Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.0%

Time: 10.4s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

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(Float64(t1 / Float64(t1 + u)) * 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[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

   1. associate-*l/ 74.9%

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

   2. *-commutative 74.9%

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

3. Simplified 74.9%

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

   1. *-commutative 74.9%

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

   2. associate-*l/ 71.6%

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

   3. times-frac 97.8%

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

   4. frac-2neg 97.8%

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

   5. remove-double-neg 97.8%

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

   6. +-commutative 97.8%

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

   7. distribute-neg-in 97.8%

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

   8. sub-neg 97.8%

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

   9. frac-2neg 97.8%

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

   10. +-commutative 97.8%

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

   11. distribute-neg-in 97.8%

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

   12. sub-neg 97.8%

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

   13. associate-*r/ 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 2: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ t_2 := t\_1 \cdot \frac{t1}{-u}\\ \mathbf{if}\;u \leq -9.4 \cdot 10^{+57}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq -5.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u - t1}}{t1}\\ \mathbf{elif}\;u \leq -1.7 \cdot 10^{-75}:\\ \;\;\;\;t1 \cdot \frac{t\_1}{-u}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{-74}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))) (t_2 (* t_1 (/ t1 (- u)))))
   (if (<= u -9.4e+57)
     t_2
     (if (<= u -5.8e-68)
       (/ (* v (/ t1 (- u t1))) t1)
       (if (<= u -1.7e-75)
         (* t1 (/ t_1 (- u)))
         (if (<= u 1.15e-74) (/ v (- (- t1) (* u 2.0))) t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double t_2 = t_1 * (t1 / -u);
	double tmp;
	if (u <= -9.4e+57) {
		tmp = t_2;
	} else if (u <= -5.8e-68) {
		tmp = (v * (t1 / (u - t1))) / t1;
	} else if (u <= -1.7e-75) {
		tmp = t1 * (t_1 / -u);
	} else if (u <= 1.15e-74) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = v / (t1 + u)
    t_2 = t_1 * (t1 / -u)
    if (u <= (-9.4d+57)) then
        tmp = t_2
    else if (u <= (-5.8d-68)) then
        tmp = (v * (t1 / (u - t1))) / t1
    else if (u <= (-1.7d-75)) then
        tmp = t1 * (t_1 / -u)
    else if (u <= 1.15d-74) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double t_2 = t_1 * (t1 / -u);
	double tmp;
	if (u <= -9.4e+57) {
		tmp = t_2;
	} else if (u <= -5.8e-68) {
		tmp = (v * (t1 / (u - t1))) / t1;
	} else if (u <= -1.7e-75) {
		tmp = t1 * (t_1 / -u);
	} else if (u <= 1.15e-74) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	t_2 = t_1 * (t1 / -u)
	tmp = 0
	if u <= -9.4e+57:
		tmp = t_2
	elif u <= -5.8e-68:
		tmp = (v * (t1 / (u - t1))) / t1
	elif u <= -1.7e-75:
		tmp = t1 * (t_1 / -u)
	elif u <= 1.15e-74:
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	t_2 = Float64(t_1 * Float64(t1 / Float64(-u)))
	tmp = 0.0
	if (u <= -9.4e+57)
		tmp = t_2;
	elseif (u <= -5.8e-68)
		tmp = Float64(Float64(v * Float64(t1 / Float64(u - t1))) / t1);
	elseif (u <= -1.7e-75)
		tmp = Float64(t1 * Float64(t_1 / Float64(-u)));
	elseif (u <= 1.15e-74)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	t_2 = t_1 * (t1 / -u);
	tmp = 0.0;
	if (u <= -9.4e+57)
		tmp = t_2;
	elseif (u <= -5.8e-68)
		tmp = (v * (t1 / (u - t1))) / t1;
	elseif (u <= -1.7e-75)
		tmp = t1 * (t_1 / -u);
	elseif (u <= 1.15e-74)
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -9.4e+57], t$95$2, If[LessEqual[u, -5.8e-68], N[(N[(v * N[(t1 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[u, -1.7e-75], N[(t1 * N[(t$95$1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.15e-74], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -9.4000000000000006e57 or 1.1499999999999999e-74 < u

   1. Initial program 79.6%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 91.8%

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

      1. associate-*r/ 91.8%

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

      2. mul-1-neg 91.8%

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

   7. Simplified 91.8%

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

   if -9.4000000000000006e57 < u < -5.8000000000000001e-68

   1. Initial program 78.6%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 71.1%

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

      1. associate-*r/ 77.4%

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

      2. add-sqr-sqrt 77.4%

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

      3. sqrt-unprod 77.4%

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

      4. sqr-neg 77.4%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 78.1%

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

   7. Applied egg-rr 78.1%

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

   if -5.8000000000000001e-68 < u < -1.70000000000000008e-75

   1. Initial program 59.1%

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

      1. associate-*l/ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 59.1%

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

      3. times-frac 99.7%

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

      4. frac-2neg 99.7%

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

      5. remove-double-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. sub-neg 99.7%

