.0) || ~((u <= 1.3e+23)))
		tmp = t_1;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.1e+126], t$95$1, If[LessEqual[u, -8e+40], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -3800000000000.0], N[Not[LessEqual[u, 1.3e+23]], $MachinePrecision]], t$95$1, N[(v / (-t1)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.1000000000000001e126 or -8.00000000000000024e40 < u < -3.8e12 or 1.29999999999999996e23 < u

   1. Initial program 72.2%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around 0 87.8%

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

      1. mul-1-neg 87.8%

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

      2. distribute-neg-frac2 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in t1 around 0 85.9%

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

   if -4.1000000000000001e126 < u < -8.00000000000000024e40

   1. Initial program 59.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 76.9%

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

   if -3.8e12 < u < 1.29999999999999996e23

   1. Initial program 65.6%

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

      1. associate-/l* 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

      4. associate-/r* 80.5%

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

      5. distribute-neg-frac2 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in t1 around inf 82.9%

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

      1. associate-*r/ 82.9%

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

      2. neg-mul-1 82.9%

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

   7. Simplified 82.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+126}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -8 \cdot 10^{+40}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq -3800000000000 \lor \neg \left(u \leq 1.3 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -2.7 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq -9.6 \cdot 10^{+40}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;u \leq -410000000000 \lor \neg \left(u \leq 2.3 \cdot 10^{+22}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ t1 (- u)) (/ v u))))
   (if (<= u -2.7e+122)
     t_1
     (if (<= u -9.6e+40)
       (/ v (- (- t1) (* u 2.0)))
       (if (or (<= u -410000000000.0) (not (<= u 2.3e+22)))
         t_1
         (/ v (- t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 / -u) * (v / u);
	double tmp;
	if (u <= -2.7e+122) {
		tmp = t_1;
	} else if (u <= -9.6e+40) {
		tmp = v / (-t1 - (u * 2.0));
	} else if ((u <= -410000000000.0) || !(u <= 2.3e+22)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (t1 / -u) * (v / u)
    if (u <= (-2.7d+122)) then
        tmp = t_1
    else if (u <= (-9.6d+40)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if ((u <= (-410000000000.0d0)) .or. (.not. (u <= 2.3d+22))) then
        tmp = t_1
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 / -u) * (v / u);
	double tmp;
	if (u <= -2.7e+122) {
		tmp = t_1;
	} else if (u <= -9.6e+40) {
		tmp = v / (-t1 - (u * 2.0));
	} else if ((u <= -410000000000.0) || !(u <= 2.3e+22)) {
		tmp = t_1;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 / -u) * (v / u)
	tmp = 0
	if u <= -2.7e+122:
		tmp = t_1
	elif u <= -9.6e+40:
		tmp = v / (-t1 - (u * 2.0))
	elif (u <= -410000000000.0) or not (u <= 2.3e+22):
		tmp = t_1
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 / Float64(-u)) * Float64(v / u))
	tmp = 0.0
	if (u <= -2.7e+122)
		tmp = t_1;
	elseif (u <= -9.6e+40)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif ((u <= -410000000000.0) || !(u <= 2.3e+22))
		tmp = t_1;
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 / -u) * (v / u);
	tmp = 0.0;
	if (u <= -2.7e+122)
		tmp = t_1;
	elseif (u <= -9.6e+40)
		tmp = v / (-t1 - (u * 2.0));
	elseif ((u <= -410000000000.0) || ~((u <= 2.3e+22)))
		tmp = t_1;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.7e+122], t$95$1, If[LessEqual[u, -9.6e+40], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -410000000000.0], N[Not[LessEqual[u, 2.3e+22]], $MachinePrecision]], t$95$1, N[(v / (-t1)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.6999999999999998e122 or -9.5999999999999999e40 < u < -4.1e11 or 2.3000000000000002e22 < u

   1. Initial program 72.2%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around 0 87.8%

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

      1. mul-1-neg 87.8%

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

      2. distribute-neg-frac2 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in t1 around 0 85.9%

