Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.1%

Time: 10.3s

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: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. times-frac 97.9%

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

   2. distribute-frac-neg 97.9%

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

   3. distribute-neg-frac2 97.9%

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

   4. +-commutative 97.9%

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

   5. distribute-neg-in 97.9%

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

   6. unsub-neg 97.9%

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

3. Simplified 97.9%

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

   1. frac-2neg 97.9%

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

   2. frac-2neg 97.9%

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

   3. frac-times 74.0%

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

   4. sub-neg 74.0%

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

   5. distribute-neg-in 74.0%

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

   6. +-commutative 74.0%

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

   7. remove-double-neg 74.0%

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

   8. frac-times 97.9%

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

   9. associate-*r/ 98.5%

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

   10. add-sqr-sqrt 50.3%

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

   11. sqrt-unprod 42.9%

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

   12. sqr-neg 42.9%

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

   13. sqrt-unprod 20.1%

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

   14. add-sqr-sqrt 38.4%

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

   15. add-sqr-sqrt 17.9%

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

   16. sqrt-unprod 58.1%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{+104}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.2e+74)
   (/ v (- t1))
   (if (<= t1 6.5e+104)
     (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))
     (/ v (- (- t1) (* u 2.0))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.2e+74) {
		tmp = v / -t1;
	} else if (t1 <= 6.5e+104) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-4.2d+74)) then
        tmp = v / -t1
    else if (t1 <= 6.5d+104) then
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.2e+74) {
		tmp = v / -t1;
	} else if (t1 <= 6.5e+104) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4.2e+74:
		tmp = v / -t1
	elif t1 <= 6.5e+104:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.2e+74)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 6.5e+104)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4.2e+74)
		tmp = v / -t1;
	elseif (t1 <= 6.5e+104)
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.2e+74], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 6.5e+104], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.1999999999999998e74

   1. Initial program 43.8%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 93.6%

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

      1. associate-*r/ 93.6%

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

      2. neg-mul-1 93.6%

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

   7. Simplified 93.6%

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

   if -4.1999999999999998e74 < t1 < 6.5000000000000005e104

   1. Initial program 84.5%

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

      1. associate-/l* 84.7%

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

      2. distribute-lft-neg-out 84.7%

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

      3. distribute-rgt-neg-in 84.7%

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

      4. associate-/r* 90.6%

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

      5. distribute-neg-frac2 90.6%

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

   3. Simplified 90.6%

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

   if 6.5000000000000005e104 < t1

   1. Initial program 61.9%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. frac-times 97.7%

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

      4. *-un-lft-identity 97.7%

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

      5. +-commutative 97.7%

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

      6. distribute-neg-in 97.7%

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

      7. sub-neg 97.7%

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

      8. frac-2neg 97.7%

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

      9. sub-neg 97.7%

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

      10. distribute-neg-in 97.7%

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

      11. +-commutative 97.7%

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

      12. remove-double-neg 97.7%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 6.1%

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

      15. sqr-neg 6.1%

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

      16. sqrt-unprod 52.8%

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

      17. add-sqr-sqrt 52.8%

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

      18. sub-neg 52.8%

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

      19. distribute-neg-in 52.8%

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

      20. +-commutative 52.8%

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

      21. add-sqr-sqrt 2.4%

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

      22. sqrt-unprod 67.3%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in u around 0 93.5%

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

      1. *-commutative 93.5%

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

   9. Simplified 93.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{+104}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.8 \cdot 10^{+43} \lor \neg \left(t1 \leq 1.8 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{1 - \frac{u}{t1}}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.8e+43) (not (<= t1 1.8e+38)))
   (/ v (- (- t1) (* u 2.0)))
   (/ (/ v (- 1.0 (/ u t1))) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.8e+43) || !(t1 <= 1.8e+38)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / (1.0 - (u / 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) :: tmp
    if ((t1 <= (-4.8d+43)) .or. (.not. (t1 <= 1.8d+38))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (v / (1.0d0 - (u / t1))) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.8e+43) || !(t1 <= 1.8e+38)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / (1.0 - (u / t1))) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.8e+43) or not (t1 <= 1.8e+38):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (v / (1.0 - (u / t1))) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.8e+43) || !(t1 <= 1.8e+38))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / Float64(1.0 - Float64(u / t1))) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.8e+43) || ~((t1 <= 1.8e+38)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (v / (1.0 - (u / t1))) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.8e+43], N[Not[LessEqual[t1, 1.8e+38]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.80000000000000046e43 or 1.79999999999999985e38 < t1