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

      9. frac-2neg 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. sub-neg 99.7%

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

      13. associate-*r/ 74.8%

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

   6. Applied egg-rr 74.8%

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

      1. associate-/l* 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. neg-mul-1 99.5%

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

      6. times-frac 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in t1 around 0 83.3%

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

       1. frac-2neg 83.3%

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

       2. associate-/r/ 100.0%

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

   13. Applied egg-rr 100.0%

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

   if -1.70000000000000008e-75 < u < 1.1499999999999999e-74

   1. Initial program 58.8%

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

      1. associate-*l/ 65.7%

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

      2. *-commutative 65.7%

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

   3. Simplified 65.7%

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

      1. associate-*r/ 58.8%

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

      2. *-commutative 58.8%

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

      3. times-frac 95.8%

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

      4. frac-2neg 95.8%

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

      5. remove-double-neg 95.8%

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

      6. +-commutative 95.8%

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

      7. distribute-neg-in 95.8%

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

      8. sub-neg 95.8%

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

      9. clear-num 95.7%

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

      10. frac-2neg 95.7%

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

      11. frac-times 99.8%

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

      12. *-un-lft-identity 99.8%

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

      13. +-commutative 99.8%

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

      14. distribute-neg-in 99.8%

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

      15. sub-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in u around 0 85.9%

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

      1. *-commutative 85.9%

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

   9. Simplified 85.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq -5.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u - t1}}{t1}\\ \mathbf{elif}\;u \leq -1.7 \cdot 10^{-75}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{-74}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;u \leq -2.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{t\_1}{\frac{u}{-t1}}\\ \mathbf{elif}\;u \leq -1.34 \cdot 10^{-67}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u - t1}}{t1}\\ \mathbf{elif}\;u \leq -4.6 \cdot 10^{-75}:\\ \;\;\;\;t1 \cdot \frac{t\_1}{-u}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))))
   (if (<= u -2.8e+58)
     (/ t_1 (/ u (- t1)))
     (if (<= u -1.34e-67)
       (/ (* v (/ t1 (- u t1))) t1)
       (if (<= u -4.6e-75)
         (* t1 (/ t_1 (- u)))
         (if (<= u 1.4e-74)
           (/ v (- (- t1) (* u 2.0)))
           (* t_1 (/ t1 (- u)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (u <= -2.8e+58) {
		tmp = t_1 / (u / -t1);
	} else if (u <= -1.34e-67) {
		tmp = (v * (t1 / (u - t1))) / t1;
	} else if (u <= -4.6e-75) {
		tmp = t1 * (t_1 / -u);
	} else if (u <= 1.4e-74) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t_1 * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (t1 + u)
    if (u <= (-2.8d+58)) then
        tmp = t_1 / (u / -t1)
    else if (u <= (-1.34d-67)) then
        tmp = (v * (t1 / (u - t1))) / t1
    else if (u <= (-4.6d-75)) then
        tmp = t1 * (t_1 / -u)
    else if (u <= 1.4d-74) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t_1 * (t1 / -u)
    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 (u <= -2.8e+58) {
		tmp = t_1 / (u / -t1);
	} else if (u <= -1.34e-67) {
		tmp = (v * (t1 / (u - t1))) / t1;
	} else if (u <= -4.6e-75) {
		tmp = t1 * (t_1 / -u);
	} else if (u <= 1.4e-74) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t_1 * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	tmp = 0
	if u <= -2.8e+58:
		tmp = t_1 / (u / -t1)
	elif u <= -1.34e-67:
		tmp = (v * (t1 / (u - t1))) / t1
	elif u <= -4.6e-75:
		tmp = t1 * (t_1 / -u)
	elif u <= 1.4e-74:
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t_1 * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if (u <= -2.8e+58)
		tmp = Float64(t_1 / Float64(u / Float64(-t1)));
	elseif (u <= -1.34e-67)
		tmp = Float64(Float64(v * Float64(t1 / Float64(u - t1))) / t1);
	elseif (u <= -4.6e-75)
		tmp = Float64(t1 * Float64(t_1 / Float64(-u)));
	elseif (u <= 1.4e-74)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t_1 * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	tmp = 0.0;
	if (u <= -2.8e+58)
		tmp = t_1 / (u / -t1);
	elseif (u <= -1.34e-67)
		tmp = (v * (t1 / (u - t1))) / t1;
	elseif (u <= -4.6e-75)
		tmp = t1 * (t_1 / -u);
	elseif (u <= 1.4e-74)
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t_1 * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.8e+58], N[(t$95$1 / N[(u / (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.34e-67], N[(N[(v * N[(t1 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[u, -4.6e-75], N[(t1 * N[(t$95$1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.4e-74], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -2.7999999999999998e58