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

   if -2.6999999999999998e122 < u < -9.5999999999999999e40

   1. Initial program 59.2%

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

      1. associate-/l* 58.5%

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

      2. distribute-lft-neg-out 58.5%

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

      3. distribute-rgt-neg-in 58.5%

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

      4. associate-/r* 74.5%

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

      5. distribute-neg-frac2 74.5%

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

   3. Simplified 74.5%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-neg-in 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.7%

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

      7. frac-2neg 99.7%

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

      8. frac-times 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. sub-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 76.9%

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

      1. *-commutative 76.9%

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

   9. Simplified 76.9%

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

   if -4.1e11 < u < 2.3000000000000002e22

   1. Initial program 65.6%

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

      1. associate-/l* 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

      4. associate-/r* 80.5%

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

      5. distribute-neg-frac2 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in t1 around inf 82.9%

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

      1. associate-*r/ 82.9%

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

      2. neg-mul-1 82.9%

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

   7. Simplified 82.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+122}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -9.6 \cdot 10^{+40}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;u \leq -410000000000 \lor \neg \left(u \leq 2.3 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -2.7 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq -1.32 \cdot 10^{+42}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;u \leq -19000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+21}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ t1 (- u)) (/ v u))))
   (if (<= u -2.7e+122)
     t_1
     (if (<= u -1.32e+42)
       (/ v (- (- t1) (* u 2.0)))
       (if (<= u -19000000000.0)
         t_1
         (if (<= u 9e+21) (/ v (- t1)) (/ (* t1 (/ v (- u))) u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 / -u) * (v / u);
	double tmp;
	if (u <= -2.7e+122) {
		tmp = t_1;
	} else if (u <= -1.32e+42) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (u <= -19000000000.0) {
		tmp = t_1;
	} else if (u <= 9e+21) {
		tmp = v / -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) :: t_1
    real(8) :: tmp
    t_1 = (t1 / -u) * (v / u)
    if (u <= (-2.7d+122)) then
        tmp = t_1
    else if (u <= (-1.32d+42)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (u <= (-19000000000.0d0)) then
        tmp = t_1
    else if (u <= 9d+21) then
        tmp = v / -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 t_1 = (t1 / -u) * (v / u);
	double tmp;
	if (u <= -2.7e+122) {
		tmp = t_1;
	} else if (u <= -1.32e+42) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (u <= -19000000000.0) {
		tmp = t_1;
	} else if (u <= 9e+21) {
		tmp = v / -t1;
	} else {
		tmp = (t1 * (v / -u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 / -u) * (v / u)
	tmp = 0
	if u <= -2.7e+122:
		tmp = t_1
	elif u <= -1.32e+42:
		tmp = v / (-t1 - (u * 2.0))
	elif u <= -19000000000.0:
		tmp = t_1
	elif u <= 9e+21:
		tmp = v / -t1
	else:
		tmp = (t1 * (v / -u)) / u
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 / Float64(-u)) * Float64(v / u))
	tmp = 0.0
	if (u <= -2.7e+122)
		tmp = t_1;
	elseif (u <= -1.32e+42)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (u <= -19000000000.0)
		tmp = t_1;
	elseif (u <= 9e+21)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 / -u) * (v / u);
	tmp = 0.0;
	if (u <= -2.7e+122)
		tmp = t_1;
	elseif (u <= -1.32e+42)
		tmp = v / (-t1 - (u * 2.0));
	elseif (u <= -19000000000.0)
		tmp = t_1;
	elseif (u <= 9e+21)
		tmp = v / -t1;
	else
		tmp = (t1 * (v / -u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.7e+122], t$95$1, If[LessEqual[u, -1.32e+42], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -19000000000.0], t$95$1, If[LessEqual[u, 9e+21], N[(v / (-t1)), $MachinePrecision], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -2.6999999999999998e122 or -1.32e42 < u < -1.9e10

   1. Initial program 69.0%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 97.0%

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

      1. mul-1-neg 97.0%

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

      2. distribute-neg-frac2 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in t1 around 0 97.0%

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

   if -2.6999999999999998e122 < u < -1.32e42

   1. Initial program 59.2%

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

      1. associate-/l* 58.5%

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

      2. distribute-lft-neg-out 58.5%

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

      3. distribute-rgt-neg-in 58.5%

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

      4. associate-/r* 74.5%

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

      5. distribute-neg-frac2 74.5%

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

   3. Simplified 74.5%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-neg-in 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.7%

