Rosa's DopplerBench

Percentage Accurate: 72.1% → 97.8%

Time: 11.3s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. associate-/l* 75.6%

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

   2. distribute-lft-neg-out 75.6%

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

   3. distribute-rgt-neg-in 75.6%

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

   4. associate-/r* 88.0%

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

   5. distribute-neg-frac2 88.0%

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

3. Simplified 88.0%

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

   1. distribute-frac-neg2 88.0%

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

   2. distribute-rgt-neg-out 88.0%

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

   3. associate-/r* 75.6%

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

   4. distribute-lft-neg-out 75.6%

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

   5. associate-/l* 75.3%

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

   6. times-frac 98.4%

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

   7. frac-2neg 98.4%

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

   8. associate-*r/ 99.2%

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

   9. add-sqr-sqrt 51.7%

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

   10. sqrt-unprod 45.0%

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

   11. sqr-neg 45.0%

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

   12. sqrt-unprod 18.1%

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

   13. add-sqr-sqrt 35.8%

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

   14. add-sqr-sqrt 17.3%

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

   15. sqrt-unprod 57.0%

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

   16. sqr-neg 57.0%

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

   17. sqrt-prod 48.6%

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

   18. add-sqr-sqrt 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-u\right) - t1}\\ \mathbf{if}\;t1 \leq -54000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 7.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{u}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-34} \lor \neg \left(t1 \leq 2.1 \cdot 10^{+22}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- u) t1))))
   (if (<= t1 -54000.0)
     t_1
     (if (<= t1 7.5e-68)
       (/ (* v (/ t1 (- u))) u)
       (if (or (<= t1 1.35e-34) (not (<= t1 2.1e+22)))
         t_1
         (/ t1 (* u (/ u (- v)))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-u - t1);
	double tmp;
	if (t1 <= -54000.0) {
		tmp = t_1;
	} else if (t1 <= 7.5e-68) {
		tmp = (v * (t1 / -u)) / u;
	} else if ((t1 <= 1.35e-34) || !(t1 <= 2.1e+22)) {
		tmp = t_1;
	} else {
		tmp = t1 / (u * (u / -v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (-u - t1)
    if (t1 <= (-54000.0d0)) then
        tmp = t_1
    else if (t1 <= 7.5d-68) then
        tmp = (v * (t1 / -u)) / u
    else if ((t1 <= 1.35d-34) .or. (.not. (t1 <= 2.1d+22))) then
        tmp = t_1
    else
        tmp = t1 / (u * (u / -v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (-u - t1);
	double tmp;
	if (t1 <= -54000.0) {
		tmp = t_1;
	} else if (t1 <= 7.5e-68) {
		tmp = (v * (t1 / -u)) / u;
	} else if ((t1 <= 1.35e-34) || !(t1 <= 2.1e+22)) {
		tmp = t_1;
	} else {
		tmp = t1 / (u * (u / -v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-u - t1)
	tmp = 0
	if t1 <= -54000.0:
		tmp = t_1
	elif t1 <= 7.5e-68:
		tmp = (v * (t1 / -u)) / u
	elif (t1 <= 1.35e-34) or not (t1 <= 2.1e+22):
		tmp = t_1
	else:
		tmp = t1 / (u * (u / -v))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-u) - t1))
	tmp = 0.0
	if (t1 <= -54000.0)
		tmp = t_1;
	elseif (t1 <= 7.5e-68)
		tmp = Float64(Float64(v * Float64(t1 / Float64(-u))) / u);
	elseif ((t1 <= 1.35e-34) || !(t1 <= 2.1e+22))
		tmp = t_1;
	else
		tmp = Float64(t1 / Float64(u * Float64(u / Float64(-v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-u - t1);
	tmp = 0.0;
	if (t1 <= -54000.0)
		tmp = t_1;
	elseif (t1 <= 7.5e-68)
		tmp = (v * (t1 / -u)) / u;
	elseif ((t1 <= 1.35e-34) || ~((t1 <= 2.1e+22)))
		tmp = t_1;
	else
		tmp = t1 / (u * (u / -v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -54000.0], t$95$1, If[LessEqual[t1, 7.5e-68], N[(N[(v * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[Or[LessEqual[t1, 1.35e-34], N[Not[LessEqual[t1, 2.1e+22]], $MachinePrecision]], t$95$1, N[(t1 / N[(u * N[(u / (-v)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -54000 or 7.50000000000000081e-68 < t1 < 1.35000000000000008e-34 or 2.0999999999999998e22 < t1