   1. Initial program 77.7%

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

      1. associate-*l/ 73.6%

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

      2. *-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*l/ 77.7%

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

      3. times-frac 97.9%

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

      4. frac-2neg 97.9%

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

      5. remove-double-neg 97.9%

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

      6. +-commutative 97.9%

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

      7. distribute-neg-in 97.9%

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

      8. sub-neg 97.9%

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

      9. frac-2neg 97.9%

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

      10. +-commutative 97.9%

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

      11. distribute-neg-in 97.9%

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

      12. sub-neg 97.9%

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

      13. associate-*r/ 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. associate-/l* 97.9%

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

      2. clear-num 97.9%

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

      3. frac-times 85.0%

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

      4. *-un-lft-identity 85.0%

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

      5. neg-mul-1 85.0%

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

      6. times-frac 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. associate-*l/ 98.0%

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

      2. associate-*r/ 98.0%

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

      3. mul-1-neg 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in t1 around 0 92.6%

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

   if -2.7999999999999998e58 < u < -1.3399999999999999e-67

   1. Initial program 78.6%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 71.1%

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

      1. associate-*r/ 77.4%

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

      2. add-sqr-sqrt 77.4%

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

      3. sqrt-unprod 77.4%

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

      4. sqr-neg 77.4%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 78.1%

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

   7. Applied egg-rr 78.1%

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

   if -1.3399999999999999e-67 < u < -4.6e-75

   1. Initial program 59.1%

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

      1. associate-*l/ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 59.1%

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

      3. times-frac 99.7%

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

      4. frac-2neg 99.7%

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

      5. remove-double-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. sub-neg 99.7%

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

      9. frac-2neg 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. sub-neg 99.7%

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

      13. associate-*r/ 74.8%

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

   6. Applied egg-rr 74.8%

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

      1. associate-/l* 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. neg-mul-1 99.5%

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

      6. times-frac 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in t1 around 0 83.3%

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

       1. frac-2neg 83.3%

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

       2. associate-/r/ 100.0%

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

   13. Applied egg-rr 100.0%

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

   if -4.6e-75 < u < 1.39999999999999994e-74

   1. Initial program 58.8%

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

      1. associate-*l/ 65.7%

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

      2. *-commutative 65.7%

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

   3. Simplified 65.7%

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

      1. associate-*r/ 58.8%

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

      2. *-commutative 58.8%

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

      3. times-frac 95.8%

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

      4. frac-2neg 95.8%

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

      5. remove-double-neg 95.8%

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

      6. +-commutative 95.8%

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

      7. distribute-neg-in 95.8%

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

      8. sub-neg 95.8%

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

      9. clear-num 95.7%

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

      10. frac-2neg 95.7%

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

      11. frac-times 99.8%

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

      12. *-un-lft-identity 99.8%

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

      13. +-commutative 99.8%

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

      14. distribute-neg-in 99.8%

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

      15. sub-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in u around 0 85.9%

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

      1. *-commutative 85.9%

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

   9. Simplified 85.9%

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

   if 1.39999999999999994e-74 < u

   1. Initial program 80.9%

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

      1. times-frac 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. distribute-neg-frac2 99.2%

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

      4. +-commutative 99.2%

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

      5. distribute-neg-in 99.2%

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

      6. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t1 around 0 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   7. Simplified 91.4%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{t1 + u} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{\frac{u}{-t1}}\\ \mathbf{elif}\;u \leq -1.34 \cdot 10^{-67}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u - t1}}{t1}\\ \mathbf{elif}\;u \leq -4.6 \cdot 10^{-75}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7 \cdot 10^{+56} \lor \neg \left(u \leq 1.4 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7e+56) (not (<= u 1.4e-74)))
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7e+56) || !(u <= 1.4e-74)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-7d+56)) .or. (.not. (u <= 1.4d-74))) then
        tmp = (v / (t1 + u)) * (t1 / -u)
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7e+56) || !(u <= 1.4e-74)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7e+56) or not (u <= 1.4e-74):
		tmp = (v / (t1 + u)) * (t1 / -u)
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7e+56) || !(u <= 1.4e-74))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7e+56) || ~((u <= 1.4e-74)))
		tmp = (v / (t1 + u)) * (t1 / -u);
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7e+56], N[Not[LessEqual[u, 1.4e-74]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.99999999999999999e56 or 1.39999999999999994e-74 < u

