Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.2%

Time: 16.3s

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: 72.7% 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: 98.2% accurate, 1.0× speedup

Math

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

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[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * (-v)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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* 80.3%

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

   5. distribute-neg-frac2 80.3%

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

3. Simplified 80.3%

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

   1. distribute-frac-neg2 80.3%

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

   2. associate-/r* 70.0%

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

   3. distribute-rgt-neg-in 70.0%

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

   4. distribute-lft-neg-out 70.0%

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

   5. associate-*r/ 70.2%

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

   6. times-frac 97.3%

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

   7. frac-2neg 97.3%

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

   8. associate-*r/ 98.7%

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

   9. add-sqr-sqrt 43.8%

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

   10. sqrt-unprod 43.9%

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

   11. sqr-neg 43.9%

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

   12. sqrt-unprod 22.8%

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

   13. add-sqr-sqrt 37.2%

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

   14. add-sqr-sqrt 21.0%

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

   15. sqrt-unprod 56.1%

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

   16. sqr-neg 56.1%

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

   17. sqrt-prod 47.6%

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

   18. add-sqr-sqrt 98.7%

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

6. Applied egg-rr 98.7%

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

Alternative 2: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.45 \cdot 10^{+120} \lor \neg \left(t1 \leq 1.22 \cdot 10^{+146}\right):\\ \;\;\;\;t\_1 \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{t\_1}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))))
   (if (or (<= t1 -1.45e+120) (not (<= t1 1.22e+146)))
     (* t_1 (+ (/ u t1) -1.0))
     (* t1 (/ t_1 (- (- u) t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if ((t1 <= -1.45e+120) || !(t1 <= 1.22e+146)) {
		tmp = t_1 * ((u / t1) + -1.0);
	} else {
		tmp = t1 * (t_1 / (-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) :: t_1
    real(8) :: tmp
    t_1 = v / (t1 + u)
    if ((t1 <= (-1.45d+120)) .or. (.not. (t1 <= 1.22d+146))) then
        tmp = t_1 * ((u / t1) + (-1.0d0))
    else
        tmp = t1 * (t_1 / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if ((t1 <= -1.45e+120) || !(t1 <= 1.22e+146)) {
		tmp = t_1 * ((u / t1) + -1.0);
	} else {
		tmp = t1 * (t_1 / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	tmp = 0
	if (t1 <= -1.45e+120) or not (t1 <= 1.22e+146):
		tmp = t_1 * ((u / t1) + -1.0)
	else:
		tmp = t1 * (t_1 / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if ((t1 <= -1.45e+120) || !(t1 <= 1.22e+146))
		tmp = Float64(t_1 * Float64(Float64(u / t1) + -1.0));
	else
		tmp = Float64(t1 * Float64(t_1 / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	tmp = 0.0;
	if ((t1 <= -1.45e+120) || ~((t1 <= 1.22e+146)))
		tmp = t_1 * ((u / t1) + -1.0);
	else
		tmp = t1 * (t_1 / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t1, -1.45e+120], N[Not[LessEqual[t1, 1.22e+146]], $MachinePrecision]], N[(t$95$1 * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(t$95$1 / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.4500000000000001e120 or 1.21999999999999991e146 < t1

   1. Initial program 46.8%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 91.5%

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

   if -1.4500000000000001e120 < t1 < 1.21999999999999991e146

   1. Initial program 80.1%

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

      1. associate-/l* 78.8%

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

      2. distribute-lft-neg-out 78.8%

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

      3. distribute-rgt-neg-in 78.8%

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

      4. associate-/r* 88.7%

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

      5. distribute-neg-frac2 88.7%

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

   3. Simplified 88.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{+120} \lor \neg \left(t1 \leq 1.22 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{u}{t1} + -1\\ \mathbf{if}\;t1 \leq -1.75 \cdot 10^{+18}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot t\_1\\ \mathbf{elif}\;t1 \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (+ (/ u t1) -1.0)))
   (if (<= t1 -1.75e+18)
     (* (/ v (+ t1 u)) t_1)
     (if (<= t1 7.8e-7)
       (/ (* v (/ t1 (- u))) (+ t1 u))
       (/ t_1 (/ (+ t1 u) v))))))