   1. Initial program 58.8%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 98.2%

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

      4. *-un-lft-identity 98.2%

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

      5. +-commutative 98.2%

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

      6. distribute-neg-in 98.2%

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

      7. sub-neg 98.2%

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

      8. frac-2neg 98.2%

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

      9. sub-neg 98.2%

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

      10. distribute-neg-in 98.2%

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

      11. +-commutative 98.2%

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

      12. remove-double-neg 98.2%

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

      13. add-sqr-sqrt 48.4%

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

      14. sqrt-unprod 24.3%

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

      15. sqr-neg 24.3%

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

      16. sqrt-unprod 23.1%

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

      17. add-sqr-sqrt 35.1%

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

      18. sub-neg 35.1%

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

      19. distribute-neg-in 35.1%

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

      20. +-commutative 35.1%

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

      21. add-sqr-sqrt 12.7%

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

      22. sqrt-unprod 49.3%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in u around 0 90.2%

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

      1. *-commutative 90.2%

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

   9. Simplified 90.2%

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

   if -4.80000000000000046e43 < t1 < 1.79999999999999985e38

   1. Initial program 85.2%

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

   3. Taylor expanded in t1 around 0 68.7%

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

      1. *-un-lft-identity 68.7%

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

      2. associate-/l* 68.8%

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

      3. add-sqr-sqrt 37.5%

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

      4. sqrt-unprod 48.7%

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

      5. sqr-neg 48.7%

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

      6. sqrt-unprod 19.5%

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

      7. add-sqr-sqrt 42.7%

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

      8. associate-/r* 43.4%

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

      9. frac-2neg 43.4%

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

      10. add-sqr-sqrt 18.7%

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

      11. sqrt-unprod 47.0%

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

      12. sqr-neg 47.0%

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

      13. sqrt-unprod 40.7%

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

      14. add-sqr-sqrt 73.8%

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

      15. distribute-neg-in 73.8%

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

      16. add-sqr-sqrt 40.2%

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

      17. sqrt-unprod 75.3%

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

      18. sqr-neg 75.3%

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

      19. sqrt-unprod 34.4%

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

      20. add-sqr-sqrt 75.4%

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

      21. sub-neg 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. *-lft-identity 75.4%

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

      2. *-commutative 75.4%

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

      3. associate-*l/ 78.7%

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

      4. associate-/r/ 78.1%

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

      5. div-sub 78.1%

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

      6. *-inverses 78.1%

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

   7. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\frac{v}{1 - \frac{u}{t1}}}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.8 \cdot 10^{+43} \lor \neg \left(t1 \leq 1.8 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{1 - \frac{u}{t1}}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+43} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.8e+43) (not (<= t1 1.4e+38)))
   (/ v (- (- t1) (* u 2.0)))
   (* (/ v (- u)) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e+43) || !(t1 <= 1.4e+38)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / -u) * (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) :: tmp
    if ((t1 <= (-2.8d+43)) .or. (.not. (t1 <= 1.4d+38))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (v / -u) * (t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e+43) || !(t1 <= 1.4e+38)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / -u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.8e+43) or not (t1 <= 1.4e+38):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (v / -u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.8e+43) || !(t1 <= 1.4e+38))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / Float64(-u)) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.8e+43) || ~((t1 <= 1.4e+38)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (v / -u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.8e+43], N[Not[LessEqual[t1, 1.4e+38]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / (-u)), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.80000000000000019e43 or 1.4e38 < t1