   1. Initial program 66.9%

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

      1. associate-/l* 66.2%

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

      2. distribute-lft-neg-out 66.2%

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

      3. distribute-rgt-neg-in 66.2%

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

      4. associate-/r* 85.0%

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

      5. distribute-neg-frac2 85.0%

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

   3. Simplified 85.0%

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

      1. distribute-frac-neg2 85.0%

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

      2. distribute-rgt-neg-out 85.0%

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

      3. associate-/r* 66.2%

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

      4. distribute-lft-neg-out 66.2%

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

      5. associate-/l* 66.9%

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

      6. times-frac 99.9%

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

      7. frac-2neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 54.1%

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

      10. sqrt-unprod 31.6%

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

      11. sqr-neg 31.6%

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

      12. sqrt-unprod 13.7%

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

      13. add-sqr-sqrt 34.4%

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

      14. add-sqr-sqrt 17.7%

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

      15. sqrt-unprod 54.9%

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

      16. sqr-neg 54.9%

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

      17. sqrt-prod 49.3%

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

      18. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 84.2%

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

      1. mul-1-neg 84.2%

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

   9. Simplified 84.2%

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

   if -54000 < t1 < 7.50000000000000081e-68

   1. Initial program 82.5%

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

      1. times-frac 96.6%

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

      2. distribute-frac-neg 96.6%

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

      3. distribute-neg-frac2 96.6%

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

      4. +-commutative 96.6%

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

      5. distribute-neg-in 96.6%

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

      6. unsub-neg 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. mul-1-neg 76.5%

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

   7. Simplified 76.5%

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

      1. clear-num 76.4%

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

      2. frac-times 75.3%

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

      3. *-commutative 75.3%

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

      4. *-un-lft-identity 75.3%

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

      5. add-sqr-sqrt 38.9%

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

      6. sqrt-unprod 47.4%

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

      7. sqr-neg 47.4%

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

      8. sqrt-unprod 22.4%

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

      9. add-sqr-sqrt 41.5%

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

      10. frac-2neg 41.5%

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

      11. distribute-neg-in 41.5%

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

      12. add-sqr-sqrt 19.1%

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

      13. sqrt-unprod 39.6%

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

      14. sqr-neg 39.6%

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

      15. sqrt-unprod 21.5%

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

      16. add-sqr-sqrt 39.7%

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

      17. sub-neg 39.7%

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

      18. add-sqr-sqrt 16.4%

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

      19. sqrt-unprod 48.9%

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

      20. sqr-neg 48.9%

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

      21. sqrt-unprod 44.9%

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

      22. add-sqr-sqrt 77.1%

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

   9. Applied egg-rr 77.1%

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

       1. *-commutative 77.1%

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

       2. associate-/r* 78.9%

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

       3. associate-/r/ 79.9%

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

       4. associate-*l/ 73.6%

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

       5. associate-*r/ 78.2%

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

       6. *-commutative 78.2%

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

       7. associate-/r/ 79.8%

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

       8. div-sub 79.8%

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

       9. *-inverses 79.8%

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

   11. Simplified 79.8%

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

   12. Taylor expanded in u around inf 76.4%

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

       1. mul-1-neg 76.4%

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

       2. *-commutative 76.4%

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

       3. associate-*r/ 82.8%

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

       4. distribute-lft-neg-out 82.8%

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

   14. Simplified 82.8%

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

   if 1.35000000000000008e-34 < t1 < 2.0999999999999998e22

   1. Initial program 92.8%

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

      1. times-frac 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. distribute-neg-frac2 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-neg-in 99.6%

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

      6. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t1 around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. mul-1-neg 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in t1 around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. clear-num 73.2%