   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 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 91.9%

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

      1. associate-*r/ 91.9%

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

      2. mul-1-neg 91.9%

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

   7. Simplified 91.9%

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

   if -6.99999999999999999e56 < u < 1.39999999999999994e-74

   1. Initial program 63.2%

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

      1. associate-*l/ 70.2%

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

      2. *-commutative 70.2%

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

   3. Simplified 70.2%

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

      1. associate-*r/ 63.2%

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

      2. *-commutative 63.2%

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

      3. times-frac 96.9%

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

      4. frac-2neg 96.9%

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

      5. remove-double-neg 96.9%

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

      6. +-commutative 96.9%

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

      7. distribute-neg-in 96.9%

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

      8. sub-neg 96.9%

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

      9. clear-num 96.9%

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

      10. frac-2neg 96.9%

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

      11. frac-times 99.8%

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

      12. *-un-lft-identity 99.8%

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

      13. +-commutative 99.8%

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

      14. distribute-neg-in 99.8%

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

      15. sub-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in u around 0 79.9%

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

      1. *-commutative 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7 \cdot 10^{+56} \lor \neg \left(u \leq 1.4 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+139} \lor \neg \left(u \leq 3.2 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.75e+139) (not (<= u 3.2e+142)))
   (/ (* v (/ t1 u)) u)
   (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+139) || !(u <= 3.2e+142)) {
		tmp = (v * (t1 / u)) / u;
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.75d+139)) .or. (.not. (u <= 3.2d+142))) then
        tmp = (v * (t1 / u)) / u
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+139) || !(u <= 3.2e+142)) {
		tmp = (v * (t1 / u)) / u;
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.75e+139) or not (u <= 3.2e+142):
		tmp = (v * (t1 / u)) / u
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.75e+139) || !(u <= 3.2e+142))
		tmp = Float64(Float64(v * Float64(t1 / u)) / u);
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.75e+139) || ~((u <= 3.2e+142)))
		tmp = (v * (t1 / u)) / u;
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.75e+139], N[Not[LessEqual[u, 3.2e+142]], $MachinePrecision]], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.74999999999999989e139 or 3.20000000000000005e142 < u

   1. Initial program 74.3%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 56.6%

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

      1. associate-*l/ 47.7%

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

      2. add-sqr-sqrt 19.4%

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

      3. sqrt-unprod 60.6%

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

      4. sqr-neg 60.6%

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

      5. sqrt-unprod 28.3%

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

      6. add-sqr-sqrt 47.7%

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

   7. Applied egg-rr 47.7%

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

   8. Taylor expanded in u around inf 43.6%

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

   9. Taylor expanded in t1 around inf 70.4%

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

       1. *-commutative 70.4%

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

       2. associate-*r/ 70.6%

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

   11. Simplified 70.6%

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

   if -1.74999999999999989e139 < u < 3.20000000000000005e142

   1. Initial program 70.5%

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

      1. associate-*l/ 75.5%

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

      2. *-commutative 75.5%

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

   3. Simplified 75.5%

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

      1. associate-*r/ 70.5%

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

      2. *-commutative 70.5%

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

      3. times-frac 96.9%

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

      4. frac-2neg 96.9%

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

      5. remove-double-neg 96.9%

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

      6. +-commutative 96.9%

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

      7. distribute-neg-in 96.9%

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

      8. sub-neg 96.9%

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

      9. clear-num 96.7%

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

      10. frac-2neg 96.7%

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

      11. frac-times 98.3%

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

      12. *-un-lft-identity 98.3%

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

      13. +-commutative 98.3%

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

      14. distribute-neg-in 98.3%

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

      15. sub-neg 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in u around 0 68.6%

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

      1. *-commutative 68.6%

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

   9. Simplified 68.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+139} \lor \neg \left(u \leq 3.2 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{u}{-t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -3.8e-37)
   (/ v (- (- t1) (* u 2.0)))
   (if (<= t1 2.2e-28) (/ (/ v u) (/ u (- t1))) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.8e-37) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 2.2e-28) {
		tmp = (v / u) / (u / -t1);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-3.8d-37)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (t1 <= 2.2d-28) then
        tmp = (v / u) / (u / -t1)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.8e-37) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 2.2e-28) {
		tmp = (v / u) / (u / -t1);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -3.8e-37:
		tmp = v / (-t1 - (u * 2.0))
	elif t1 <= 2.2e-28:
		tmp = (v / u) / (u / -t1)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -3.8e-37)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (t1 <= 2.2e-28)
		tmp = Float64(Float64(v / u) / Float64(u / Float64(-t1)));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -3.8e-37)
		tmp = v / (-t1 - (u * 2.0));
	elseif (t1 <= 2.2e-28)
		tmp = (v / u) / (u / -t1);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -3.8e-37], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.2e-28], N[(N[(v / u), $MachinePrecision] / N[(u / (-t1)), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.8000000000000004e-37