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

      7. frac-2neg 99.7%

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

      8. frac-times 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. sub-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 76.9%

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

      1. *-commutative 76.9%

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

   9. Simplified 76.9%

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

   if -1.9e10 < u < 9e21

   1. Initial program 65.6%

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

      1. associate-/l* 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

      4. associate-/r* 80.5%

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

      5. distribute-neg-frac2 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in t1 around inf 82.9%

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

      1. associate-*r/ 82.9%

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

      2. neg-mul-1 82.9%

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

   7. Simplified 82.9%

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

   if 9e21 < u

   1. Initial program 74.9%

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

      1. times-frac 96.6%

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

      2. distribute-frac-neg 96.6%

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

      3. distribute-neg-frac2 96.6%

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

      4. +-commutative 96.6%

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

      5. distribute-neg-in 96.6%

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

      6. unsub-neg 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around 0 80.2%

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

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac2 80.2%

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

   7. Simplified 80.2%

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

      1. distribute-frac-neg2 80.2%

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

      2. distribute-frac-neg 80.2%

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

      3. associate-*l/ 83.4%

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

      4. frac-2neg 83.4%

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

      5. add-sqr-sqrt 43.5%

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

      6. sqrt-unprod 57.7%

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

      7. sqr-neg 57.7%

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

      8. sqrt-unprod 29.6%

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

      9. add-sqr-sqrt 59.7%

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

      10. +-commutative 59.7%

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

      11. distribute-neg-in 59.7%

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

      12. sub-neg 59.7%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 73.2%

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

      15. sqr-neg 73.2%

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

      16. sqrt-unprod 83.2%

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

      17. add-sqr-sqrt 83.2%

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

   9. Applied egg-rr 83.2%

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

   10. Taylor expanded in t1 around 0 74.8%

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

       1. mul-1-neg 74.8%

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

       2. associate-*r/ 80.0%

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

       3. distribute-rgt-neg-in 80.0%

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

       4. distribute-neg-frac2 80.0%

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

   12. Simplified 80.0%

       \[\leadsto \frac{\color{blue}{t1 \cdot \frac{v}{-u}}}{u} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+122}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -1.32 \cdot 10^{+42}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;u \leq -19000000000:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+21}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.5e+173)
   (/ v (- t1))
   (if (<= t1 1.85e+151)
     (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))
     (/ v (- (- t1) (* u 2.0))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.5e+173) {
		tmp = v / -t1;
	} else if (t1 <= 1.85e+151) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-4.5d+173)) then
        tmp = v / -t1
    else if (t1 <= 1.85d+151) then
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.5e+173) {
		tmp = v / -t1;
	} else if (t1 <= 1.85e+151) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4.5e+173:
		tmp = v / -t1
	elif t1 <= 1.85e+151:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.5e+173)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 1.85e+151)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4.5e+173)
		tmp = v / -t1;
	elseif (t1 <= 1.85e+151)
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.5e+173], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 1.85e+151], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.5000000000000002e173

   1. Initial program 27.6%

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

      1. associate-/l* 30.2%

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

      2. distribute-lft-neg-out 30.2%

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

      3. distribute-rgt-neg-in 30.2%

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

      4. associate-/r* 64.4%

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

      5. distribute-neg-frac2 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in t1 around inf 96.0%

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

      1. associate-*r/ 96.0%

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

      2. neg-mul-1 96.0%

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

   7. Simplified 96.0%

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

   if -4.5000000000000002e173 < t1 < 1.8499999999999999e151

   1. Initial program 77.3%

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

      1. associate-/l* 78.9%

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

      2. distribute-lft-neg-out 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. associate-/r* 90.0%

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

      5. distribute-neg-frac2 90.0%

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

   3. Simplified 90.0%

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

   if 1.8499999999999999e151 < t1

   1. Initial program 42.8%

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

      1. associate-/l* 44.3%

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

      2. distribute-lft-neg-out 44.3%

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

      3. distribute-rgt-neg-in 44.3%

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

      4. associate-/r* 57.1%

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

      5. distribute-neg-frac2 57.1%

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

   3. Simplified 57.1%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. sub-neg 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 92.4%