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

      3. frac-2neg 73.2%

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

      4. frac-times 73.4%

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

      5. *-un-lft-identity 73.4%

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

      6. remove-double-neg 73.4%

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

   10. Applied egg-rr 73.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -54000:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 7.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{u}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-34} \lor \neg \left(t1 \leq 2.1 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{if}\;u \leq -5.5 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6.6 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ t1 (* u (/ u (- v))))))
   (if (<= u -5.5e-8)
     t_1
     (if (<= u 2.8e+14)
       (/ (- v) t1)
       (if (<= u 6.6e+72)
         t_1
         (if (<= u 3.5e+81) (/ v (- (- u) t1)) (* (/ t1 (- u)) (/ v u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 / (u * (u / -v));
	double tmp;
	if (u <= -5.5e-8) {
		tmp = t_1;
	} else if (u <= 2.8e+14) {
		tmp = -v / t1;
	} else if (u <= 6.6e+72) {
		tmp = t_1;
	} else if (u <= 3.5e+81) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t1 / (u * (u / -v))
    if (u <= (-5.5d-8)) then
        tmp = t_1
    else if (u <= 2.8d+14) then
        tmp = -v / t1
    else if (u <= 6.6d+72) then
        tmp = t_1
    else if (u <= 3.5d+81) then
        tmp = v / (-u - t1)
    else
        tmp = (t1 / -u) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 / (u * (u / -v));
	double tmp;
	if (u <= -5.5e-8) {
		tmp = t_1;
	} else if (u <= 2.8e+14) {
		tmp = -v / t1;
	} else if (u <= 6.6e+72) {
		tmp = t_1;
	} else if (u <= 3.5e+81) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 / (u * (u / -v))
	tmp = 0
	if u <= -5.5e-8:
		tmp = t_1
	elif u <= 2.8e+14:
		tmp = -v / t1
	elif u <= 6.6e+72:
		tmp = t_1
	elif u <= 3.5e+81:
		tmp = v / (-u - t1)
	else:
		tmp = (t1 / -u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 / Float64(u * Float64(u / Float64(-v))))
	tmp = 0.0
	if (u <= -5.5e-8)
		tmp = t_1;
	elseif (u <= 2.8e+14)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 6.6e+72)
		tmp = t_1;
	elseif (u <= 3.5e+81)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 / (u * (u / -v));
	tmp = 0.0;
	if (u <= -5.5e-8)
		tmp = t_1;
	elseif (u <= 2.8e+14)
		tmp = -v / t1;
	elseif (u <= 6.6e+72)
		tmp = t_1;
	elseif (u <= 3.5e+81)
		tmp = v / (-u - t1);
	else
		tmp = (t1 / -u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 / N[(u * N[(u / (-v)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.5e-8], t$95$1, If[LessEqual[u, 2.8e+14], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 6.6e+72], t$95$1, If[LessEqual[u, 3.5e+81], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -5.5000000000000003e-8 or 2.8e14 < u < 6.6e72

   1. Initial program 86.2%

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

      1. times-frac 98.7%

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

      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 84.5%

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

      1. associate-*r/ 84.5%

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

      2. mul-1-neg 84.5%

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

   7. Simplified 84.5%

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

   8. Taylor expanded in t1 around 0 77.2%

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

      1. *-commutative 77.2%

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

      2. clear-num 78.3%

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

      3. frac-2neg 78.3%

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

      4. frac-times 82.1%

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

      5. *-un-lft-identity 82.1%

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

      6. remove-double-neg 82.1%

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

   10. Applied egg-rr 82.1%

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

   if -5.5000000000000003e-8 < u < 2.8e14

   1. Initial program 67.5%

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

      1. associate-/l* 68.0%

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

      2. distribute-lft-neg-out 68.0%

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

      3. distribute-rgt-neg-in 68.0%

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

      4. associate-/r* 83.0%

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

      5. distribute-neg-frac2 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in t1 around inf 79.2%