C

double code(double u, double v, double t1) {
	double t_1 = (u / t1) + -1.0;
	double tmp;
	if (t1 <= -1.75e+18) {
		tmp = (v / (t1 + u)) * t_1;
	} else if (t1 <= 7.8e-7) {
		tmp = (v * (t1 / -u)) / (t1 + u);
	} else {
		tmp = t_1 / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (u / t1) + (-1.0d0)
    if (t1 <= (-1.75d+18)) then
        tmp = (v / (t1 + u)) * t_1
    else if (t1 <= 7.8d-7) then
        tmp = (v * (t1 / -u)) / (t1 + u)
    else
        tmp = t_1 / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (u / t1) + -1.0;
	double tmp;
	if (t1 <= -1.75e+18) {
		tmp = (v / (t1 + u)) * t_1;
	} else if (t1 <= 7.8e-7) {
		tmp = (v * (t1 / -u)) / (t1 + u);
	} else {
		tmp = t_1 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (u / t1) + -1.0
	tmp = 0
	if t1 <= -1.75e+18:
		tmp = (v / (t1 + u)) * t_1
	elif t1 <= 7.8e-7:
		tmp = (v * (t1 / -u)) / (t1 + u)
	else:
		tmp = t_1 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(u / t1) + -1.0)
	tmp = 0.0
	if (t1 <= -1.75e+18)
		tmp = Float64(Float64(v / Float64(t1 + u)) * t_1);
	elseif (t1 <= 7.8e-7)
		tmp = Float64(Float64(v * Float64(t1 / Float64(-u))) / Float64(t1 + u));
	else
		tmp = Float64(t_1 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (u / t1) + -1.0;
	tmp = 0.0;
	if (t1 <= -1.75e+18)
		tmp = (v / (t1 + u)) * t_1;
	elseif (t1 <= 7.8e-7)
		tmp = (v * (t1 / -u)) / (t1 + u);
	else
		tmp = t_1 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[t1, -1.75e+18], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t1, 7.8e-7], N[(N[(v * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.75e18

   1. Initial program 56.6%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.1%

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

   if -1.75e18 < t1 < 7.80000000000000049e-7

   1. Initial program 79.8%

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

      1. associate-/l* 77.7%

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

      2. distribute-lft-neg-out 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. associate-/r* 86.6%

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

      5. distribute-neg-frac2 86.6%

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

   3. Simplified 86.6%

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

      1. distribute-frac-neg2 86.6%

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

      2. associate-/r* 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. distribute-lft-neg-out 77.7%

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

      5. associate-*r/ 79.8%

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

      6. times-frac 94.9%

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

      7. frac-2neg 94.9%

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

      8. associate-*r/ 97.6%

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

      9. add-sqr-sqrt 45.5%

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

      10. sqrt-unprod 57.1%

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

      11. sqr-neg 57.1%

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

      12. sqrt-unprod 21.0%

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

      13. add-sqr-sqrt 36.2%

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

      14. add-sqr-sqrt 24.5%

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

      15. sqrt-unprod 59.5%

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

      16. sqr-neg 59.5%

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

      17. sqrt-prod 40.9%

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

      18. add-sqr-sqrt 97.6%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in t1 around 0 80.3%

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

   if 7.80000000000000049e-7 < t1

   1. Initial program 61.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 84.5%

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

      1. clear-num 85.9%

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

      2. un-div-inv 85.9%

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

      3. sub-neg 85.9%

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

      4. metadata-eval 85.9%

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

   7. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{\frac{\frac{u}{t1} + -1}{\frac{t1 + u}{v}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.75 \cdot 10^{+18}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{u}{t1} + -1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.7e+17)
   (* (/ v (+ t1 u)) (+ (/ u t1) -1.0))
   (if (<= t1 2.2e-7)
     (/ (* v (/ t1 (- u))) (+ t1 u))
     (/ -1.0 (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.7e+17) {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	} else if (t1 <= 2.2e-7) {
		tmp = (v * (t1 / -u)) / (t1 + u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.7d+17)) then
        tmp = (v / (t1 + u)) * ((u / t1) + (-1.0d0))
    else if (t1 <= 2.2d-7) then
        tmp = (v * (t1 / -u)) / (t1 + u)
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.7e+17) {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	} else if (t1 <= 2.2e-7) {
		tmp = (v * (t1 / -u)) / (t1 + u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.7e+17:
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0)
	elif t1 <= 2.2e-7:
		tmp = (v * (t1 / -u)) / (t1 + u)
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.7e+17)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(u / t1) + -1.0));
	elseif (t1 <= 2.2e-7)
		tmp = Float64(Float64(v * Float64(t1 / Float64(-u))) / Float64(t1 + u));
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.7e+17)
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	elseif (t1 <= 2.2e-7)
		tmp = (v * (t1 / -u)) / (t1 + u);
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.7e+17], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.2e-7], N[(N[(v * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.7e17