   1. Initial program 55.3%

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

      1. associate-*l/ 60.3%

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

      2. *-commutative 60.3%

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

   3. Simplified 60.3%

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

      1. associate-*r/ 55.3%

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

      2. *-commutative 55.3%

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

      3. times-frac 99.9%

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

      4. frac-2neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. clear-num 99.9%

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

      10. frac-2neg 99.9%

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

      11. frac-times 92.9%

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

      12. *-un-lft-identity 92.9%

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

      13. +-commutative 92.9%

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

      14. distribute-neg-in 92.9%

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

      15. sub-neg 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in u around 0 83.0%

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

      1. *-commutative 83.0%

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

   9. Simplified 83.0%

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

   if -3.8000000000000004e-37 < t1 < 2.19999999999999996e-28

   1. Initial program 86.7%

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

      1. associate-*l/ 88.8%

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

      2. *-commutative 88.8%

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

   3. Simplified 88.8%

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

      1. *-commutative 88.8%

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

      2. associate-*l/ 86.7%

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

      3. times-frac 95.6%

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

      4. frac-2neg 95.6%

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

      5. remove-double-neg 95.6%

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

      6. +-commutative 95.6%

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

      7. distribute-neg-in 95.6%

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

      8. sub-neg 95.6%

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

      9. frac-2neg 95.6%

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

      10. +-commutative 95.6%

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

      11. distribute-neg-in 95.6%

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

      12. sub-neg 95.6%

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

      13. associate-*r/ 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. associate-/l* 95.6%

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

      2. clear-num 95.3%

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

      3. frac-times 95.3%

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

      4. *-un-lft-identity 95.3%

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

      5. neg-mul-1 95.3%

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

      6. times-frac 95.3%

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

   8. Applied egg-rr 95.3%

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

      1. associate-*l/ 95.4%

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

      2. associate-*r/ 95.4%

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

      3. mul-1-neg 95.4%

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

   10. Simplified 95.4%

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

   11. Taylor expanded in t1 around 0 80.8%

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

   12. Taylor expanded in t1 around 0 83.5%

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

       1. mul-1-neg 83.5%

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

       2. distribute-neg-frac2 83.5%

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

   14. Simplified 83.5%

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

   if 2.19999999999999996e-28 < t1

   1. Initial program 59.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 82.2%

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

      1. associate-*l/ 82.2%

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

      2. add-sqr-sqrt 48.6%

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

      3. sqrt-unprod 82.2%

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

      4. sqr-neg 82.2%

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

      5. sqrt-unprod 33.3%

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

      6. add-sqr-sqrt 82.3%

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

   7. Applied egg-rr 82.3%

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

   8. Taylor expanded in t1 around 0 82.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{u}{-t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+139} \lor \neg \left(u \leq 3.1 \cdot 10^{+156}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.75e+139) (not (<= u 3.1e+156)))
   (/ (* v (/ t1 u)) u)
   (/ v (- (- u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+139) || !(u <= 3.1e+156)) {
		tmp = (v * (t1 / u)) / u;
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.75d+139)) .or. (.not. (u <= 3.1d+156))) then
        tmp = (v * (t1 / u)) / u
    else
        tmp = v / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+139) || !(u <= 3.1e+156)) {
		tmp = (v * (t1 / u)) / u;
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.75e+139) or not (u <= 3.1e+156):
		tmp = (v * (t1 / u)) / u
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.75e+139) || !(u <= 3.1e+156))
		tmp = Float64(Float64(v * Float64(t1 / u)) / u);
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.75e+139) || ~((u <= 3.1e+156)))
		tmp = (v * (t1 / u)) / u;
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.75e+139], N[Not[LessEqual[u, 3.1e+156]], $MachinePrecision]], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.74999999999999989e139 or 3.1000000000000002e156 < u