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

      1. associate-*r/ 79.2%

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

      2. neg-mul-1 79.2%

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

   7. Simplified 79.2%

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

   if 6.6e72 < u < 3.5e81

   1. Initial program 44.4%

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

      1. associate-/l* 59.3%

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

      2. distribute-lft-neg-out 59.3%

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

      3. distribute-rgt-neg-in 59.3%

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

      4. associate-/r* 73.2%

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

      5. distribute-neg-frac2 73.2%

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

   3. Simplified 73.2%

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

      1. distribute-frac-neg2 73.2%

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

      2. distribute-rgt-neg-out 73.2%

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

      3. associate-/r* 59.3%

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

      4. distribute-lft-neg-out 59.3%

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

      5. associate-/l* 44.4%

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

      6. times-frac 100.0%

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

      7. frac-2neg 100.0%

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

      8. associate-*r/ 100.0%

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

      9. add-sqr-sqrt 27.8%

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

      10. sqrt-unprod 29.9%

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

      11. sqr-neg 29.9%

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

      12. sqrt-unprod 29.8%

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

      13. add-sqr-sqrt 30.5%

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

      14. add-sqr-sqrt 0.7%

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

      15. sqrt-unprod 59.3%

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

      16. sqr-neg 59.3%

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

      17. sqrt-prod 71.4%

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

      18. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 87.2%

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

      1. mul-1-neg 87.2%

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

   9. Simplified 87.2%

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

   if 3.5e81 < u

   1. Initial program 84.2%

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

      1. 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 0 96.8%

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

      1. associate-*r/ 96.8%

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

      2. mul-1-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in t1 around 0 94.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 6.6 \cdot 10^{+72}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := \frac{v}{t\_1}\\ \mathbf{if}\;t1 \leq -2.3 \cdot 10^{+130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -1.7 \cdot 10^{-108}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{elif}\;t1 \leq 3.4 \cdot 10^{+163}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (/ v t_1)))
   (if (<= t1 -2.3e+130)
     t_2
     (if (<= t1 -1.7e-108)
       (* v (/ t1 (* (+ t1 u) t_1)))
       (if (<= t1 3.4e+163) (* t1 (/ (/ v (+ t1 u)) t_1)) t_2)))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / t_1;
	double tmp;
	if (t1 <= -2.3e+130) {
		tmp = t_2;
	} else if (t1 <= -1.7e-108) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else if (t1 <= 3.4e+163) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} 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 = -u - t1
    t_2 = v / t_1
    if (t1 <= (-2.3d+130)) then
        tmp = t_2
    else if (t1 <= (-1.7d-108)) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else if (t1 <= 3.4d+163) then
        tmp = t1 * ((v / (t1 + u)) / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / t_1;
	double tmp;
	if (t1 <= -2.3e+130) {
		tmp = t_2;
	} else if (t1 <= -1.7e-108) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else if (t1 <= 3.4e+163) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	t_2 = v / t_1
	tmp = 0
	if t1 <= -2.3e+130:
		tmp = t_2
	elif t1 <= -1.7e-108:
		tmp = v * (t1 / ((t1 + u) * t_1))
	elif t1 <= 3.4e+163:
		tmp = t1 * ((v / (t1 + u)) / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(v / t_1)
	tmp = 0.0
	if (t1 <= -2.3e+130)
		tmp = t_2;
	elseif (t1 <= -1.7e-108)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	elseif (t1 <= 3.4e+163)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	t_2 = v / t_1;
	tmp = 0.0;
	if (t1 <= -2.3e+130)
		tmp = t_2;
	elseif (t1 <= -1.7e-108)
		tmp = v * (t1 / ((t1 + u) * t_1));
	elseif (t1 <= 3.4e+163)
		tmp = t1 * ((v / (t1 + u)) / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(v / t$95$1), $MachinePrecision]}, If[LessEqual[t1, -2.3e+130], t$95$2, If[LessEqual[t1, -1.7e-108], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.4e+163], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.30000000000000021e130 or 3.4000000000000001e163 < t1

   1. Initial program 46.1%

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

      1. associate-/l* 47.5%

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

      2. distribute-lft-neg-out 47.5%

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

      3. distribute-rgt-neg-in 47.5%

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

      4. associate-/r* 81.1%

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

      5. distribute-neg-frac2 81.1%

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

   3. Simplified 81.1%

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

      1. distribute-frac-neg2 81.1%

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

      2. distribute-rgt-neg-out 81.1%

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

      3. associate-/r* 47.5%

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

      4. distribute-lft-neg-out 47.5%

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

      5. associate-/l* 46.1%

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

      6. times-frac 100.0%

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

      7. frac-2neg 100.0%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 65.8%

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

      10. sqrt-unprod 9.9%

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

      11. sqr-neg 9.9%

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

      12. sqrt-unprod 16.7%

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

      13. add-sqr-sqrt 43.4%

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

      14. add-sqr-sqrt 20.2%

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

      15. sqrt-unprod 44.2%

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

      16. sqr-neg 44.2%

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

      17. sqrt-prod 41.7%

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

      18. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 92.6%

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

      1. mul-1-neg 92.6%

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

   9. Simplified 92.6%

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

   if -2.30000000000000021e130 < t1 < -1.70000000000000001e-108

   1. Initial program 97.9%

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

      1. associate-*l/ 97.8%

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

      2. *-commutative 97.8%

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

   3. Simplified 97.8%

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

   if -1.70000000000000001e-108 < t1 < 3.4000000000000001e163

   1. Initial program 80.1%

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

      1. associate-/l* 85.0%

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

      2. distribute-lft-neg-out 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

      4. associate-/r* 92.5%

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

      5. distribute-neg-frac2 92.5%

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

   3. Simplified 92.5%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.3 \cdot 10^{+130}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -1.7 \cdot 10^{-108}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;t1 \leq 3.4 \cdot 10^{+163}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{-74} \lor \neg \left(u \leq 1.8 \cdot 10^{-159}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.2e-74) (not (<= u 1.8e-159)))
   (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e-74) || !(u <= 1.8e-159)) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.2d-74)) .or. (.not. (u <= 1.8d-159))) then
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e-74) || !(u <= 1.8e-159)) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.2e-74) or not (u <= 1.8e-159):
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.2e-74) || !(u <= 1.8e-159))
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.2e-74) || ~((u <= 1.8e-159)))
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.2e-74], N[Not[LessEqual[u, 1.8e-159]], $MachinePrecision]], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.1999999999999999e-74 or 1.80000000000000011e-159 < u

   1. Initial program 77.9%

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

      1. associate-/l* 79.9%

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

      2. distribute-lft-neg-out 79.9%

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

      3. distribute-rgt-neg-in 79.9%

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

      4. associate-/r* 92.1%

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

      5. distribute-neg-frac2 92.1%

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

   3. Simplified 92.1%

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

   if -3.1999999999999999e-74 < u < 1.80000000000000011e-159

   1. Initial program 69.7%

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

      1. associate-/l* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 79.3%

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

      5. distribute-neg-frac2 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in t1 around inf 91.1%

