Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.0%

Time: 13.4s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. associate-/l* 73.0%

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

   2. distribute-lft-neg-out 73.0%

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

   3. distribute-rgt-neg-in 73.0%

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

   4. associate-/r* 82.0%

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

   5. distribute-neg-frac2 82.0%

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

3. Simplified 82.0%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

   3. distribute-neg-in 97.4%

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

   4. sub-neg 97.4%

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

   5. associate-*l/ 97.8%

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

   6. frac-2neg 97.8%

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

   7. associate-*r/ 98.3%

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

   8. remove-double-neg 98.3%

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

   9. sub-neg 98.3%

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

   10. distribute-neg-in 98.3%

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

   11. +-commutative 98.3%

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

   12. frac-2neg 98.3%

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

   13. add-sqr-sqrt 50.6%

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

   14. sqrt-unprod 42.4%

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

   15. sqr-neg 42.4%

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

   16. sqrt-unprod 19.5%

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

   17. add-sqr-sqrt 40.8%

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

   18. add-sqr-sqrt 17.8%

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

   19. sqrt-unprod 58.5%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 2: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{+157}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1e+134)
   (/ v (- (* u (- 2.0)) t1))
   (if (<= t1 1.25e+157) (* t1 (/ (/ v (+ t1 u)) (- (- u) t1))) (/ v (- t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1e+134) {
		tmp = v / ((u * -2.0) - t1);
	} else if (t1 <= 1.25e+157) {
		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 (t1 <= (-1d+134)) then
        tmp = v / ((u * -2.0d0) - t1)
    else if (t1 <= 1.25d+157) 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 (t1 <= -1e+134) {
		tmp = v / ((u * -2.0) - t1);
	} else if (t1 <= 1.25e+157) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1e+134:
		tmp = v / ((u * -2.0) - t1)
	elif t1 <= 1.25e+157:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1e+134)
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	elseif (t1 <= 1.25e+157)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1e+134)
		tmp = v / ((u * -2.0) - t1);
	elseif (t1 <= 1.25e+157)
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1e+134], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.25e+157], 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 3 regimes

2. if t1 < -9.99999999999999921e133

   1. Initial program 35.0%

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

      1. associate-/l* 38.7%

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

      2. distribute-lft-neg-out 38.7%

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

      3. distribute-rgt-neg-in 38.7%

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

      4. associate-/r* 51.8%

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

      5. distribute-neg-frac2 51.8%

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

   3. Simplified 51.8%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. clear-num 99.8%

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

      8. frac-times 97.3%

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

      9. *-un-lft-identity 97.3%

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

      10. +-commutative 97.3%

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

      11. distribute-neg-in 97.3%

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

      12. sub-neg 97.3%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in u around 0 85.7%

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

      1. *-commutative 85.7%

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

   9. Simplified 85.7%

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

   if -9.99999999999999921e133 < t1 < 1.24999999999999994e157

   1. Initial program 84.4%

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

      1. associate-/l* 85.3%

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

      2. distribute-lft-neg-out 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

      4. associate-/r* 91.6%

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

      5. distribute-neg-frac2 91.6%

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

   3. Simplified 91.6%

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

   if 1.24999999999999994e157 < t1

   1. Initial program 46.5%

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

      1. associate-/l* 47.7%

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

      2. distribute-lft-neg-out 47.7%

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

      3. distribute-rgt-neg-in 47.7%

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

      4. associate-/r* 65.3%

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

      5. distribute-neg-frac2 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in t1 around inf 90.2%

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

      1. associate-*r/ 90.2%

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

      2. neg-mul-1 90.2%

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

   7. Simplified 90.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{+157}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+56} \lor \neg \left(u \leq 1.4 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -8.5e+56) (not (<= u 1.4e-74)))
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (/ v (- (* u (- 2.0)) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.5e+56) || !(u <= 1.4e-74)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = v / ((u * -2.0) - 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 <= (-8.5d+56)) .or. (.not. (u <= 1.4d-74))) then
        tmp = (v / (t1 + u)) * (t1 / -u)
    else
        tmp = v / ((u * -2.0d0) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.5e+56) || !(u <= 1.4e-74)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -8.5e+56) or not (u <= 1.4e-74):
		tmp = (v / (t1 + u)) * (t1 / -u)
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -8.5e+56) || !(u <= 1.4e-74))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	else
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -8.5e+56) || ~((u <= 1.4e-74)))
		tmp = (v / (t1 + u)) * (t1 / -u);
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -8.5e+56], N[Not[LessEqual[u, 1.4e-74]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -8.4999999999999998e56 or 1.39999999999999994e-74 < u