   1. Initial program 74.3%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 56.6%

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

      1. associate-*l/ 47.7%

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

      2. add-sqr-sqrt 19.4%

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

      3. sqrt-unprod 60.6%

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

      4. sqr-neg 60.6%

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

      5. sqrt-unprod 28.3%

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

      6. add-sqr-sqrt 47.7%

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

   7. Applied egg-rr 47.7%

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

   8. Taylor expanded in u around inf 43.6%

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

   9. Taylor expanded in t1 around inf 70.4%

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

       1. *-commutative 70.4%

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

       2. associate-*r/ 70.6%

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

   11. Simplified 70.6%

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

   if -1.74999999999999989e139 < u < 3.1000000000000002e156

   1. Initial program 70.5%

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

      1. associate-*l/ 75.5%

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

      2. *-commutative 75.5%

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

   3. Simplified 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*l/ 70.5%

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

      3. times-frac 96.9%

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

      4. frac-2neg 96.9%

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

      5. remove-double-neg 96.9%

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

      6. +-commutative 96.9%

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

      7. distribute-neg-in 96.9%

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

      8. sub-neg 96.9%

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

      9. frac-2neg 96.9%

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

      10. +-commutative 96.9%

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

      11. distribute-neg-in 96.9%

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

      12. sub-neg 96.9%

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

      13. associate-*r/ 97.6%

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

   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 inf 68.0%

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

      1. mul-1-neg 68.0%

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

   9. Simplified 68.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+139} \lor \neg \left(u \leq 3.1 \cdot 10^{+156}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+138} \lor \neg \left(u \leq 2.6 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.4e+138) (not (<= u 2.6e+142)))
   (* (/ v u) -0.5)
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+138) || !(u <= 2.6e+142)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.4d+138)) .or. (.not. (u <= 2.6d+142))) then
        tmp = (v / u) * (-0.5d0)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+138) || !(u <= 2.6e+142)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.4e+138) or not (u <= 2.6e+142):
		tmp = (v / u) * -0.5
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.4e+138) || !(u <= 2.6e+142))
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.4e+138) || ~((u <= 2.6e+142)))
		tmp = (v / u) * -0.5;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.4e+138], N[Not[LessEqual[u, 2.6e+142]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.4e138 or 2.60000000000000021e142 < u

   1. Initial program 74.9%

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

      1. associate-*l/ 72.9%

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

      2. *-commutative 72.9%

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

   3. Simplified 72.9%

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

      1. associate-*r/ 74.9%

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

      2. *-commutative 74.9%

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

      3. times-frac 98.6%

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

      4. frac-2neg 98.6%

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

      5. remove-double-neg 98.6%

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

      6. +-commutative 98.6%

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

      7. distribute-neg-in 98.6%

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

      8. sub-neg 98.6%

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

      9. clear-num 98.6%

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

      10. frac-2neg 98.6%

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

      11. frac-times 84.6%

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

      12. *-un-lft-identity 84.6%

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

      13. +-commutative 84.6%

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

      14. distribute-neg-in 84.6%

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

      15. sub-neg 84.6%

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

   6. Applied egg-rr 84.6%

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

   7. Taylor expanded in u around 0 48.2%

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

      1. *-commutative 48.2%

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

   9. Simplified 48.2%

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

   10. Taylor expanded in t1 around 0 45.5%

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

       1. *-commutative 45.5%

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

   12. Simplified 45.5%

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

   if -1.4e138 < u < 2.60000000000000021e142

   1. Initial program 70.1%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in t1 around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+138} \lor \neg \left(u \leq 2.6 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{+141}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.2e+62)
   (/ 1.0 (/ u v))
   (if (<= u 3e+141) (/ v (- t1)) (* (/ v u) -0.5))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.2e+62) {
		tmp = 1.0 / (u / v);
	} else if (u <= 3e+141) {
		tmp = v / -t1;
	} else {
		tmp = (v / u) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.2d+62)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 3d+141) then
        tmp = v / -t1
    else
        tmp = (v / u) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.2e+62) {
		tmp = 1.0 / (u / v);
	} else if (u <= 3e+141) {
		tmp = v / -t1;
	} else {
		tmp = (v / u) * -0.5;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.2e+62:
		tmp = 1.0 / (u / v)
	elif u <= 3e+141:
		tmp = v / -t1
	else:
		tmp = (v / u) * -0.5
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.2e+62)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 3e+141)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(v / u) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.2e+62)
		tmp = 1.0 / (u / v);
	elseif (u <= 3e+141)
		tmp = v / -t1;
	else
		tmp = (v / u) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.2e+62], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3e+141], N[(v / (-t1)), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.2e62

   1. Initial program 76.8%

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

      1. times-frac 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-neg-frac2 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. unsub-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around inf 56.2%