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

      1. associate-*r/ 91.1%

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

      2. neg-mul-1 91.1%

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

   7. Simplified 91.1%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{-74} \lor \neg \left(u \leq 1.8 \cdot 10^{-159}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{-9} \lor \neg \left(u \leq 1.75 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.7e-9) (not (<= u 1.75e+14)))
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.7e-9) || !(u <= 1.75e+14)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.7d-9)) .or. (.not. (u <= 1.75d+14))) then
        tmp = (v / (t1 + u)) * (t1 / -u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.7e-9) || !(u <= 1.75e+14)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.7e-9) or not (u <= 1.75e+14):
		tmp = (v / (t1 + u)) * (t1 / -u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.7e-9) || !(u <= 1.75e+14))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.7e-9) || ~((u <= 1.75e+14)))
		tmp = (v / (t1 + u)) * (t1 / -u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.7e-9], N[Not[LessEqual[u, 1.75e+14]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.7e-9 or 1.75e14 < u

   1. Initial program 83.2%

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

      1. times-frac 99.2%

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

      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 85.7%

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

      1. associate-*r/ 85.7%

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

      2. mul-1-neg 85.7%

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

   7. Simplified 85.7%

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

   if -3.7e-9 < u < 1.75e14

   1. Initial program 67.5%

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

      1. associate-/l* 68.0%

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

      2. distribute-lft-neg-out 68.0%

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

      3. distribute-rgt-neg-in 68.0%

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

      4. associate-/r* 83.0%

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

      5. distribute-neg-frac2 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in t1 around inf 79.2%

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

      1. associate-*r/ 79.2%

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

      2. neg-mul-1 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{-9} \lor \neg \left(u \leq 1.75 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-9} \lor \neg \left(u \leq 65000000000000\right):\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.8e-9) (not (<= u 65000000000000.0)))
   (/ (/ v (- t1 u)) (/ u t1))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e-9) || !(u <= 65000000000000.0)) {
		tmp = (v / (t1 - u)) / (u / t1);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.8d-9)) .or. (.not. (u <= 65000000000000.0d0))) then
        tmp = (v / (t1 - u)) / (u / t1)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e-9) || !(u <= 65000000000000.0)) {
		tmp = (v / (t1 - u)) / (u / t1);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.8e-9) or not (u <= 65000000000000.0):
		tmp = (v / (t1 - u)) / (u / t1)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.8e-9) || !(u <= 65000000000000.0))
		tmp = Float64(Float64(v / Float64(t1 - u)) / Float64(u / t1));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.8e-9) || ~((u <= 65000000000000.0)))
		tmp = (v / (t1 - u)) / (u / t1);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.8e-9], N[Not[LessEqual[u, 65000000000000.0]], $MachinePrecision]], N[(N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / N[(u / t1), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.79999999999999982e-9 or 6.5e13 < u

   1. Initial program 83.2%

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

      1. times-frac 99.2%

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

      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 85.7%

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

      1. associate-*r/ 85.7%

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

      2. mul-1-neg 85.7%

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

   7. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. clear-num 85.6%

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

      3. un-div-inv 85.6%

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

      4. frac-2neg 85.6%

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

      5. add-sqr-sqrt 41.5%

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

      6. sqrt-unprod 58.3%

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

      7. sqr-neg 58.3%

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

      8. sqrt-unprod 31.5%

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

      9. add-sqr-sqrt 57.8%

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

      10. distribute-neg-in 57.8%

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

      11. add-sqr-sqrt 28.3%

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

      12. sqrt-unprod 57.9%

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

      13. sqr-neg 57.9%

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

      14. sqrt-unprod 29.4%

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

      15. add-sqr-sqrt 57.9%

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

      16. sub-neg 57.9%

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

      17. add-sqr-sqrt 28.5%

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

      18. sqrt-unprod 57.3%

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

      19. sqr-neg 57.3%

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

      20. sqrt-unprod 42.6%

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

      21. add-sqr-sqrt 85.4%

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

   9. Applied egg-rr 85.4%

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

   if -5.79999999999999982e-9 < u < 6.5e13

   1. Initial program 67.5%

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

      1. associate-/l* 68.0%

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

      2. distribute-lft-neg-out 68.0%

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

      3. distribute-rgt-neg-in 68.0%

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

      4. associate-/r* 83.0%

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

      5. distribute-neg-frac2 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in t1 around inf 79.2%