   1. Initial program 58.8%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 98.2%

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

      4. *-un-lft-identity 98.2%

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

      5. +-commutative 98.2%

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

      6. distribute-neg-in 98.2%

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

      7. sub-neg 98.2%

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

      8. frac-2neg 98.2%

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

      9. sub-neg 98.2%

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

      10. distribute-neg-in 98.2%

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

      11. +-commutative 98.2%

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

      12. remove-double-neg 98.2%

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

      13. add-sqr-sqrt 48.4%

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

      14. sqrt-unprod 24.3%

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

      15. sqr-neg 24.3%

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

      16. sqrt-unprod 23.1%

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

      17. add-sqr-sqrt 35.1%

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

      18. sub-neg 35.1%

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

      19. distribute-neg-in 35.1%

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

      20. +-commutative 35.1%

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

      21. add-sqr-sqrt 12.7%

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

      22. sqrt-unprod 49.3%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in u around 0 90.2%

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

      1. *-commutative 90.2%

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

   9. Simplified 90.2%

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

   if -2.80000000000000019e43 < t1 < 1.4e38

   1. Initial program 85.2%

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

   3. Taylor expanded in t1 around 0 68.7%

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

      1. *-commutative 68.7%

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

      2. times-frac 76.2%

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

      3. frac-2neg 76.2%

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

      4. add-sqr-sqrt 35.0%

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

      5. sqrt-unprod 48.3%

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

      6. sqr-neg 48.3%

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

      7. sqrt-unprod 24.8%

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

      8. add-sqr-sqrt 43.5%

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

      9. distribute-neg-in 43.5%

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

      10. add-sqr-sqrt 23.9%

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

      11. sqrt-unprod 41.9%

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

      12. sqr-neg 41.9%

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

      13. sqrt-unprod 18.6%

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

      14. add-sqr-sqrt 41.8%

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

      15. sub-neg 41.8%

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

      16. add-sqr-sqrt 23.2%

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

      17. sqrt-unprod 54.0%

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

      18. sqr-neg 54.0%

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

      19. sqrt-unprod 35.8%

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

      20. add-sqr-sqrt 77.8%

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

   5. Applied egg-rr 77.8%

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

   6. Taylor expanded in t1 around 0 78.0%

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

      1. neg-mul-1 78.0%

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

      2. distribute-neg-frac2 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+43} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{+43} \lor \neg \left(t1 \leq 6.6 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.5e+43) (not (<= t1 6.6e+37)))
   (/ v (- (- u) t1))
   (* (/ v (- u)) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.5e+43) || !(t1 <= 6.6e+37)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v / -u) * (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) :: tmp
    if ((t1 <= (-5.5d+43)) .or. (.not. (t1 <= 6.6d+37))) then
        tmp = v / (-u - t1)
    else
        tmp = (v / -u) * (t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.5e+43) || !(t1 <= 6.6e+37)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v / -u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.5e+43) or not (t1 <= 6.6e+37):
		tmp = v / (-u - t1)
	else:
		tmp = (v / -u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.5e+43) || !(t1 <= 6.6e+37))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(v / Float64(-u)) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.5e+43) || ~((t1 <= 6.6e+37)))
		tmp = v / (-u - t1);
	else
		tmp = (v / -u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.5e+43], N[Not[LessEqual[t1, 6.6e+37]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / (-u)), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.49999999999999989e43 or 6.6000000000000002e37 < t1