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

      1. associate-*l/ 47.6%

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

      2. add-sqr-sqrt 47.6%

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

      3. sqrt-unprod 54.8%

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

      4. sqr-neg 54.8%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 47.8%

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

   7. Applied egg-rr 47.8%

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

      1. clear-num 49.6%

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

      2. inv-pow 49.6%

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

   9. Applied egg-rr 49.6%

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

       1. unpow-1 49.6%

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

       2. associate-*r/ 51.7%

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

       3. *-commutative 51.7%

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

       4. associate-/l* 49.9%

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

       5. *-inverses 49.9%

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

       6. *-rgt-identity 49.9%

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

   11. Simplified 49.9%

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

   12. Taylor expanded in u around inf 41.8%

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

   if -4.2e62 < u < 2.9999999999999999e141

   1. Initial program 68.5%

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

      1. associate-*l/ 74.6%

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

      2. *-commutative 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in t1 around inf 67.6%

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

      1. associate-*r/ 67.6%

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

      2. neg-mul-1 67.6%

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

   7. Simplified 67.6%

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

   if 2.9999999999999999e141 < u

   1. Initial program 77.5%

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

      1. associate-*l/ 78.2%

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

      2. *-commutative 78.2%

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

   3. Simplified 78.2%

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

      1. associate-*r/ 77.5%

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

      2. *-commutative 77.5%

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

      3. times-frac 99.8%

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

      4. frac-2neg 99.8%

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

      5. remove-double-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-in 99.8%

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

      8. sub-neg 99.8%

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

      9. clear-num 99.8%

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

      10. frac-2neg 99.8%

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

      11. frac-times 86.8%

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

      12. *-un-lft-identity 86.8%

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

      13. +-commutative 86.8%

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

      14. distribute-neg-in 86.8%

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

      15. sub-neg 86.8%

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

   6. Applied egg-rr 86.8%

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

   7. Taylor expanded in u around 0 51.6%

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

      1. *-commutative 51.6%

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

   9. Simplified 51.6%

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

   10. Taylor expanded in t1 around 0 49.4%

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

       1. *-commutative 49.4%

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

   12. Simplified 49.4%

       \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{+141}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+138} \lor \neg \left(u \leq 1.45 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.1e+138) (not (<= u 1.45e+151))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.1e+138) || !(u <= 1.45e+151)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.1d+138)) .or. (.not. (u <= 1.45d+151))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.1e+138) || !(u <= 1.45e+151)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.1e+138) or not (u <= 1.45e+151):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.1e+138) || !(u <= 1.45e+151))
		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 <= -3.1e+138) || ~((u <= 1.45e+151)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.1e+138], N[Not[LessEqual[u, 1.45e+151]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.0999999999999998e138 or 1.45000000000000009e151 < u

   1. Initial program 74.9%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around inf 56.4%

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

      1. associate-*l/ 47.7%

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

      2. add-sqr-sqrt 20.2%

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

      3. sqrt-unprod 60.3%

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

      4. sqr-neg 60.3%

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

      5. sqrt-unprod 27.6%

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

      6. add-sqr-sqrt 47.7%

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

   7. Applied egg-rr 47.7%

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

   8. Taylor expanded in t1 around 0 45.4%

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

   if -3.0999999999999998e138 < u < 1.45000000000000009e151

   1. Initial program 70.1%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in t1 around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+138} \lor \neg \left(u \leq 1.45 \cdot 10^{+151}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.15 \cdot 10^{+138} \lor \neg \left(u \leq 2.4 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.15e+138) (not (<= u 2.4e+143))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.15e+138) || !(u <= 2.4e+143)) {
		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 <= (-2.15d+138)) .or. (.not. (u <= 2.4d+143))) 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 <= -2.15e+138) || !(u <= 2.4e+143)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.15e+138) or not (u <= 2.4e+143):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.15e+138) || !(u <= 2.4e+143))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.15e+138) || ~((u <= 2.4e+143)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.15e+138], N[Not[LessEqual[u, 2.4e+143]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.1499999999999999e138 or 2.3999999999999998e143 < u

   1. Initial program 74.9%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around inf 56.4%

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

   6. Taylor expanded in t1 around 0 45.5%

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

      1. associate-*r/ 45.5%

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

      2. mul-1-neg 45.5%

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

   8. Simplified 45.5%

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

   if -2.1499999999999999e138 < u < 2.3999999999999998e143

   1. Initial program 70.1%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in t1 around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.15 \cdot 10^{+138} \lor \neg \left(u \leq 2.4 \cdot 10^{+143}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 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 71.6%

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

   1. times-frac 97.8%

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

   2. distribute-frac-neg 97.8%

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

   3. distribute-neg-frac2 97.8%

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

   4. +-commutative 97.8%

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

   5. distribute-neg-in 97.8%

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

   6. unsub-neg 97.8%

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

3. Simplified 97.8%

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

5. Final simplification 97.8%

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

Alternative 13: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (/ v (+ t1 u)) (- -1.0 (/ u t1))))