   1. Initial program 79.8%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 91.9%

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

      1. associate-*r/ 91.9%

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

      2. mul-1-neg 91.9%

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

   7. Simplified 91.9%

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

   if -8.4999999999999998e56 < u < 1.39999999999999994e-74

   1. Initial program 63.2%

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

      1. associate-/l* 64.6%

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

      2. distribute-lft-neg-out 64.6%

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

      3. distribute-rgt-neg-in 64.6%

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

      4. associate-/r* 73.1%

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

      5. distribute-neg-frac2 73.1%

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

   3. Simplified 73.1%

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

      1. associate-*r/ 95.0%

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

      2. +-commutative 95.0%

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

      3. distribute-neg-in 95.0%

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

      4. sub-neg 95.0%

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

      5. associate-*l/ 96.9%

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

      6. frac-2neg 96.9%

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

      7. clear-num 96.9%

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

      8. frac-times 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. distribute-neg-in 99.8%

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

      12. sub-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in u around 0 79.9%

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

      1. *-commutative 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+56} \lor \neg \left(u \leq 1.4 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;u \leq -5.4 \cdot 10^{+56}:\\ \;\;\;\;\frac{t\_1}{\frac{u}{-t1}}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{-74}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))))
   (if (<= u -5.4e+56)
     (/ t_1 (/ u (- t1)))
     (if (<= u 1.15e-74) (/ v (- (* u (- 2.0)) t1)) (* t_1 (/ t1 (- u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (u <= -5.4e+56) {
		tmp = t_1 / (u / -t1);
	} else if (u <= 1.15e-74) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_1 * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (t1 + u)
    if (u <= (-5.4d+56)) then
        tmp = t_1 / (u / -t1)
    else if (u <= 1.15d-74) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = t_1 * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (u <= -5.4e+56) {
		tmp = t_1 / (u / -t1);
	} else if (u <= 1.15e-74) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t_1 * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	tmp = 0
	if u <= -5.4e+56:
		tmp = t_1 / (u / -t1)
	elif u <= 1.15e-74:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t_1 * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if (u <= -5.4e+56)
		tmp = Float64(t_1 / Float64(u / Float64(-t1)));
	elseif (u <= 1.15e-74)
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(t_1 * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	tmp = 0.0;
	if (u <= -5.4e+56)
		tmp = t_1 / (u / -t1);
	elseif (u <= 1.15e-74)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t_1 * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -5.4e+56], N[(t$95$1 / N[(u / (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.15e-74], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.40000000000000019e56

   1. Initial program 78.1%

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

      1. associate-/l* 78.2%

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

      2. distribute-lft-neg-out 78.2%

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

      3. distribute-rgt-neg-in 78.2%

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

      4. associate-/r* 87.2%

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

      5. distribute-neg-frac2 87.2%

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

   3. Simplified 87.2%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-*l/ 98.0%

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

      6. frac-2neg 98.0%

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

      7. clear-num 97.9%

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

      8. frac-times 85.3%

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

      9. *-un-lft-identity 85.3%

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

      10. +-commutative 85.3%

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

      11. distribute-neg-in 85.3%

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

      12. sub-neg 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. neg-mul-1 85.3%

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

      2. metadata-eval 85.3%

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

      3. times-frac 97.9%

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

      4. metadata-eval 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. associate-*l/ 98.0%

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

      2. neg-mul-1 98.0%

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

      3. distribute-neg-frac2 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in t1 around 0 92.7%