   1. Initial program 56.6%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.1%

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

   if -1.7e17 < t1 < 2.2000000000000001e-7

   1. Initial program 79.8%

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

      1. associate-/l* 77.7%

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

      2. distribute-lft-neg-out 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. associate-/r* 86.6%

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

      5. distribute-neg-frac2 86.6%

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

   3. Simplified 86.6%

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

      1. distribute-frac-neg2 86.6%

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

      2. associate-/r* 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. distribute-lft-neg-out 77.7%

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

      5. associate-*r/ 79.8%

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

      6. times-frac 94.9%

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

      7. frac-2neg 94.9%

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

      8. associate-*r/ 97.6%

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

      9. add-sqr-sqrt 45.5%

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

      10. sqrt-unprod 57.1%

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

      11. sqr-neg 57.1%

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

      12. sqrt-unprod 21.0%

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

      13. add-sqr-sqrt 36.2%

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

      14. add-sqr-sqrt 24.5%

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

      15. sqrt-unprod 59.5%

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

      16. sqr-neg 59.5%

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

      17. sqrt-prod 40.9%

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

      18. add-sqr-sqrt 97.6%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in t1 around 0 80.3%

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

   if 2.2000000000000001e-7 < t1

   1. Initial program 61.5%

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

      1. clear-num 99.0%

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

      2. inv-pow 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow-1 99.0%

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

   8. Simplified 99.0%

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

      1. frac-2neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.0%

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

      4. sub-neg 99.0%

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

      5. add-sqr-sqrt 52.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. sqr-neg 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 43.2%

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

      12. sqr-neg 43.2%

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

      13. sqrt-unprod 42.8%

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

      14. add-sqr-sqrt 42.8%

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

      15. +-commutative 42.8%

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

      16. distribute-frac-neg2 42.8%

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

      17. add-sqr-sqrt 21.9%

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

      18. sqrt-unprod 56.5%

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

   10. Applied egg-rr 99.0%

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

   11. Taylor expanded in t1 around inf 85.7%

       \[\leadsto \frac{\color{blue}{-1}}{\frac{t1 + u}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -8.5e+18)
   (/ v (- t1))
   (if (<= t1 2.7e-5)
     (/ (* v (/ t1 (- u))) (+ t1 u))
     (/ -1.0 (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -8.5e+18) {
		tmp = v / -t1;
	} else if (t1 <= 2.7e-5) {
		tmp = (v * (t1 / -u)) / (t1 + u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-8.5d+18)) then
        tmp = v / -t1
    else if (t1 <= 2.7d-5) then
        tmp = (v * (t1 / -u)) / (t1 + u)
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -8.5e+18) {
		tmp = v / -t1;
	} else if (t1 <= 2.7e-5) {
		tmp = (v * (t1 / -u)) / (t1 + u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -8.5e+18:
		tmp = v / -t1
	elif t1 <= 2.7e-5:
		tmp = (v * (t1 / -u)) / (t1 + u)
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -8.5e+18)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 2.7e-5)
		tmp = Float64(Float64(v * Float64(t1 / Float64(-u))) / Float64(t1 + u));
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -8.5e+18)
		tmp = v / -t1;
	elseif (t1 <= 2.7e-5)
		tmp = (v * (t1 / -u)) / (t1 + u);
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -8.5e+18], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 2.7e-5], N[(N[(v * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.5e18