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

      1. *-commutative 92.4%

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

   9. Simplified 92.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;u \leq -98000000000:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t\_1}\\ \mathbf{elif}\;u \leq 4.65 \cdot 10^{+21}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= u -98000000000.0)
     (/ (* v (/ t1 u)) t_1)
     (if (<= u 4.65e+21) (/ v (- t1)) (* t1 (/ (/ v u) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (u <= -98000000000.0) {
		tmp = (v * (t1 / u)) / t_1;
	} else if (u <= 4.65e+21) {
		tmp = v / -t1;
	} else {
		tmp = t1 * ((v / u) / t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = -u - t1
    if (u <= (-98000000000.0d0)) then
        tmp = (v * (t1 / u)) / t_1
    else if (u <= 4.65d+21) then
        tmp = v / -t1
    else
        tmp = t1 * ((v / u) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (u <= -98000000000.0) {
		tmp = (v * (t1 / u)) / t_1;
	} else if (u <= 4.65e+21) {
		tmp = v / -t1;
	} else {
		tmp = t1 * ((v / u) / t_1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if u <= -98000000000.0:
		tmp = (v * (t1 / u)) / t_1
	elif u <= 4.65e+21:
		tmp = v / -t1
	else:
		tmp = t1 * ((v / u) / t_1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (u <= -98000000000.0)
		tmp = Float64(Float64(v * Float64(t1 / u)) / t_1);
	elseif (u <= 4.65e+21)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (u <= -98000000000.0)
		tmp = (v * (t1 / u)) / t_1;
	elseif (u <= 4.65e+21)
		tmp = v / -t1;
	else
		tmp = t1 * ((v / u) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[u, -98000000000.0], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[u, 4.65e+21], N[(v / (-t1)), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if u < -9.8e10

   1. Initial program 67.1%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. distribute-neg-frac2 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in v around 0 65.1%

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

      1. mul-1-neg 65.1%

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

      2. associate-/r* 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r/ 86.5%

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

   10. Simplified 86.5%

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

   if -9.8e10 < u < 4.65e21

   1. Initial program 65.6%

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

      1. associate-/l* 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

      4. associate-/r* 80.5%

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

      5. distribute-neg-frac2 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in t1 around inf 82.9%

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

      1. associate-*r/ 82.9%

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

      2. neg-mul-1 82.9%

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

   7. Simplified 82.9%

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

   if 4.65e21 < u

   1. Initial program 74.9%

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

      1. associate-/l* 75.5%

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

      2. distribute-lft-neg-out 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

      4. associate-/r* 85.7%

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

      5. distribute-neg-frac2 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in t1 around 0 81.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -98000000000:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 4.65 \cdot 10^{+21}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{-24}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 1.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6e-24)
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (if (<= u 1.9e+22) (/ v (- t1)) (* t1 (/ (/ v u) (- (- u) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e-24) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else if (u <= 1.9e+22) {
		tmp = v / -t1;
	} else {
		tmp = t1 * ((v / 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 (u <= (-6d-24)) then
        tmp = (v / (t1 + u)) * (t1 / -u)
    else if (u <= 1.9d+22) then
        tmp = v / -t1
    else
        tmp = t1 * ((v / u) / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e-24) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else if (u <= 1.9e+22) {
		tmp = v / -t1;
	} else {
		tmp = t1 * ((v / u) / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6e-24:
		tmp = (v / (t1 + u)) * (t1 / -u)
	elif u <= 1.9e+22:
		tmp = v / -t1
	else:
		tmp = t1 * ((v / u) / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6e-24)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	elseif (u <= 1.9e+22)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6e-24)
		tmp = (v / (t1 + u)) * (t1 / -u);
	elseif (u <= 1.9e+22)
		tmp = v / -t1;
	else
		tmp = t1 * ((v / u) / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6e-24], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.9e+22], N[(v / (-t1)), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.99999999999999991e-24

   1. Initial program 68.6%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 90.0%

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

      1. mul-1-neg 90.0%

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

      2. distribute-neg-frac2 90.0%

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

   7. Simplified 90.0%

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

   if -5.99999999999999991e-24 < u < 1.9000000000000002e22

   1. Initial program 64.8%

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

      1. associate-/l* 68.4%

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

      2. distribute-lft-neg-out 68.4%

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

      3. distribute-rgt-neg-in 68.4%

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

      4. associate-/r* 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in t1 around inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. neg-mul-1 83.2%