   1. Initial program 58.8%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 58.8%

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

      4. sub-neg 58.8%

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

      5. distribute-neg-in 58.8%

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

      6. +-commutative 58.8%

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

      7. remove-double-neg 58.8%

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

      8. frac-times 99.9%

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

      9. associate-*r/ 99.9%

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

      10. add-sqr-sqrt 49.2%

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

      11. sqrt-unprod 25.1%

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

      12. sqr-neg 25.1%

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

      13. sqrt-unprod 23.1%

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

      14. add-sqr-sqrt 35.0%

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

      15. add-sqr-sqrt 12.7%

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

      16. sqrt-unprod 49.3%

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

   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 89.6%

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

      1. mul-1-neg 89.6%

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

   9. Simplified 89.6%

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

   if -5.49999999999999989e43 < t1 < 6.6000000000000002e37

   1. Initial program 85.2%

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

   3. Taylor expanded in t1 around 0 68.7%

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

      1. *-commutative 68.7%

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

      2. times-frac 76.2%

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

      3. frac-2neg 76.2%

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

      4. add-sqr-sqrt 35.0%

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

      5. sqrt-unprod 48.3%

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

      6. sqr-neg 48.3%

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

      7. sqrt-unprod 24.8%

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

      8. add-sqr-sqrt 43.5%

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

      9. distribute-neg-in 43.5%

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

      10. add-sqr-sqrt 23.9%

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

      11. sqrt-unprod 41.9%

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

      12. sqr-neg 41.9%

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

      13. sqrt-unprod 18.6%

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

      14. add-sqr-sqrt 41.8%

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

      15. sub-neg 41.8%

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

      16. add-sqr-sqrt 23.2%

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

      17. sqrt-unprod 54.0%

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

      18. sqr-neg 54.0%

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

      19. sqrt-unprod 35.8%

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

      20. add-sqr-sqrt 77.8%

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

   5. Applied egg-rr 77.8%

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

   6. Taylor expanded in t1 around 0 78.0%

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

      1. neg-mul-1 78.0%

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

      2. distribute-neg-frac2 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{+43} \lor \neg \left(t1 \leq 6.6 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+43} \lor \neg \left(t1 \leq 6.5 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3e+43) (not (<= t1 6.5e+39)))
   (/ v (- (- u) t1))
   (* v (/ t1 (* u (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e+43) || !(t1 <= 6.5e+39)) {
		tmp = v / (-u - t1);
	} else {
		tmp = v * (t1 / (u * -u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3d+43)) .or. (.not. (t1 <= 6.5d+39))) then
        tmp = v / (-u - t1)
    else
        tmp = v * (t1 / (u * -u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e+43) || !(t1 <= 6.5e+39)) {
		tmp = v / (-u - t1);
	} else {
		tmp = v * (t1 / (u * -u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3e+43) or not (t1 <= 6.5e+39):
		tmp = v / (-u - t1)
	else:
		tmp = v * (t1 / (u * -u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3e+43) || !(t1 <= 6.5e+39))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(-u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3e+43) || ~((t1 <= 6.5e+39)))
		tmp = v / (-u - t1);
	else
		tmp = v * (t1 / (u * -u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3e+43], N[Not[LessEqual[t1, 6.5e+39]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(t1 / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.00000000000000017e43 or 6.5000000000000001e39 < t1

   1. Initial program 58.8%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 58.8%

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

      4. sub-neg 58.8%

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

      5. distribute-neg-in 58.8%

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

      6. +-commutative 58.8%

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

      7. remove-double-neg 58.8%

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

      8. frac-times 99.9%

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

      9. associate-*r/ 99.9%

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

      10. add-sqr-sqrt 49.2%

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

      11. sqrt-unprod 25.1%

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

      12. sqr-neg 25.1%

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

      13. sqrt-unprod 23.1%

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

      14. add-sqr-sqrt 35.0%

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

      15. add-sqr-sqrt 12.7%

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

      16. sqrt-unprod 49.3%

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

   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 89.6%

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

      1. mul-1-neg 89.6%

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

   9. Simplified 89.6%

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

   if -3.00000000000000017e43 < t1 < 6.5000000000000001e39

   1. Initial program 85.2%

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

      1. associate-*l/ 82.8%

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

      2. *-commutative 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t1 around 0 65.7%