   1. Initial program 56.6%

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

      1. associate-/l* 59.6%

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

      2. distribute-lft-neg-out 59.6%

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

      3. distribute-rgt-neg-in 59.6%

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

      4. associate-/r* 69.2%

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

      5. distribute-neg-frac2 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 88.0%

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

      1. associate-*r/ 88.0%

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

      2. neg-mul-1 88.0%

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

   7. Simplified 88.0%

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

   if -8.5e18 < t1 < 2.6999999999999999e-5

   1. Initial program 79.8%

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

      1. associate-/l* 77.7%

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

      2. distribute-lft-neg-out 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. associate-/r* 86.6%

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

      5. distribute-neg-frac2 86.6%

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

   3. Simplified 86.6%

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

      1. distribute-frac-neg2 86.6%

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

      2. associate-/r* 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. distribute-lft-neg-out 77.7%

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

      5. associate-*r/ 79.8%

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

      6. times-frac 94.9%

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

      7. frac-2neg 94.9%

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

      8. associate-*r/ 97.6%

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

      9. add-sqr-sqrt 45.5%

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

      10. sqrt-unprod 57.1%

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

      11. sqr-neg 57.1%

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

      12. sqrt-unprod 21.0%

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

      13. add-sqr-sqrt 36.2%

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

      14. add-sqr-sqrt 24.5%

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

      15. sqrt-unprod 59.5%

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

      16. sqr-neg 59.5%

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

      17. sqrt-prod 40.9%

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

      18. add-sqr-sqrt 97.6%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in t1 around 0 80.3%

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

   if 2.6999999999999999e-5 < t1

   1. Initial program 61.5%

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

      1. clear-num 99.0%

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

      2. inv-pow 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow-1 99.0%

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

   8. Simplified 99.0%

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

      1. frac-2neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.0%

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

      4. sub-neg 99.0%

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

      5. add-sqr-sqrt 52.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. sqr-neg 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 43.2%

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

      12. sqr-neg 43.2%

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

      13. sqrt-unprod 42.8%

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

      14. add-sqr-sqrt 42.8%

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

      15. +-commutative 42.8%

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

      16. distribute-frac-neg2 42.8%

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

      17. add-sqr-sqrt 21.9%

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

      18. sqrt-unprod 56.5%

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

   10. Applied egg-rr 99.0%

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

   11. Taylor expanded in t1 around inf 85.7%

       \[\leadsto \frac{\color{blue}{-1}}{\frac{t1 + u}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 0.000106:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -7.4e+19)
   (/ v (- t1))
   (if (<= t1 0.000106)
     (* (/ t1 (+ t1 u)) (/ v (- u)))
     (/ -1.0 (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -7.4e+19) {
		tmp = v / -t1;
	} else if (t1 <= 0.000106) {
		tmp = (t1 / (t1 + u)) * (v / -u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-7.4d+19)) then
        tmp = v / -t1
    else if (t1 <= 0.000106d0) then
        tmp = (t1 / (t1 + u)) * (v / -u)
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -7.4e+19) {
		tmp = v / -t1;
	} else if (t1 <= 0.000106) {
		tmp = (t1 / (t1 + u)) * (v / -u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -7.4e+19:
		tmp = v / -t1
	elif t1 <= 0.000106:
		tmp = (t1 / (t1 + u)) * (v / -u)
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -7.4e+19)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 0.000106)
		tmp = Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(-u)));
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -7.4e+19)
		tmp = v / -t1;
	elseif (t1 <= 0.000106)
		tmp = (t1 / (t1 + u)) * (v / -u);
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -7.4e+19], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 0.000106], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / (-u)), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -7.4e19

   1. Initial program 56.6%

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

      1. associate-/l* 59.6%

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

      2. distribute-lft-neg-out 59.6%

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

      3. distribute-rgt-neg-in 59.6%

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

      4. associate-/r* 69.2%

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

      5. distribute-neg-frac2 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 88.0%

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

      1. associate-*r/ 88.0%

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

      2. neg-mul-1 88.0%

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

   7. Simplified 88.0%

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

   if -7.4e19 < t1 < 1.06e-4

   1. Initial program 79.8%

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

      1. times-frac 94.9%

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

      2. distribute-frac-neg 94.9%

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

      3. distribute-neg-frac2 94.9%

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

      4. +-commutative 94.9%

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

      5. distribute-neg-in 94.9%

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

      6. unsub-neg 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in t1 around 0 77.6%