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

   7. Simplified 83.2%

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

   if 1.9000000000000002e22 < u

   1. Initial program 74.9%

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

      1. associate-/l* 75.5%

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

      2. distribute-lft-neg-out 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

      4. associate-/r* 85.7%

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

      5. distribute-neg-frac2 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in t1 around 0 81.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{-24}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 1.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 1.46 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 - u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.3e-35)
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (if (<= u 1.46e+22) (/ v (- t1)) (/ (* t1 (/ v (- t1 u))) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e-35) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else if (u <= 1.46e+22) {
		tmp = v / -t1;
	} else {
		tmp = (t1 * (v / (t1 - u))) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.3d-35)) then
        tmp = (v / (t1 + u)) * (t1 / -u)
    else if (u <= 1.46d+22) then
        tmp = v / -t1
    else
        tmp = (t1 * (v / (t1 - u))) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e-35) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else if (u <= 1.46e+22) {
		tmp = v / -t1;
	} else {
		tmp = (t1 * (v / (t1 - u))) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.3e-35:
		tmp = (v / (t1 + u)) * (t1 / -u)
	elif u <= 1.46e+22:
		tmp = v / -t1
	else:
		tmp = (t1 * (v / (t1 - u))) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.3e-35)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	elseif (u <= 1.46e+22)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 - u))) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.3e-35)
		tmp = (v / (t1 + u)) * (t1 / -u);
	elseif (u <= 1.46e+22)
		tmp = v / -t1;
	else
		tmp = (t1 * (v / (t1 - u))) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.3e-35], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.46e+22], N[(v / (-t1)), $MachinePrecision], N[(N[(t1 * N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.30000000000000002e-35

   1. Initial program 68.6%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 90.0%

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

      1. mul-1-neg 90.0%

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

      2. distribute-neg-frac2 90.0%

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

   7. Simplified 90.0%

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

   if -1.30000000000000002e-35 < u < 1.46e22

   1. Initial program 64.8%

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

      1. associate-/l* 68.4%

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

      2. distribute-lft-neg-out 68.4%

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

      3. distribute-rgt-neg-in 68.4%

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

      4. associate-/r* 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in t1 around inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. neg-mul-1 83.2%