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

   6. Taylor expanded in t1 around 0 68.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+43} \lor \neg \left(t1 \leq 6.5 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-186}:\\ \;\;\;\;v \cdot \frac{-1}{t1 - u}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{v}{\frac{u}{t1}}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e-186)
   (* v (/ -1.0 (- t1 u)))
   (if (<= t1 2e-136) (/ (/ v (/ u t1)) t1) (/ v (- (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e-186) {
		tmp = v * (-1.0 / (t1 - u));
	} else if (t1 <= 2e-136) {
		tmp = (v / (u / t1)) / t1;
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.2d-186)) then
        tmp = v * ((-1.0d0) / (t1 - u))
    else if (t1 <= 2d-136) then
        tmp = (v / (u / t1)) / t1
    else
        tmp = v / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e-186) {
		tmp = v * (-1.0 / (t1 - u));
	} else if (t1 <= 2e-136) {
		tmp = (v / (u / t1)) / t1;
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e-186:
		tmp = v * (-1.0 / (t1 - u))
	elif t1 <= 2e-136:
		tmp = (v / (u / t1)) / t1
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e-186)
		tmp = Float64(v * Float64(-1.0 / Float64(t1 - u)));
	elseif (t1 <= 2e-136)
		tmp = Float64(Float64(v / Float64(u / t1)) / t1);
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e-186)
		tmp = v * (-1.0 / (t1 - u));
	elseif (t1 <= 2e-136)
		tmp = (v / (u / t1)) / t1;
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e-186], N[(v * N[(-1.0 / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2e-136], N[(N[(v / N[(u / t1), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.20000000000000002e-186

   1. Initial program 70.7%

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

      1. times-frac 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-neg-frac2 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. unsub-neg 99.0%

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

   3. Simplified 99.0%

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

      1. frac-2neg 99.0%

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

      2. frac-2neg 99.0%

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

      3. frac-times 70.7%

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

      4. sub-neg 70.7%

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

      5. distribute-neg-in 70.7%

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

      6. +-commutative 70.7%

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

      7. remove-double-neg 70.7%

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

      8. frac-times 99.0%

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

      9. associate-*r/ 99.8%

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

      10. add-sqr-sqrt 99.4%

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

      11. sqrt-unprod 64.0%

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

      12. sqr-neg 64.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 33.8%

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

      15. add-sqr-sqrt 24.7%

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

      16. sqrt-unprod 39.0%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t1 around inf 68.0%

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

      1. mul-1-neg 68.0%

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

   9. Simplified 68.0%

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

       1. div-inv 67.8%

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

       2. *-commutative 67.8%

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

       3. frac-2neg 67.8%

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

       4. metadata-eval 67.8%

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

       5. distribute-neg-in 67.8%

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

       6. add-sqr-sqrt 67.7%

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

       7. sqrt-unprod 45.6%

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

       8. sqr-neg 45.6%

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

       9. sqrt-unprod 0.0%

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

       10. add-sqr-sqrt 21.4%

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

       11. sub-neg 21.4%

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

       12. add-sqr-sqrt 8.0%

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

       13. sqrt-unprod 33.3%

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

       14. sqr-neg 33.3%

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

       15. sqrt-unprod 31.8%

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

       16. add-sqr-sqrt 68.2%

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

   11. Applied egg-rr 68.2%

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

   if -1.20000000000000002e-186 < t1 < 2e-136

   1. Initial program 80.2%

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

      1. times-frac 93.6%

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

      2. distribute-frac-neg 93.6%

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

      3. distribute-neg-frac2 93.6%

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

      4. +-commutative 93.6%

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

      5. distribute-neg-in 93.6%

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

      6. unsub-neg 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in t1 around inf 37.1%

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

      1. add-sqr-sqrt 21.3%

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

      2. add-sqr-sqrt 18.0%

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

      3. difference-of-squares 18.0%

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

      4. add-sqr-sqrt 18.0%

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

      5. sqrt-unprod 17.2%

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

      6. sqr-neg 17.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 6.1%

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

      11. sqr-neg 6.1%

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

      12. sqrt-unprod 4.8%

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

      13. add-sqr-sqrt 4.8%

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

   7. Applied egg-rr 4.8%

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

      1. difference-of-squares 4.8%

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

      2. rem-square-sqrt 23.2%

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

      3. rem-square-sqrt 36.3%

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

   9. Simplified 36.3%

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

   10. Taylor expanded in u around inf 29.5%

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

       1. associate-*r/ 50.2%

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

       2. clear-num 50.2%

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

       3. associate-*l/ 50.2%

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

       4. *-un-lft-identity 50.2%

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

   12. Applied egg-rr 50.2%

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

   if 2e-136 < t1

   1. Initial program 73.0%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 73.0%

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

      4. sub-neg 73.0%

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

      5. distribute-neg-in 73.0%

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

      6. +-commutative 73.0%

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

      7. remove-double-neg 73.0%

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

      8. frac-times 99.9%

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

      9. associate-*r/ 98.8%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 15.5%