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

   if 1.06e-4 < t1

   1. Initial program 61.5%

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

      1. clear-num 99.0%

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

      2. inv-pow 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow-1 99.0%

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

   8. Simplified 99.0%

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

      1. frac-2neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.0%

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

      4. sub-neg 99.0%

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

      5. add-sqr-sqrt 52.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. sqr-neg 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 43.2%

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

      12. sqr-neg 43.2%

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

      13. sqrt-unprod 42.8%

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

      14. add-sqr-sqrt 42.8%

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

      15. +-commutative 42.8%

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

      16. distribute-frac-neg2 42.8%

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

      17. add-sqr-sqrt 21.9%

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

      18. sqrt-unprod 56.5%

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

   10. Applied egg-rr 99.0%

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

   11. Taylor expanded in t1 around inf 85.7%

       \[\leadsto \frac{\color{blue}{-1}}{\frac{t1 + u}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 0.000106:\\ \;\;\;\;\frac{t1}{t1 + u} \cdot \frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{t1 \cdot \left(-v\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.7e+17)
   (/ v (- t1))
   (if (<= t1 4.4e-7) (/ (/ (* t1 (- v)) u) u) (/ -1.0 (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.7e+17) {
		tmp = v / -t1;
	} else if (t1 <= 4.4e-7) {
		tmp = ((t1 * -v) / u) / u;
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.7d+17)) then
        tmp = v / -t1
    else if (t1 <= 4.4d-7) then
        tmp = ((t1 * -v) / u) / u
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.7e+17) {
		tmp = v / -t1;
	} else if (t1 <= 4.4e-7) {
		tmp = ((t1 * -v) / u) / u;
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.7e+17:
		tmp = v / -t1
	elif t1 <= 4.4e-7:
		tmp = ((t1 * -v) / u) / u
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.7e+17)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 4.4e-7)
		tmp = Float64(Float64(Float64(t1 * Float64(-v)) / u) / u);
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.7e+17)
		tmp = v / -t1;
	elseif (t1 <= 4.4e-7)
		tmp = ((t1 * -v) / u) / u;
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.7e+17], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 4.4e-7], N[(N[(N[(t1 * (-v)), $MachinePrecision] / u), $MachinePrecision] / u), $MachinePrecision], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.7e17

   1. Initial program 56.6%

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

      1. associate-/l* 59.6%

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

      2. distribute-lft-neg-out 59.6%

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

      3. distribute-rgt-neg-in 59.6%

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

      4. associate-/r* 69.2%

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

      5. distribute-neg-frac2 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 88.0%

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

      1. associate-*r/ 88.0%

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

      2. neg-mul-1 88.0%

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

   7. Simplified 88.0%

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

   if -1.7e17 < t1 < 4.4000000000000002e-7

   1. Initial program 79.8%

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

      1. associate-/l* 77.7%

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

      2. distribute-lft-neg-out 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. associate-/r* 86.6%

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

      5. distribute-neg-frac2 86.6%

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

   3. Simplified 86.6%

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

      1. distribute-frac-neg2 86.6%

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

      2. associate-/r* 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. distribute-lft-neg-out 77.7%

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

      5. associate-*r/ 79.8%

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

      6. times-frac 94.9%

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

      7. frac-2neg 94.9%

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

      8. associate-*r/ 97.6%

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

      9. add-sqr-sqrt 45.5%

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

      10. sqrt-unprod 57.1%

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

      11. sqr-neg 57.1%

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

      12. sqrt-unprod 21.0%

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

      13. add-sqr-sqrt 36.2%

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

      14. add-sqr-sqrt 24.5%

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

      15. sqrt-unprod 59.5%

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

      16. sqr-neg 59.5%

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

      17. sqrt-prod 40.9%

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

      18. add-sqr-sqrt 97.6%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in t1 around 0 75.9%