C

double code(double u, double v, double t1) {
	return (v / (t1 + u)) / (-1.0 - (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 / (t1 + u)) / ((-1.0d0) - (u / t1))
end function

Java

public static double code(double u, double v, double t1) {
	return (v / (t1 + u)) / (-1.0 - (u / t1));
}

Python

def code(u, v, t1):
	return (v / (t1 + u)) / (-1.0 - (u / t1))

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(t1 + u)) / Float64(-1.0 - Float64(u / t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / (t1 + u)) / (-1.0 - (u / t1));
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

   1. associate-*l/ 74.9%

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

   2. *-commutative 74.9%

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

3. Simplified 74.9%

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

   1. *-commutative 74.9%

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

   2. associate-*l/ 71.6%

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

   3. times-frac 97.8%

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

   4. frac-2neg 97.8%

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

   5. remove-double-neg 97.8%

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

   6. +-commutative 97.8%

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

   7. distribute-neg-in 97.8%

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

   8. sub-neg 97.8%

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

   9. frac-2neg 97.8%

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

   10. +-commutative 97.8%

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

   11. distribute-neg-in 97.8%

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

   12. sub-neg 97.8%

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

   13. associate-*r/ 98.3%

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

6. Applied egg-rr 98.3%

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

   1. associate-/l* 97.8%

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

   2. clear-num 97.6%

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

   3. frac-times 94.3%

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

   4. *-un-lft-identity 94.3%

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

   5. neg-mul-1 94.3%

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

   6. times-frac 97.6%

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

8. Applied egg-rr 97.6%

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

   1. associate-*l/ 97.7%

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

   2. associate-*r/ 97.7%

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

   3. mul-1-neg 97.7%

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

10. Simplified 97.7%

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

11. Taylor expanded in t1 around inf 97.7%

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

12. Taylor expanded in v around 0 94.3%

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

    1. mul-1-neg 94.3%

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

    2. associate-/l/ 97.7%

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

    3. distribute-frac-neg2 97.7%

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

    4. +-commutative 97.7%

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

    5. distribute-neg-in 97.7%

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

    6. metadata-eval 97.7%

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

    7. unsub-neg 97.7%

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

14. Simplified 97.7%

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

15. Final simplification 97.7%

    \[\leadsto \frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}} \]
16. Add Preprocessing

Alternative 14: 61.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. associate-*l/ 74.9%

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

   2. *-commutative 74.9%

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

3. Simplified 74.9%

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

   1. *-commutative 74.9%

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

   2. associate-*l/ 71.6%

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

   3. times-frac 97.8%

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

   4. frac-2neg 97.8%

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

   5. remove-double-neg 97.8%

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

   6. +-commutative 97.8%

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

   7. distribute-neg-in 97.8%

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

   8. sub-neg 97.8%

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

   9. frac-2neg 97.8%

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

   10. +-commutative 97.8%

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

   11. distribute-neg-in 97.8%

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

   12. sub-neg 97.8%

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

   13. associate-*r/ 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in t1 around inf 61.9%

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

   1. mul-1-neg 61.9%

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

9. Simplified 61.9%

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

10. Final simplification 61.9%

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

Alternative 15: 61.3% 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 71.6%

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

   1. times-frac 97.8%

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

   2. distribute-frac-neg 97.8%

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

   3. distribute-neg-frac2 97.8%

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

   4. +-commutative 97.8%

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

   5. distribute-neg-in 97.8%

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

   6. unsub-neg 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in t1 around inf 65.2%

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

   1. associate-*l/ 61.7%

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

   2. add-sqr-sqrt 33.1%

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

   3. sqrt-unprod 65.8%

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

   4. sqr-neg 65.8%

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

   5. sqrt-unprod 27.9%

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

   6. add-sqr-sqrt 60.3%

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

7. Applied egg-rr 60.3%

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

8. Taylor expanded in t1 around 0 60.4%

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

9. Final simplification 60.4%

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

Alternative 16: 16.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. times-frac 97.8%

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

   2. distribute-frac-neg 97.8%

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

   3. distribute-neg-frac2 97.8%

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

   4. +-commutative 97.8%

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

   5. distribute-neg-in 97.8%

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

   6. unsub-neg 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in t1 around inf 65.2%

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

   1. associate-*l/ 61.7%

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

   2. add-sqr-sqrt 33.1%

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

   3. sqrt-unprod 65.8%

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

   4. sqr-neg 65.8%

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

   5. sqrt-unprod 27.9%

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

   6. add-sqr-sqrt 60.3%

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

7. Applied egg-rr 60.3%

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

8. Taylor expanded in t1 around 0 20.0%

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

9. Final simplification 20.0%

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

Reproduce

herbie shell --seed 2024076 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))