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

      12. sqr-neg 15.5%

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

      13. sqrt-unprod 38.1%

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

      14. add-sqr-sqrt 38.1%

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

      15. add-sqr-sqrt 9.7%

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

      16. sqrt-unprod 73.6%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in t1 around inf 72.3%

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

      1. mul-1-neg 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-186}:\\ \;\;\;\;v \cdot \frac{-1}{t1 - u}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{v}{\frac{u}{t1}}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+215}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.15e+215) (/ v u) (if (<= u 1.1e+141) (/ v (- t1)) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+215) {
		tmp = v / u;
	} else if (u <= 1.1e+141) {
		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 (u <= (-1.15d+215)) then
        tmp = v / u
    else if (u <= 1.1d+141) 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 (u <= -1.15e+215) {
		tmp = v / u;
	} else if (u <= 1.1e+141) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.15e+215:
		tmp = v / u
	elif u <= 1.1e+141:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.15e+215)
		tmp = Float64(v / u);
	elseif (u <= 1.1e+141)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.15e+215)
		tmp = v / u;
	elseif (u <= 1.1e+141)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.15e+215], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.1e+141], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.1500000000000001e215

   1. Initial program 92.8%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 63.0%

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

      1. add-sqr-sqrt 63.0%

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

      2. add-sqr-sqrt 39.9%

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

      3. difference-of-squares 39.9%

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

      4. add-sqr-sqrt 39.9%

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

      5. sqrt-unprod 61.5%

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

      6. sqr-neg 61.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. sqr-neg 0.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. difference-of-squares 0.0%

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

      2. rem-square-sqrt 40.0%

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

      3. rem-square-sqrt 63.0%

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

   9. Simplified 63.0%

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

   10. Taylor expanded in t1 around 0 41.9%

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

   if -1.1500000000000001e215 < u < 1.1e141

   1. Initial program 73.3%

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

      1. times-frac 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. distribute-neg-frac2 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-neg-in 97.3%

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

      6. unsub-neg 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in t1 around inf 61.2%

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

      1. associate-*r/ 61.2%

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

      2. neg-mul-1 61.2%

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

   7. Simplified 61.2%

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

   if 1.1e141 < u

   1. Initial program 71.2%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 37.7%

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

   6. Taylor expanded in t1 around 0 30.6%

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

      1. associate-*r/ 30.6%

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

      2. mul-1-neg 30.6%

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

   8. Simplified 30.6%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+215}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{+141}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+31} \lor \neg \left(t1 \leq 2 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7e+31) (not (<= t1 2e+137))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7e+31) || !(t1 <= 2e+137)) {
		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 <= (-7d+31)) .or. (.not. (t1 <= 2d+137))) 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 <= -7e+31) || !(t1 <= 2e+137)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7e+31) or not (t1 <= 2e+137):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7e+31) || !(t1 <= 2e+137))
		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 <= -7e+31) || ~((t1 <= 2e+137)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7e+31], N[Not[LessEqual[t1, 2e+137]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7e31 or 2.0000000000000001e137 < t1