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

      1. associate-*r/ 75.9%

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

      2. associate-*r* 75.9%

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

      3. mul-1-neg 75.9%

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

   9. Simplified 75.9%

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

   10. Taylor expanded in t1 around 0 77.6%

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

   if 4.4000000000000002e-7 < t1

   1. Initial program 61.5%

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

      1. clear-num 99.0%

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

      2. inv-pow 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow-1 99.0%

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

   8. Simplified 99.0%

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

      1. frac-2neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.0%

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

      4. sub-neg 99.0%

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

      5. add-sqr-sqrt 52.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. sqr-neg 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 43.2%

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

      12. sqr-neg 43.2%

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

      13. sqrt-unprod 42.8%

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

      14. add-sqr-sqrt 42.8%

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

      15. +-commutative 42.8%

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

      16. distribute-frac-neg2 42.8%

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

      17. add-sqr-sqrt 21.9%

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

      18. sqrt-unprod 56.5%

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

   10. Applied egg-rr 99.0%

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

   11. Taylor expanded in t1 around inf 85.7%

       \[\leadsto \frac{\color{blue}{-1}}{\frac{t1 + u}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.7 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{t1 \cdot \left(-v\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e+18)
   (/ v (- t1))
   (if (<= t1 1.5e-7) (* t1 (/ (/ v u) (- u))) (/ -1.0 (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+18) {
		tmp = v / -t1;
	} else if (t1 <= 1.5e-7) {
		tmp = t1 * ((v / u) / -u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.2d+18)) then
        tmp = v / -t1
    else if (t1 <= 1.5d-7) then
        tmp = t1 * ((v / u) / -u)
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+18) {
		tmp = v / -t1;
	} else if (t1 <= 1.5e-7) {
		tmp = t1 * ((v / u) / -u);
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e+18:
		tmp = v / -t1
	elif t1 <= 1.5e-7:
		tmp = t1 * ((v / u) / -u)
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e+18)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 1.5e-7)
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e+18)
		tmp = v / -t1;
	elseif (t1 <= 1.5e-7)
		tmp = t1 * ((v / u) / -u);
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e+18], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 1.5e-7], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.2e18

   1. Initial program 56.6%

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

      1. associate-/l* 59.6%

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

      2. distribute-lft-neg-out 59.6%

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

      3. distribute-rgt-neg-in 59.6%

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

      4. associate-/r* 69.2%

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

      5. distribute-neg-frac2 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in t1 around inf 88.0%

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

      1. associate-*r/ 88.0%

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

      2. neg-mul-1 88.0%

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

   7. Simplified 88.0%

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

   if -1.2e18 < t1 < 1.4999999999999999e-7

   1. Initial program 79.8%

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

      1. associate-/l* 77.7%

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

      2. distribute-lft-neg-out 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. associate-/r* 86.6%

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

      5. distribute-neg-frac2 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in t1 around 0 73.3%

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

   6. Taylor expanded in t1 around 0 75.0%

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

   if 1.4999999999999999e-7 < t1

   1. Initial program 61.5%

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

      1. clear-num 99.0%

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

      2. inv-pow 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow-1 99.0%

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

   8. Simplified 99.0%

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

      1. frac-2neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.0%

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

      4. sub-neg 99.0%

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

      5. add-sqr-sqrt 52.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. sqr-neg 0.0%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 43.2%

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

      12. sqr-neg 43.2%

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

      13. sqrt-unprod 42.8%

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

      14. add-sqr-sqrt 42.8%

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

      15. +-commutative 42.8%

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

      16. distribute-frac-neg2 42.8%

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

      17. add-sqr-sqrt 21.9%

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

      18. sqrt-unprod 56.5%

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

   10. Applied egg-rr 99.0%

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

   11. Taylor expanded in t1 around inf 85.7%

       \[\leadsto \frac{\color{blue}{-1}}{\frac{t1 + u}{v}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{+162} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -8e+162) (not (<= t1 8.5e+108))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8e+162) || !(t1 <= 8.5e+108)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-8d+162)) .or. (.not. (t1 <= 8.5d+108))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8e+162) || !(t1 <= 8.5e+108)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -8e+162) or not (t1 <= 8.5e+108):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -8e+162) || !(t1 <= 8.5e+108))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -8e+162) || ~((t1 <= 8.5e+108)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -8e+162], N[Not[LessEqual[t1, 8.5e+108]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.9999999999999995e162 or 8.50000000000000016e108 < t1