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

      1. associate-*r/ 79.2%

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

      2. neg-mul-1 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-9} \lor \neg \left(u \leq 65000000000000\right):\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{-7} \lor \neg \left(u \leq 2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.7e-7) (not (<= u 2e+14)))
   (* (/ t1 (- u)) (/ v u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e-7) || !(u <= 2e+14)) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.7d-7)) .or. (.not. (u <= 2d+14))) then
        tmp = (t1 / -u) * (v / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e-7) || !(u <= 2e+14)) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.7e-7) or not (u <= 2e+14):
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.7e-7) || !(u <= 2e+14))
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.7e-7) || ~((u <= 2e+14)))
		tmp = (t1 / -u) * (v / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.7e-7], N[Not[LessEqual[u, 2e+14]], $MachinePrecision]], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.70000000000000009e-7 or 2e14 < u

   1. Initial program 83.2%

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

      1. times-frac 99.2%

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

      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 85.7%

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

      1. associate-*r/ 85.7%

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

      2. mul-1-neg 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in t1 around 0 80.4%

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

   if -2.70000000000000009e-7 < u < 2e14

   1. Initial program 67.5%

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

      1. associate-/l* 68.0%

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

      2. distribute-lft-neg-out 68.0%

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

      3. distribute-rgt-neg-in 68.0%

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

      4. associate-/r* 83.0%

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

      5. distribute-neg-frac2 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in t1 around inf 79.2%

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

      1. associate-*r/ 79.2%

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

      2. neg-mul-1 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{-7} \lor \neg \left(u \leq 2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+58} \lor \neg \left(u \leq 6.8 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9.4e+58) (not (<= u 6.8e+84)))
   (/ t1 (* u (/ u v)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.4e+58) || !(u <= 6.8e+84)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-9.4d+58)) .or. (.not. (u <= 6.8d+84))) then
        tmp = t1 / (u * (u / v))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.4e+58) || !(u <= 6.8e+84)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9.4e+58) or not (u <= 6.8e+84):
		tmp = t1 / (u * (u / v))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9.4e+58) || !(u <= 6.8e+84))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9.4e+58) || ~((u <= 6.8e+84)))
		tmp = t1 / (u * (u / v));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9.4e+58], N[Not[LessEqual[u, 6.8e+84]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -9.39999999999999944e58 or 6.7999999999999996e84 < u

   1. Initial program 83.7%

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

      1. times-frac 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t1 around 0 93.4%

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

      1. associate-*r/ 93.4%

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

      2. mul-1-neg 93.4%

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

   7. Simplified 93.4%

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

   8. Taylor expanded in t1 around 0 88.0%

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

      1. *-commutative 88.0%

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

      2. clear-num 88.0%

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

      3. frac-times 89.8%

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

      4. *-un-lft-identity 89.8%

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

      5. add-sqr-sqrt 43.8%

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

      6. sqrt-unprod 62.0%

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

      7. sqr-neg 62.0%

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

      8. sqrt-unprod 37.2%

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

      9. add-sqr-sqrt 71.9%

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

   10. Applied egg-rr 71.9%

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

   if -9.39999999999999944e58 < u < 6.7999999999999996e84

   1. Initial program 70.9%

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

      1. associate-/l* 71.8%

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

      2. distribute-lft-neg-out 71.8%

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

      3. distribute-rgt-neg-in 71.8%

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

      4. associate-/r* 85.0%

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

      5. distribute-neg-frac2 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in t1 around inf 70.6%

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

      1. associate-*r/ 70.6%

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

      2. neg-mul-1 70.6%

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

   7. Simplified 70.6%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+58} \lor \neg \left(u \leq 6.8 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+146} \lor \neg \left(u \leq 9.5 \cdot 10^{+150}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.35e+146) (not (<= u 9.5e+150))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.35e+146) || !(u <= 9.5e+150)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.35d+146)) .or. (.not. (u <= 9.5d+150))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.35e+146) || !(u <= 9.5e+150)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.35e+146) or not (u <= 9.5e+150):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.35e+146) || !(u <= 9.5e+150))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.35e+146) || ~((u <= 9.5e+150)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.35e+146], N[Not[LessEqual[u, 9.5e+150]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.34999999999999994e146 or 9.5000000000000001e150 < u