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

   if -5.40000000000000019e56 < u < 1.1499999999999999e-74

   1. Initial program 63.2%

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

      1. associate-/l* 64.6%

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

      2. distribute-lft-neg-out 64.6%

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

      3. distribute-rgt-neg-in 64.6%

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

      4. associate-/r* 73.1%

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

      5. distribute-neg-frac2 73.1%

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

   3. Simplified 73.1%

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

      1. associate-*r/ 95.0%

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

      2. +-commutative 95.0%

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

      3. distribute-neg-in 95.0%

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

      4. sub-neg 95.0%

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

      5. associate-*l/ 96.9%

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

      6. frac-2neg 96.9%

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

      7. clear-num 96.9%

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

      8. frac-times 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. distribute-neg-in 99.8%

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

      12. sub-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in u around 0 79.9%

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

      1. *-commutative 79.9%

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

   9. Simplified 79.9%

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

   if 1.1499999999999999e-74 < u

   1. Initial program 80.9%

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

      1. times-frac 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. distribute-neg-frac2 99.2%

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

      4. +-commutative 99.2%

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

      5. distribute-neg-in 99.2%

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

      6. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t1 around 0 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.4 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{\frac{u}{-t1}}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{-74}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.4 \cdot 10^{+56} \lor \neg \left(u \leq 1.4 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.4e+56) (not (<= u 1.4e-74)))
   (/ (* t1 (/ v u)) (- t1 u))
   (/ v (- (* u (- 2.0)) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.4e+56) || !(u <= 1.4e-74)) {
		tmp = (t1 * (v / u)) / (t1 - u);
	} else {
		tmp = v / ((u * -2.0) - 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 <= (-5.4d+56)) .or. (.not. (u <= 1.4d-74))) then
        tmp = (t1 * (v / u)) / (t1 - u)
    else
        tmp = v / ((u * -2.0d0) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.4e+56) || !(u <= 1.4e-74)) {
		tmp = (t1 * (v / u)) / (t1 - u);
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.4e+56) or not (u <= 1.4e-74):
		tmp = (t1 * (v / u)) / (t1 - u)
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.4e+56) || !(u <= 1.4e-74))
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	else
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.4e+56) || ~((u <= 1.4e-74)))
		tmp = (t1 * (v / u)) / (t1 - u);
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.4e+56], N[Not[LessEqual[u, 1.4e-74]], $MachinePrecision]], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.40000000000000019e56 or 1.39999999999999994e-74 < u

   1. Initial program 79.8%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 91.9%

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

      1. associate-*r/ 91.9%

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

      2. mul-1-neg 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in v around 0 77.6%

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

      1. mul-1-neg 77.6%

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

      2. associate-/l* 78.9%

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

      3. distribute-lft-neg-in 78.9%

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

   10. Simplified 78.9%

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

       1. *-commutative 78.9%

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

       2. frac-2neg 78.9%

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

       3. associate-*l/ 77.6%

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

       4. add-sqr-sqrt 38.0%

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

       5. sqrt-unprod 62.4%

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

       6. sqr-neg 62.4%

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

       7. sqrt-unprod 32.3%

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

       8. add-sqr-sqrt 62.2%

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

       9. add-sqr-sqrt 33.3%

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

       10. sqrt-unprod 56.2%

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

       11. sqr-neg 56.2%

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

       12. sqrt-unprod 34.7%

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

       13. add-sqr-sqrt 77.6%

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

       14. distribute-rgt-neg-in 77.6%

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

       15. distribute-neg-in 77.6%

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

       16. add-sqr-sqrt 42.8%

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

       17. sqrt-unprod 78.3%

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

       18. sqr-neg 78.3%

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

       19. sqrt-unprod 34.7%

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

       20. add-sqr-sqrt 77.6%

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

       21. sub-neg 77.6%

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

   12. Applied egg-rr 77.6%

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

       1. associate-/l* 75.9%

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

   14. Simplified 75.9%

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

   15. Taylor expanded in v around 0 77.6%

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

       1. associate-/r* 82.1%

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

       2. associate-*r/ 87.2%

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

   17. Simplified 87.2%

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

   if -5.40000000000000019e56 < u < 1.39999999999999994e-74

   1. Initial program 63.2%

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

      1. associate-/l* 64.6%

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

      2. distribute-lft-neg-out 64.6%

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

      3. distribute-rgt-neg-in 64.6%

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

      4. associate-/r* 73.1%

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

      5. distribute-neg-frac2 73.1%

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

   3. Simplified 73.1%

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

      1. associate-*r/ 95.0%

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

      2. +-commutative 95.0%

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

      3. distribute-neg-in 95.0%

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

      4. sub-neg 95.0%

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

      5. associate-*l/ 96.9%

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

      6. frac-2neg 96.9%

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

      7. clear-num 96.9%

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

      8. frac-times 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. distribute-neg-in 99.8%