   1. Initial program 45.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 90.2%

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

   6. Taylor expanded in u around inf 41.8%

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

   if -7.9999999999999995e162 < t1 < 8.50000000000000016e108

   1. Initial program 80.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf 44.2%

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

      1. associate-/l* 42.2%

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

      2. add-sqr-sqrt 20.5%

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

      3. sqrt-unprod 38.3%

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

      4. sqr-neg 38.3%

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

      5. sqrt-unprod 11.8%

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

      6. add-sqr-sqrt 21.1%

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

      7. *-commutative 21.1%

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

      8. associate-/r* 20.6%

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

   5. Applied egg-rr 20.6%

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

      1. associate-*r/ 19.9%

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

      2. associate-*l/ 25.5%

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

      3. *-commutative 25.5%

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

      4. times-frac 22.2%

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

      5. associate-*r/ 20.3%

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

      6. associate-/r* 20.0%

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

      7. *-inverses 20.0%

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

   7. Simplified 20.0%

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

   8. Taylor expanded in t1 around 0 21.3%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{+162} \lor \neg \left(t1 \leq 8.5 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. times-frac 97.3%

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

   2. distribute-frac-neg 97.3%

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

   3. distribute-neg-frac2 97.3%

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

   4. +-commutative 97.3%

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

   5. distribute-neg-in 97.3%

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

   6. unsub-neg 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 11: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 1.8e+153) (/ v (- (- u) t1)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1.8e+153) {
		tmp = v / (-u - t1);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (v <= 1.8d+153) then
        tmp = v / (-u - t1)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1.8e+153) {
		tmp = v / (-u - t1);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 1.8e+153:
		tmp = v / (-u - t1)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 1.8e+153)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 1.8e+153)
		tmp = v / (-u - t1);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 1.8e+153], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.8e153

   1. Initial program 72.3%

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

      1. associate-/l* 72.5%

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

      2. distribute-lft-neg-out 72.5%

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

      3. distribute-rgt-neg-in 72.5%

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

      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. Step-by-step derivation

      1. distribute-frac-neg2 80.8%

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

      2. associate-/r* 72.5%

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

      3. distribute-rgt-neg-in 72.5%

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

      4. distribute-lft-neg-out 72.5%

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

      5. associate-*r/ 72.3%

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

      6. times-frac 98.2%

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

      7. frac-2neg 98.2%

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

      8. associate-*r/ 99.0%

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

      9. add-sqr-sqrt 44.4%

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

      10. sqrt-unprod 45.6%

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

      11. sqr-neg 45.6%

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

      12. sqrt-unprod 24.8%

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

      13. add-sqr-sqrt 40.7%

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

      14. add-sqr-sqrt 23.2%

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

      15. sqrt-unprod 58.3%

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

      16. sqr-neg 58.3%

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

      17. sqrt-prod 47.0%

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

      18. add-sqr-sqrt 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in t1 around inf 62.0%

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

      1. mul-1-neg 62.0%

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

   9. Simplified 62.0%

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

   if 1.8e153 < v

   1. Initial program 51.3%

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

      1. associate-/l* 48.7%

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

      2. distribute-lft-neg-out 48.7%

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

      3. distribute-rgt-neg-in 48.7%

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

      4. associate-/r* 75.7%

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

      5. distribute-neg-frac2 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in t1 around inf 48.3%

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

      1. associate-*r/ 48.3%

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

      2. neg-mul-1 48.3%

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

   7. Simplified 48.3%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.8e+49) (/ v (+ t1 u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.8e+49) {
		tmp = v / (t1 + u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.8d+49)) then
        tmp = v / (t1 + u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.8e+49) {
		tmp = v / (t1 + u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.8e+49:
		tmp = v / (t1 + u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.8e+49)
		tmp = Float64(v / Float64(t1 + u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.8e+49)
		tmp = v / (t1 + u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.8e+49], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.7999999999999998e49