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

   7. Simplified 83.2%

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

   if 1.46e22 < u

   1. Initial program 74.9%

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

      1. times-frac 96.6%

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

      2. distribute-frac-neg 96.6%

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

      3. distribute-neg-frac2 96.6%

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

      4. +-commutative 96.6%

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

      5. distribute-neg-in 96.6%

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

      6. unsub-neg 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around 0 80.2%

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

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac2 80.2%

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

   7. Simplified 80.2%

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

      1. distribute-frac-neg2 80.2%

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

      2. distribute-frac-neg 80.2%

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

      3. associate-*l/ 83.4%

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

      4. frac-2neg 83.4%

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

      5. add-sqr-sqrt 43.5%

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

      6. sqrt-unprod 57.7%

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

      7. sqr-neg 57.7%

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

      8. sqrt-unprod 29.6%

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

      9. add-sqr-sqrt 59.7%

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

      10. +-commutative 59.7%

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

      11. distribute-neg-in 59.7%

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

      12. sub-neg 59.7%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 73.2%

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

      15. sqr-neg 73.2%

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

      16. sqrt-unprod 83.2%

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

      17. add-sqr-sqrt 83.2%

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

   9. Applied egg-rr 83.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 1.46 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 - u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2100000000000:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2100000000000.0)
   (/ (* v (/ t1 u)) (- (- u) t1))
   (if (<= u 5.2e+21) (/ v (- t1)) (/ (* t1 (/ v (- u))) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2100000000000.0) {
		tmp = (v * (t1 / u)) / (-u - t1);
	} else if (u <= 5.2e+21) {
		tmp = v / -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 (u <= (-2100000000000.0d0)) then
        tmp = (v * (t1 / u)) / (-u - t1)
    else if (u <= 5.2d+21) then
        tmp = v / -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 (u <= -2100000000000.0) {
		tmp = (v * (t1 / u)) / (-u - t1);
	} else if (u <= 5.2e+21) {
		tmp = v / -t1;
	} else {
		tmp = (t1 * (v / -u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2100000000000.0:
		tmp = (v * (t1 / u)) / (-u - t1)
	elif u <= 5.2e+21:
		tmp = v / -t1
	else:
		tmp = (t1 * (v / -u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2100000000000.0)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(Float64(-u) - t1));
	elseif (u <= 5.2e+21)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2100000000000.0)
		tmp = (v * (t1 / u)) / (-u - t1);
	elseif (u <= 5.2e+21)
		tmp = v / -t1;
	else
		tmp = (t1 * (v / -u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2100000000000.0], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 5.2e+21], N[(v / (-t1)), $MachinePrecision], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.1e12

   1. Initial program 67.1%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. distribute-neg-frac2 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in v around 0 65.1%

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

      1. mul-1-neg 65.1%

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

      2. associate-/r* 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r/ 86.5%

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

   10. Simplified 86.5%

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

   if -2.1e12 < u < 5.2e21

   1. Initial program 65.6%

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

      1. associate-/l* 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

      4. associate-/r* 80.5%

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

      5. distribute-neg-frac2 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in t1 around inf 82.9%

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

      1. associate-*r/ 82.9%

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

      2. neg-mul-1 82.9%

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

   7. Simplified 82.9%

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

   if 5.2e21 < u

   1. Initial program 74.9%

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

      1. times-frac 96.6%

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

      2. distribute-frac-neg 96.6%

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

      3. distribute-neg-frac2 96.6%

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

      4. +-commutative 96.6%

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

      5. distribute-neg-in 96.6%

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

      6. unsub-neg 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around 0 80.2%

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

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac2 80.2%

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

   7. Simplified 80.2%

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

      1. distribute-frac-neg2 80.2%

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

      2. distribute-frac-neg 80.2%

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

      3. associate-*l/ 83.4%

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

      4. frac-2neg 83.4%

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

      5. add-sqr-sqrt 43.5%

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

      6. sqrt-unprod 57.7%

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

      7. sqr-neg 57.7%

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

      8. sqrt-unprod 29.6%

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

      9. add-sqr-sqrt 59.7%

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

      10. +-commutative 59.7%

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

      11. distribute-neg-in 59.7%

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

      12. sub-neg 59.7%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 73.2%

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

      15. sqr-neg 73.2%

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

      16. sqrt-unprod 83.2%

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

      17. add-sqr-sqrt 83.2%

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

   9. Applied egg-rr 83.2%

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

   10. Taylor expanded in t1 around 0 74.8%

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

       1. mul-1-neg 74.8%

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

       2. associate-*r/ 80.0%

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

       3. distribute-rgt-neg-in 80.0%

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

       4. distribute-neg-frac2 80.0%

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

   12. Simplified 80.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2100000000000:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+81} \lor \neg \left(u \leq 2.2 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.4e+81) (not (<= u 2.2e+100)))
   (/ t1 (* u (/ u v)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.4e+81) || !(u <= 2.2e+100)) {
		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 <= (-6.4d+81)) .or. (.not. (u <= 2.2d+100))) 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 <= -6.4e+81) || !(u <= 2.2e+100)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.4e+81) or not (u <= 2.2e+100):
		tmp = t1 / (u * (u / v))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.4e+81) || !(u <= 2.2e+100))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.4e+81) || ~((u <= 2.2e+100)))
		tmp = t1 / (u * (u / v));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.4e+81], N[Not[LessEqual[u, 2.2e+100]], $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 < -6.4e81 or 2.2000000000000001e100 < u

   1. Initial program 69.8%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around 0 90.4%

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

      1. mul-1-neg 90.4%

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

      2. distribute-neg-frac2 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in t1 around 0 87.2%