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

      12. sub-neg 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in u around 0 79.9%

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

      1. *-commutative 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.4 \cdot 10^{+56} \lor \neg \left(u \leq 1.4 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6.2 \cdot 10^{-37} \lor \neg \left(t1 \leq 2.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -6.2e-37) (not (<= t1 2.6e-16)))
   (/ v (- (- u) t1))
   (* (/ t1 (- u)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6.2e-37) || !(t1 <= 2.6e-16)) {
		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) :: tmp
    if ((t1 <= (-6.2d-37)) .or. (.not. (t1 <= 2.6d-16))) 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 tmp;
	if ((t1 <= -6.2e-37) || !(t1 <= 2.6e-16)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -6.2e-37) or not (t1 <= 2.6e-16):
		tmp = v / (-u - t1)
	else:
		tmp = (t1 / -u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -6.2e-37) || !(t1 <= 2.6e-16))
		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)
	tmp = 0.0;
	if ((t1 <= -6.2e-37) || ~((t1 <= 2.6e-16)))
		tmp = v / (-u - t1);
	else
		tmp = (t1 / -u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -6.2e-37], N[Not[LessEqual[t1, 2.6e-16]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.19999999999999987e-37 or 2.5999999999999998e-16 < t1

   1. Initial program 56.6%

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

      1. associate-/l* 59.0%

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

      2. distribute-lft-neg-out 59.0%

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

      3. distribute-rgt-neg-in 59.0%

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

      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. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. remove-double-neg 99.9%

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

      9. sub-neg 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. +-commutative 99.9%

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

      12. frac-2neg 99.9%

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

      13. add-sqr-sqrt 51.7%

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

      14. sqrt-unprod 29.3%

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

      15. sqr-neg 29.3%

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

      16. sqrt-unprod 18.1%

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

      17. add-sqr-sqrt 36.4%

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

      18. add-sqr-sqrt 16.6%

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

      19. sqrt-unprod 49.1%

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

   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 82.9%

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

      1. mul-1-neg 82.9%

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

   9. Simplified 82.9%

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

   if -6.19999999999999987e-37 < t1 < 2.5999999999999998e-16

   1. Initial program 86.9%

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

      1. times-frac 95.7%

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

      2. distribute-frac-neg 95.7%

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

      3. distribute-neg-frac2 95.7%

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

      4. +-commutative 95.7%

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

      5. distribute-neg-in 95.7%

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

      6. unsub-neg 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in t1 around 0 80.5%

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

      1. associate-*r/ 80.5%

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

      2. mul-1-neg 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in t1 around 0 83.2%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.2 \cdot 10^{-37} \lor \neg \left(t1 \leq 2.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-37} \lor \neg \left(t1 \leq 2.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.5e-37) (not (<= t1 2.6e-16)))
   (/ v (- (* u (- 2.0)) t1))
   (* (/ t1 (- u)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.5e-37) || !(t1 <= 2.6e-16)) {
		tmp = v / ((u * -2.0) - 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) :: tmp
    if ((t1 <= (-7.5d-37)) .or. (.not. (t1 <= 2.6d-16))) then
        tmp = v / ((u * -2.0d0) - 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 tmp;
	if ((t1 <= -7.5e-37) || !(t1 <= 2.6e-16)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / -u) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7.5e-37) or not (t1 <= 2.6e-16):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (t1 / -u) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.5e-37) || !(t1 <= 2.6e-16))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -7.5e-37) || ~((t1 <= 2.6e-16)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (t1 / -u) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7.5e-37], N[Not[LessEqual[t1, 2.6e-16]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.5000000000000004e-37 or 2.5999999999999998e-16 < t1