   1. Initial program 80.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 96.5%

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

      1. associate-*r/ 96.5%

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

      2. mul-1-neg 96.5%

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

   7. Simplified 96.5%

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

      1. clear-num 96.5%

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

      2. frac-times 89.6%

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

      3. *-commutative 89.6%

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

      4. *-un-lft-identity 89.6%

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

      5. add-sqr-sqrt 49.2%

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

      6. sqrt-unprod 67.7%

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

      7. sqr-neg 67.7%

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

      8. sqrt-unprod 37.2%

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

      9. add-sqr-sqrt 79.5%

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

      10. frac-2neg 79.5%

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

      11. distribute-neg-in 79.5%

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

      12. add-sqr-sqrt 42.3%

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

      13. sqrt-unprod 79.6%

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

      14. sqr-neg 79.6%

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

      15. sqrt-unprod 37.2%

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

      16. add-sqr-sqrt 79.4%

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

      17. sub-neg 79.4%

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

      18. add-sqr-sqrt 32.2%

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

      19. sqrt-unprod 64.3%

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

      20. sqr-neg 64.3%

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

      21. sqrt-unprod 52.1%

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

      22. add-sqr-sqrt 89.6%

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

   9. Applied egg-rr 89.6%

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

       1. *-commutative 89.6%

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

       2. associate-/r* 96.5%

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

       3. associate-/r/ 96.5%

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

       4. associate-*l/ 88.7%

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

       5. associate-*r/ 96.5%

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

       6. *-commutative 96.5%

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

       7. associate-/r/ 96.5%

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

       8. div-sub 96.5%

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

       9. *-inverses 96.5%

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

   11. Simplified 96.5%

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

   12. Taylor expanded in u around 0 39.8%

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

   if -1.34999999999999994e146 < u < 9.5000000000000001e150

   1. Initial program 73.6%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 86.7%

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

      5. distribute-neg-frac2 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in t1 around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   7. Simplified 64.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+146} \lor \neg \left(u \leq 9.5 \cdot 10^{+150}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{+146} \lor \neg \left(u \leq 2.2 \cdot 10^{+150}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.5e+146) (not (<= u 2.2e+150))) (/ v (- u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.5e+146) || !(u <= 2.2e+150)) {
		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 <= (-4.5d+146)) .or. (.not. (u <= 2.2d+150))) 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 <= -4.5e+146) || !(u <= 2.2e+150)) {
		tmp = v / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.5e+146) or not (u <= 2.2e+150):
		tmp = v / -u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.5e+146) || !(u <= 2.2e+150))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.5e+146) || ~((u <= 2.2e+150)))
		tmp = v / -u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.5e+146], N[Not[LessEqual[u, 2.2e+150]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.50000000000000026e146 or 2.19999999999999999e150 < u

   1. Initial program 80.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 96.5%

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

      1. associate-*r/ 96.5%

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

      2. mul-1-neg 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in t1 around inf 40.2%

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

      1. associate-*r/ 40.2%

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

      2. mul-1-neg 40.2%

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

   10. Simplified 40.2%

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

   if -4.50000000000000026e146 < u < 2.19999999999999999e150

   1. Initial program 73.6%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 86.7%

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

      5. distribute-neg-frac2 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in t1 around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   7. Simplified 64.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.5 \cdot 10^{+146} \lor \neg \left(u \leq 2.2 \cdot 10^{+150}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 23.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{+160} \lor \neg \left(t1 \leq 6.2 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.05e+160) (not (<= t1 6.2e+73))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.05e+160) || !(t1 <= 6.2e+73)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.05d+160)) .or. (.not. (t1 <= 6.2d+73))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.05e+160) || !(t1 <= 6.2e+73)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.05e+160) or not (t1 <= 6.2e+73):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.05e+160) || !(t1 <= 6.2e+73))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.05e+160) || ~((t1 <= 6.2e+73)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.05e+160], N[Not[LessEqual[t1, 6.2e+73]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.04999999999999998e160 or 6.1999999999999999e73 < t1

   1. Initial program 49.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 88.1%

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

   6. Taylor expanded in u around inf 38.1%

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

   if -1.04999999999999998e160 < t1 < 6.1999999999999999e73

   1. Initial program 85.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 0 66.2%

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

      1. associate-*r/ 66.2%

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

      2. mul-1-neg 66.2%

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

   7. Simplified 66.2%

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

      1. clear-num 66.2%

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

      2. frac-times 64.5%

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

      3. *-commutative 64.5%

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

      4. *-un-lft-identity 64.5%

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

      5. add-sqr-sqrt 33.6%

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

      6. sqrt-unprod 42.3%

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

      7. sqr-neg 42.3%

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

      8. sqrt-unprod 18.7%

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

      9. add-sqr-sqrt 37.3%

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

      10. frac-2neg 37.3%

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

      11. distribute-neg-in 37.3%

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

      12. add-sqr-sqrt 18.7%

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

      13. sqrt-unprod 36.1%

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

      14. sqr-neg 36.1%

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

      15. sqrt-unprod 18.0%

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

      16. add-sqr-sqrt 36.4%

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

      17. sub-neg 36.4%

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

      18. add-sqr-sqrt 16.6%

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

      19. sqrt-unprod 42.2%

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

      20. sqr-neg 42.2%

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

      21. sqrt-unprod 35.1%

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

      22. add-sqr-sqrt 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. *-commutative 65.5%