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

      1. *-commutative 87.2%

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

      2. clear-num 89.2%

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

      3. frac-times 85.8%

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

      4. *-un-lft-identity 85.8%

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

      5. add-sqr-sqrt 46.9%

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

      6. sqrt-unprod 69.1%

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

      7. sqr-neg 69.1%

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

      8. sqrt-unprod 32.5%

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

      9. add-sqr-sqrt 67.7%

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

   10. Applied egg-rr 67.7%

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

   if -6.4e81 < u < 2.2000000000000001e100

   1. Initial program 67.1%

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

      1. associate-/l* 69.5%

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

      2. distribute-lft-neg-out 69.5%

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

      3. distribute-rgt-neg-in 69.5%

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

      4. associate-/r* 80.8%

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

      5. distribute-neg-frac2 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in t1 around inf 75.8%

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

      1. associate-*r/ 75.8%

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

      2. neg-mul-1 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.4 \cdot 10^{+81} \lor \neg \left(u \leq 2.2 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.15 \cdot 10^{+130} \lor \neg \left(u \leq 1.16 \cdot 10^{+131}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.15e+130) (not (<= u 1.16e+131)))
   (* (/ v u) -0.5)
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.15e+130) || !(u <= 1.16e+131)) {
		tmp = (v / u) * -0.5;
	} 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 <= (-2.15d+130)) .or. (.not. (u <= 1.16d+131))) then
        tmp = (v / u) * (-0.5d0)
    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 <= -2.15e+130) || !(u <= 1.16e+131)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.15e+130) or not (u <= 1.16e+131):
		tmp = (v / u) * -0.5
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.15e+130) || !(u <= 1.16e+131))
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.15e+130) || ~((u <= 1.16e+131)))
		tmp = (v / u) * -0.5;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.15e+130], N[Not[LessEqual[u, 1.16e+131]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.14999999999999992e130 or 1.16e131 < u

   1. Initial program 68.5%

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

      1. associate-/l* 69.2%

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

      2. distribute-lft-neg-out 69.2%

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

      3. distribute-rgt-neg-in 69.2%

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

      4. associate-/r* 88.1%

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

      5. distribute-neg-frac2 88.1%

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

   3. Simplified 88.1%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-neg-in 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-*l/ 98.7%

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

      6. clear-num 98.7%

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

      7. frac-2neg 98.7%

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

      8. frac-times 84.4%

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

      9. *-un-lft-identity 84.4%

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

      10. +-commutative 84.4%

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

      11. distribute-neg-in 84.4%

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

      12. sub-neg 84.4%

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

   6. Applied egg-rr 84.4%

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

   7. Taylor expanded in t1 around inf 41.2%

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

      1. *-commutative 41.2%

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

   9. Simplified 41.2%

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

   10. Taylor expanded in t1 around 0 35.5%

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

   if -2.14999999999999992e130 < u < 1.16e131

   1. Initial program 67.8%

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

      1. associate-/l* 70.0%

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

      2. distribute-lft-neg-out 70.0%

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

      3. distribute-rgt-neg-in 70.0%

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

      4. associate-/r* 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in t1 around inf 74.5%

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

   7. Simplified 74.5%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.15 \cdot 10^{+130} \lor \neg \left(u \leq 1.16 \cdot 10^{+131}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+81} \lor \neg \left(u \leq 4.4 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9e+81) (not (<= u 4.4e+133))) (/ -0.5 (/ u v)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9e+81) || !(u <= 4.4e+133)) {
		tmp = -0.5 / (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 <= (-9d+81)) .or. (.not. (u <= 4.4d+133))) then
        tmp = (-0.5d0) / (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 <= -9e+81) || !(u <= 4.4e+133)) {
		tmp = -0.5 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9e+81) or not (u <= 4.4e+133):
		tmp = -0.5 / (u / v)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9e+81) || !(u <= 4.4e+133))
		tmp = Float64(-0.5 / Float64(u / v));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9e+81) || ~((u <= 4.4e+133)))
		tmp = -0.5 / (u / v);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9e+81], N[Not[LessEqual[u, 4.4e+133]], $MachinePrecision]], N[(-0.5 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -9.00000000000000034e81 or 4.4e133 < u

   1. Initial program 69.5%

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

      1. associate-/l* 69.9%

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

      2. distribute-lft-neg-out 69.9%

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

      3. distribute-rgt-neg-in 69.9%

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

      4. associate-/r* 87.6%

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

      5. distribute-neg-frac2 87.6%

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

   3. Simplified 87.6%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-neg-in 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-*l/ 98.7%