   1. Initial program 56.6%

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

      1. associate-/l* 59.0%

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

      2. distribute-lft-neg-out 59.0%

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

      3. distribute-rgt-neg-in 59.0%

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

      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. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. clear-num 99.8%

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

      8. frac-times 93.3%

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

      9. *-un-lft-identity 93.3%

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

      10. +-commutative 93.3%

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

      11. distribute-neg-in 93.3%

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

      12. sub-neg 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in u around 0 83.1%

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

      1. *-commutative 83.1%

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

   9. Simplified 83.1%

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

   if -7.5000000000000004e-37 < t1 < 2.5999999999999998e-16

   1. Initial program 86.9%

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

      1. times-frac 95.7%

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

      2. distribute-frac-neg 95.7%

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

      3. distribute-neg-frac2 95.7%

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

      4. +-commutative 95.7%

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

      5. distribute-neg-in 95.7%

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

      6. unsub-neg 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in t1 around 0 80.5%

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

      1. associate-*r/ 80.5%

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

      2. mul-1-neg 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in t1 around 0 83.2%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-37} \lor \neg \left(t1 \leq 2.6 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+57} \lor \neg \left(u \leq 10^{+129}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.2e+57) (not (<= u 1e+129))) (/ v (+ t1 u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.2e+57) || !(u <= 1e+129)) {
		tmp = v / (t1 + u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.2d+57)) .or. (.not. (u <= 1d+129))) then
        tmp = v / (t1 + u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.2e+57) || !(u <= 1e+129)) {
		tmp = v / (t1 + u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.2e+57) or not (u <= 1e+129):
		tmp = v / (t1 + u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.2e+57) || !(u <= 1e+129))
		tmp = Float64(v / Float64(t1 + u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.2e+57) || ~((u <= 1e+129)))
		tmp = v / (t1 + u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.2e+57], N[Not[LessEqual[u, 1e+129]], $MachinePrecision]], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.19999999999999982e57 or 1e129 < u

   1. Initial program 78.1%

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

      1. associate-/l* 78.4%

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

      2. distribute-lft-neg-out 78.4%

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

      3. distribute-rgt-neg-in 78.4%

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

      4. associate-/r* 89.2%

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

      5. distribute-neg-frac2 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in t1 around inf 48.5%

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

      1. associate-*r/ 49.4%

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

      2. clear-num 50.4%

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

      3. add-sqr-sqrt 23.2%

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

      4. sqrt-unprod 58.9%

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

      5. sqr-neg 58.9%

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

      6. sqrt-unprod 26.1%

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

      7. add-sqr-sqrt 48.2%

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

   7. Applied egg-rr 48.2%

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

      1. associate-/r/ 47.3%

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

      2. associate-*r* 55.5%

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

      3. associate-*l/ 55.5%

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

      4. *-lft-identity 55.5%

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

      5. times-frac 49.8%

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

      6. *-commutative 49.8%

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

      7. times-frac 47.5%

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

      8. *-inverses 47.5%

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

      9. *-lft-identity 47.5%

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

   9. Simplified 47.5%

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

   if -4.19999999999999982e57 < u < 1e129

   1. Initial program 67.8%

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

      1. associate-/l* 69.7%

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

      2. distribute-lft-neg-out 69.7%

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

      3. distribute-rgt-neg-in 69.7%

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

      4. associate-/r* 77.6%

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

      5. distribute-neg-frac2 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in t1 around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   7. Simplified 68.0%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+57} \lor \neg \left(u \leq 10^{+129}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+138} \lor \neg \left(u \leq 10^{+151}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.8e+138) (not (<= u 1e+151))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e+138) || !(u <= 1e+151)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.8d+138)) .or. (.not. (u <= 1d+151))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e+138) || !(u <= 1e+151)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.8e+138) or not (u <= 1e+151):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.8e+138) || !(u <= 1e+151))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.8e+138) || ~((u <= 1e+151)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.8e+138], N[Not[LessEqual[u, 1e+151]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.80000000000000019e138 or 1.00000000000000002e151 < u