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

       2. associate-/r* 67.5%

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

       3. associate-/r/ 67.4%

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

       4. associate-*l/ 62.8%

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

       5. associate-*r/ 67.1%

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

       6. *-commutative 67.1%

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

       7. associate-/r/ 67.3%

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

       8. div-sub 67.3%

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

       9. *-inverses 67.3%

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

   11. Simplified 67.3%

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

   12. Taylor expanded in u around 0 18.1%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.05 \cdot 10^{+160} \lor \neg \left(t1 \leq 6.2 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.9% 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 75.3%

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

   1. times-frac 98.4%

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

   2. distribute-frac-neg 98.4%

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

   3. distribute-neg-frac2 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-neg-in 98.4%

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

   6. unsub-neg 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 14: 61.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. associate-/l* 75.6%

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

   2. distribute-lft-neg-out 75.6%

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

   3. distribute-rgt-neg-in 75.6%

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

   4. associate-/r* 88.0%

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

   5. distribute-neg-frac2 88.0%

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

3. Simplified 88.0%

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

   1. distribute-frac-neg2 88.0%

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

   2. distribute-rgt-neg-out 88.0%

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

   3. associate-/r* 75.6%

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

   4. distribute-lft-neg-out 75.6%

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

   5. associate-/l* 75.3%

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

   6. times-frac 98.4%

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

   7. frac-2neg 98.4%

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

   8. associate-*r/ 99.2%

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

   9. add-sqr-sqrt 51.7%

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

   10. sqrt-unprod 45.0%

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

   11. sqr-neg 45.0%

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

   12. sqrt-unprod 18.1%

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

   13. add-sqr-sqrt 35.8%

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

   14. add-sqr-sqrt 17.3%

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

   15. sqrt-unprod 57.0%

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

   16. sqr-neg 57.0%

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

   17. sqrt-prod 48.6%

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

   18. add-sqr-sqrt 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in t1 around inf 59.6%

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

   1. mul-1-neg 59.6%

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

9. Simplified 59.6%

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

10. Final simplification 59.6%

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

Alternative 15: 61.4% 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 75.3%

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

   1. associate-/l* 75.6%

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

   2. distribute-lft-neg-out 75.6%

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

   3. distribute-rgt-neg-in 75.6%

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

   4. associate-/r* 88.0%

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

   5. distribute-neg-frac2 88.0%

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

3. Simplified 88.0%

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

   1. distribute-frac-neg2 88.0%

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

   2. distribute-rgt-neg-out 88.0%

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

   3. associate-/r* 75.6%

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

   4. distribute-lft-neg-out 75.6%

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

   5. associate-/l* 75.3%

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

   6. times-frac 98.4%

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

   7. frac-2neg 98.4%

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

   8. associate-*r/ 99.2%

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

   9. add-sqr-sqrt 51.7%

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

   10. sqrt-unprod 45.0%

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

   11. sqr-neg 45.0%

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

   12. sqrt-unprod 18.1%

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

   13. add-sqr-sqrt 35.8%

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

   14. add-sqr-sqrt 17.3%

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

   15. sqrt-unprod 57.0%

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

   16. sqr-neg 57.0%

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

   17. sqrt-prod 48.6%

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

   18. add-sqr-sqrt 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in t1 around inf 59.6%

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

   1. mul-1-neg 59.6%

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

9. Simplified 59.6%

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

    1. distribute-frac-neg 59.6%

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

    2. *-un-lft-identity 59.6%

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

    3. associate-*l/ 59.4%

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

    4. distribute-lft-neg-in 59.4%

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

11. Applied egg-rr 59.4%

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

    1. add-sqr-sqrt 30.9%

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

    2. sqrt-unprod 43.9%

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

    3. sqr-neg 43.9%

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

    4. sqrt-unprod 11.9%

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

    5. add-sqr-sqrt 22.4%

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

    6. frac-2neg 22.4%

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

    7. metadata-eval 22.4%

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

    8. associate-*l/ 22.4%

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

    9. neg-mul-1 22.4%

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

    10. distribute-neg-in 22.4%

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

    11. add-sqr-sqrt 11.8%

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

    12. sqrt-unprod 33.8%

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

    13. sqr-neg 33.8%

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

    14. sqrt-unprod 27.5%

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

    15. add-sqr-sqrt 59.6%

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

    16. sub-neg 59.6%

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

13. Applied egg-rr 59.6%

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

14. Final simplification 59.6%

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

Alternative 16: 13.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. times-frac 98.4%

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

   2. distribute-frac-neg 98.4%

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

   3. distribute-neg-frac2 98.4%

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

   4. +-commutative 98.4%

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

   5. distribute-neg-in 98.4%

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

   6. unsub-neg 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in t1 around inf 56.1%

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

6. Taylor expanded in u around inf 13.5%

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

7. Final simplification 13.5%

   \[\leadsto \frac{v}{t1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))