   1. Initial program 80.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf 47.3%

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

      1. associate-/l* 47.4%

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

      2. add-sqr-sqrt 18.5%

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

      3. sqrt-unprod 43.5%

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

      4. sqr-neg 43.5%

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

      5. sqrt-unprod 29.0%

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

      6. add-sqr-sqrt 47.4%

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

      7. *-commutative 47.4%

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

      8. associate-/r* 46.9%

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

   5. Applied egg-rr 46.9%

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

      1. associate-*r/ 45.5%

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

      2. associate-*l/ 54.3%

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

      3. times-frac 47.4%

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

      4. *-commutative 47.4%

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

      5. times-frac 45.7%

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

      6. *-inverses 45.7%

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

      7. *-rgt-identity 45.7%

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

   7. Simplified 45.7%

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

   if -2.7999999999999998e49 < u

   1. Initial program 66.8%

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

      1. associate-/l* 66.5%

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

      2. distribute-lft-neg-out 66.5%

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

      3. distribute-rgt-neg-in 66.5%

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

      4. associate-/r* 75.7%

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

      5. distribute-neg-frac2 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in t1 around inf 60.3%

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

      1. associate-*r/ 60.3%

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

      2. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.02e+74) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.02e+74) {
		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 <= (-1.02d+74)) 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 <= -1.02e+74) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.02e+74:
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.02e+74)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.02e+74)
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.02e+74], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.02000000000000005e74

   1. Initial program 79.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf 47.9%

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

      1. associate-/l* 48.0%

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

      2. add-sqr-sqrt 19.5%

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

      3. sqrt-unprod 45.5%

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

      4. sqr-neg 45.5%

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

      5. sqrt-unprod 28.6%

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

      6. add-sqr-sqrt 48.0%

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

      7. *-commutative 48.0%

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

      8. associate-/r* 47.4%

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

   5. Applied egg-rr 47.4%

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

      1. associate-*r/ 46.0%

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

      2. associate-*l/ 55.2%

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

      3. *-commutative 55.2%

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

      4. times-frac 48.0%

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

      5. associate-*r/ 46.7%

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

      6. associate-/r* 46.2%

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

      7. *-inverses 46.2%

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

   7. Simplified 46.2%

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

   8. Taylor expanded in t1 around 0 43.4%

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

   if -1.02000000000000005e74 < u

   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* 67.1%

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

      2. distribute-lft-neg-out 67.1%

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

      3. distribute-rgt-neg-in 67.1%

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

      4. associate-/r* 76.1%

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

      5. distribute-neg-frac2 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in t1 around inf 60.0%

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

      1. associate-*r/ 60.0%

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

      2. neg-mul-1 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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* 80.3%

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

   5. distribute-neg-frac2 80.3%

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

3. Simplified 80.3%

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

   1. distribute-frac-neg2 80.3%

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

   2. associate-/r* 70.0%

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

   3. distribute-rgt-neg-in 70.0%

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

   4. distribute-lft-neg-out 70.0%

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

   5. associate-*r/ 70.2%

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

   6. times-frac 97.3%

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

   7. frac-2neg 97.3%

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

   8. associate-*r/ 98.7%

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

   9. add-sqr-sqrt 43.8%

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

   10. sqrt-unprod 43.9%

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

   11. sqr-neg 43.9%

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

   12. sqrt-unprod 22.8%

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

   13. add-sqr-sqrt 37.2%

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

   14. add-sqr-sqrt 21.0%

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

   15. sqrt-unprod 56.1%

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

   16. sqr-neg 56.1%

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

   17. sqrt-prod 47.6%

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

   18. add-sqr-sqrt 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in t1 around inf 59.4%

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

   1. mul-1-neg 59.4%

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

9. Simplified 59.4%

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

    1. add-sqr-sqrt 25.7%

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

    2. sqrt-unprod 66.6%

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

    3. sqr-neg 66.6%

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

    4. sqrt-unprod 34.5%

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

    5. add-sqr-sqrt 60.5%

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

    6. sub-neg 60.5%

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

11. Applied egg-rr 60.5%

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

12. Final simplification 60.5%

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

Alternative 15: 14.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. times-frac 97.3%

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

   2. distribute-frac-neg 97.3%

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

   3. distribute-neg-frac2 97.3%

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

   4. +-commutative 97.3%

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

   5. distribute-neg-in 97.3%

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

   6. unsub-neg 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in t1 around inf 54.1%

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

6. Taylor expanded in u around inf 15.6%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))