   1. Initial program 74.9%

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

      1. associate-/l* 75.3%

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

      2. distribute-lft-neg-out 75.3%

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

      3. distribute-rgt-neg-in 75.3%

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

      4. associate-/r* 88.3%

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

      5. distribute-neg-frac2 88.3%

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

   3. Simplified 88.3%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-*l/ 98.6%

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

      6. frac-2neg 98.6%

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

      7. associate-*r/ 98.6%

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

      8. remove-double-neg 98.6%

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

      9. sub-neg 98.6%

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

      10. distribute-neg-in 98.6%

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

      11. +-commutative 98.6%

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

      12. frac-2neg 98.6%

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

      13. add-sqr-sqrt 51.2%

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

      14. sqrt-unprod 69.4%

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

      15. sqr-neg 69.4%

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

      16. sqrt-unprod 35.4%

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

      17. add-sqr-sqrt 72.5%

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

      18. add-sqr-sqrt 29.4%

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

      19. sqrt-unprod 72.9%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in t1 around inf 48.0%

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

      1. mul-1-neg 48.0%

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

   9. Simplified 48.0%

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

       1. div-inv 48.0%

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

       2. add-sqr-sqrt 23.3%

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

       3. sqrt-unprod 47.3%

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

       4. sqr-neg 47.3%

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

       5. sqrt-unprod 24.6%

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

       6. add-sqr-sqrt 46.5%

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

       7. *-commutative 46.5%

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

   11. Applied egg-rr 46.5%

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

   12. Taylor expanded in t1 around 0 45.4%

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

   if -5.80000000000000019e138 < u < 1.00000000000000002e151

   1. Initial program 70.1%

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

      1. associate-/l* 71.9%

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

      2. distribute-lft-neg-out 71.9%

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

      3. distribute-rgt-neg-in 71.9%

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

      4. associate-/r* 79.1%

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

      5. distribute-neg-frac2 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in t1 around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+138} \lor \neg \left(u \leq 10^{+151}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+138} \lor \neg \left(u \leq 3.2 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.2e+138) (not (<= u 3.2e+142))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+138) || !(u <= 3.2e+142)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.2d+138)) .or. (.not. (u <= 3.2d+142))) then
        tmp = v / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+138) || !(u <= 3.2e+142)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.2e+138) or not (u <= 3.2e+142):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.2e+138) || !(u <= 3.2e+142))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.2e+138) || ~((u <= 3.2e+142)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.2e+138], N[Not[LessEqual[u, 3.2e+142]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.2000000000000001e138 or 3.20000000000000005e142 < u

   1. Initial program 74.9%

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

      1. associate-/l* 75.3%

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

      2. distribute-lft-neg-out 75.3%

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

      3. distribute-rgt-neg-in 75.3%

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

      4. associate-/r* 88.3%

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

      5. distribute-neg-frac2 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in t1 around inf 47.8%

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

   6. Taylor expanded in t1 around 0 45.5%

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

      1. associate-*r/ 45.5%

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

      2. mul-1-neg 45.5%

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

   8. Simplified 45.5%

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

   if -2.2000000000000001e138 < u < 3.20000000000000005e142

   1. Initial program 70.1%

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

      1. associate-/l* 71.9%

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

      2. distribute-lft-neg-out 71.9%

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

      3. distribute-rgt-neg-in 71.9%

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

      4. associate-/r* 79.1%

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

      5. distribute-neg-frac2 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in t1 around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+138} \lor \neg \left(u \leq 3.2 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{+121} \lor \neg \left(t1 \leq 1.6 \cdot 10^{+63}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9e+121) (not (<= t1 1.6e+63))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9e+121) || !(t1 <= 1.6e+63)) {
		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 <= (-9d+121)) .or. (.not. (t1 <= 1.6d+63))) 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 <= -9e+121) || !(t1 <= 1.6e+63)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9e+121) or not (t1 <= 1.6e+63):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9e+121) || !(t1 <= 1.6e+63))
		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 <= -9e+121) || ~((t1 <= 1.6e+63)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9e+121], N[Not[LessEqual[t1, 1.6e+63]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.0000000000000007e121 or 1.60000000000000006e63 < t1