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

      6. clear-num 98.8%

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

      7. frac-2neg 98.8%

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

      8. frac-times 85.5%

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

      9. *-un-lft-identity 85.5%

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

      10. +-commutative 85.5%

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

      11. distribute-neg-in 85.5%

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

      12. sub-neg 85.5%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in t1 around inf 43.2%

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

      1. *-commutative 43.2%

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

   9. Simplified 43.2%

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

   10. Taylor expanded in t1 around 0 35.8%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   11. Step-by-step derivation

       1. clear-num 37.1%

          \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{u}{v}}} \]

       2. un-div-inv 37.1%

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

   12. Applied egg-rr 37.1%

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

   if -9.00000000000000034e81 < u < 4.4e133

   1. Initial program 67.3%

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

      1. associate-/l* 69.7%

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

      2. distribute-lft-neg-out 69.7%

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

      3. distribute-rgt-neg-in 69.7%

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

      4. associate-/r* 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in t1 around inf 75.2%

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

      1. associate-*r/ 75.2%

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

      2. neg-mul-1 75.2%

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

   7. Simplified 75.2%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+81} \lor \neg \left(u \leq 4.4 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+126} \lor \neg \left(u \leq 4.3 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e+126) (not (<= u 4.3e+133))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+126) || !(u <= 4.3e+133)) {
		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 <= (-3.4d+126)) .or. (.not. (u <= 4.3d+133))) 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 <= -3.4e+126) || !(u <= 4.3e+133)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e+126) or not (u <= 4.3e+133):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e+126) || !(u <= 4.3e+133))
		tmp = Float64(v / 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 <= -3.4e+126) || ~((u <= 4.3e+133)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e+126], N[Not[LessEqual[u, 4.3e+133]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.39999999999999989e126 or 4.29999999999999994e133 < u

   1. Initial program 68.5%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. distribute-neg-frac2 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in t1 around inf 35.5%

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

      1. mul-1-neg 35.5%

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

      2. distribute-neg-frac2 35.5%

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

   10. Simplified 35.5%

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

   if -3.39999999999999989e126 < u < 4.29999999999999994e133

   1. Initial program 67.8%

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

      1. associate-/l* 70.0%

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

      2. distribute-lft-neg-out 70.0%

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

      3. distribute-rgt-neg-in 70.0%

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

      4. associate-/r* 80.7%

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

      5. distribute-neg-frac2 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in t1 around inf 74.5%

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

   7. Simplified 74.5%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+126} \lor \neg \left(u \leq 4.3 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. times-frac 98.4%

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

   2. distribute-frac-neg 98.4%

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

   3. distribute-neg-frac2 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-neg-in 98.4%

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

   6. unsub-neg 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in t1 around inf 63.0%

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

6. Final simplification 63.0%

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

Alternative 15: 17.1% 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 68.1%

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

   1. times-frac 98.4%

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

   2. distribute-frac-neg 98.4%

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

   3. distribute-neg-frac2 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-neg-in 98.4%

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

   6. unsub-neg 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in t1 around 0 52.4%

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

   1. mul-1-neg 52.4%

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

   2. distribute-neg-frac2 52.4%

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

7. Simplified 52.4%

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

   1. distribute-frac-neg2 52.4%

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

   2. distribute-frac-neg 52.4%

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

   3. associate-*l/ 55.0%

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

   4. frac-2neg 55.0%

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

   5. add-sqr-sqrt 28.4%

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

   6. sqrt-unprod 37.6%

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

   7. sqr-neg 37.6%

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

   8. sqrt-unprod 18.2%

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

   9. add-sqr-sqrt 36.5%

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

   10. +-commutative 36.5%

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

   11. distribute-neg-in 36.5%

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

   12. sub-neg 36.5%

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

   13. add-sqr-sqrt 17.0%

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

   14. sqrt-unprod 43.3%

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

   15. sqr-neg 43.3%

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

   16. sqrt-unprod 29.0%

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

   17. add-sqr-sqrt 56.1%

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

9. Applied egg-rr 56.1%

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

10. Taylor expanded in t1 around inf 17.9%

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

11. Final simplification 17.9%

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

Reproduce

herbie shell --seed 2024039 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))