   1. Initial program 45.9%

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

      1. times-frac 99.9%

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

      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 85.5%

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

   6. Taylor expanded in u around inf 39.4%

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

   if -9.0000000000000007e121 < t1 < 1.60000000000000006e63

   1. Initial program 85.8%

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

      1. associate-/l* 86.2%

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

      2. distribute-lft-neg-out 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. associate-/r* 91.4%

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

      5. distribute-neg-frac2 91.4%

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

   3. Simplified 91.4%

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

      1. associate-*r/ 96.0%

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

      2. +-commutative 96.0%

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

      3. distribute-neg-in 96.0%

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

      4. sub-neg 96.0%

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

      5. associate-*l/ 96.7%

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

      6. frac-2neg 96.7%

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

      7. associate-*r/ 97.4%

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

      8. remove-double-neg 97.4%

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

      9. sub-neg 97.4%

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

      10. distribute-neg-in 97.4%

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

      11. +-commutative 97.4%

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

      12. frac-2neg 97.4%

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

      13. add-sqr-sqrt 54.9%

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

      14. sqrt-unprod 60.7%

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

      15. sqr-neg 60.7%

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

      16. sqrt-unprod 16.9%

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

      17. add-sqr-sqrt 40.9%

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

      18. add-sqr-sqrt 18.6%

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

      19. sqrt-unprod 63.7%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in t1 around inf 48.7%

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

      1. mul-1-neg 48.7%

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

   9. Simplified 48.7%

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

       1. div-inv 48.6%

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

       2. add-sqr-sqrt 27.4%

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

       3. sqrt-unprod 34.9%

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

       4. sqr-neg 34.9%

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

       5. sqrt-unprod 8.8%

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

       6. add-sqr-sqrt 21.7%

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

       7. *-commutative 21.7%

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

   11. Applied egg-rr 21.7%

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

   12. Taylor expanded in t1 around 0 23.4%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{+121} \lor \neg \left(t1 \leq 1.6 \cdot 10^{+63}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. times-frac 97.8%

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

   2. distribute-frac-neg 97.8%

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

   3. distribute-neg-frac2 97.8%

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

   4. +-commutative 97.8%

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

   5. distribute-neg-in 97.8%

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

   6. unsub-neg 97.8%

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

3. Simplified 97.8%

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

5. Final simplification 97.8%

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

Alternative 13: 61.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. associate-/l* 73.0%

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

   2. distribute-lft-neg-out 73.0%

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

   3. distribute-rgt-neg-in 73.0%

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

   4. associate-/r* 82.0%

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

   5. distribute-neg-frac2 82.0%

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

3. Simplified 82.0%

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

   1. associate-*r/ 97.4%

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

   2. +-commutative 97.4%

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

   3. distribute-neg-in 97.4%

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

   4. sub-neg 97.4%

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

   5. associate-*l/ 97.8%

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

   6. frac-2neg 97.8%

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

   7. associate-*r/ 98.3%

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

   8. remove-double-neg 98.3%

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

   9. sub-neg 98.3%

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

   10. distribute-neg-in 98.3%

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

   11. +-commutative 98.3%

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

   12. frac-2neg 98.3%

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

   13. add-sqr-sqrt 50.6%

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

   14. sqrt-unprod 42.4%

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

   15. sqr-neg 42.4%

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

   16. sqrt-unprod 19.5%

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

   17. add-sqr-sqrt 40.8%

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

   18. add-sqr-sqrt 17.8%

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

   19. sqrt-unprod 58.5%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in t1 around inf 61.9%

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

   1. mul-1-neg 61.9%

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

9. Simplified 61.9%

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

10. Final simplification 61.9%

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

Alternative 14: 14.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. times-frac 97.8%

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

   2. distribute-frac-neg 97.8%

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

   3. distribute-neg-frac2 97.8%

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

   4. +-commutative 97.8%

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

   5. distribute-neg-in 97.8%

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

   6. unsub-neg 97.8%

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

3. Simplified 97.8%

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

5. Taylor expanded in t1 around inf 52.4%

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

6. Taylor expanded in u around inf 16.2%

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

7. Final simplification 16.2%

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

Reproduce

herbie shell --seed 2024076 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))