   1. Initial program 56.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 87.3%

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

      1. associate-*r/ 87.3%

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

      2. neg-mul-1 87.3%

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

   7. Simplified 87.3%

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

      1. distribute-frac-neg 87.3%

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

      2. div-inv 87.1%

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

      3. distribute-rgt-neg-in 87.1%

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

      4. frac-2neg 87.1%

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

      5. metadata-eval 87.1%

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

      6. add-sqr-sqrt 52.8%

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

      7. sqrt-unprod 52.5%

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

      8. sqr-neg 52.5%

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

      9. sqrt-unprod 21.4%

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

      10. add-sqr-sqrt 36.3%

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

   9. Applied egg-rr 36.3%

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

       1. distribute-rgt-neg-out 36.3%

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

       2. *-commutative 36.3%

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

       3. associate-*l/ 36.3%

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

       4. mul-1-neg 36.3%

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

       5. distribute-neg-frac 36.3%

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

       6. remove-double-neg 36.3%

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

   11. Simplified 36.3%

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

   if -7e31 < t1 < 2.0000000000000001e137

   1. Initial program 84.1%

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

      1. times-frac 96.6%

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

      2. distribute-frac-neg 96.6%

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

      3. distribute-neg-frac2 96.6%

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

      4. +-commutative 96.6%

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

      5. distribute-neg-in 96.6%

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

      6. unsub-neg 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around inf 46.8%

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

      1. add-sqr-sqrt 23.5%

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

      2. add-sqr-sqrt 16.0%

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

      3. difference-of-squares 16.0%

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

      4. add-sqr-sqrt 16.0%

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

      5. sqrt-unprod 18.7%

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

      6. sqr-neg 18.7%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 13.8%

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

      11. sqr-neg 13.8%

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

      12. sqrt-unprod 12.6%

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

      13. add-sqr-sqrt 12.6%

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

   7. Applied egg-rr 12.6%

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

      1. difference-of-squares 12.6%

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

      2. rem-square-sqrt 28.8%

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

      3. rem-square-sqrt 46.7%

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

   9. Simplified 46.7%

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

   10. Taylor expanded in t1 around 0 15.2%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+31} \lor \neg \left(t1 \leq 2 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. times-frac 97.9%

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

   2. distribute-frac-neg 97.9%

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

   3. distribute-neg-frac2 97.9%

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

   4. +-commutative 97.9%

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

   5. distribute-neg-in 97.9%

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

   6. unsub-neg 97.9%

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

3. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 11: 40.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.45 \cdot 10^{-197}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 1.45e-197) (/ v (+ t1 u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1.45e-197) {
		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 (v <= 1.45d-197) 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 (v <= 1.45e-197) {
		tmp = v / (t1 + u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 1.45e-197:
		tmp = v / (t1 + u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 1.45e-197)
		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 (v <= 1.45e-197)
		tmp = v / (t1 + u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 1.45e-197], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.45000000000000011e-197

   1. Initial program 78.2%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around inf 62.8%

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

      1. add-sqr-sqrt 38.8%

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

      2. sqrt-unprod 50.5%

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

      3. mul-1-neg 50.5%

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

      4. mul-1-neg 50.5%

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

      5. sqr-neg 50.5%

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

      6. sqrt-unprod 26.2%

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

      7. add-sqr-sqrt 28.1%

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

   7. Applied egg-rr 28.1%

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

   if 1.45000000000000011e-197 < v

   1. Initial program 67.2%

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

      1. times-frac 96.5%

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

      2. distribute-frac-neg 96.5%

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

      3. distribute-neg-frac2 96.5%

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

      4. +-commutative 96.5%

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

      5. distribute-neg-in 96.5%

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

      6. unsub-neg 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in t1 around inf 44.6%

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

      1. associate-*r/ 44.6%

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

      2. neg-mul-1 44.6%

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

   7. Simplified 44.6%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.45 \cdot 10^{-197}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+215}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.15e+215) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+215) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.15d+215)) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+215) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.15e+215:
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.15e+215)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.15e+215)
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.15e+215], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.1500000000000001e215

   1. Initial program 92.8%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 63.0%

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

      1. add-sqr-sqrt 63.0%

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

      2. add-sqr-sqrt 39.9%

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

      3. difference-of-squares 39.9%

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

      4. add-sqr-sqrt 39.9%

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

      5. sqrt-unprod 61.5%

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

      6. sqr-neg 61.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. sqr-neg 0.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. difference-of-squares 0.0%

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

      2. rem-square-sqrt 40.0%

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

      3. rem-square-sqrt 63.0%

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

   9. Simplified 63.0%

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

   10. Taylor expanded in t1 around 0 41.9%

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

   if -1.1500000000000001e215 < u

   1. Initial program 73.0%

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

      1. times-frac 97.7%

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

      2. distribute-frac-neg 97.7%

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

      3. distribute-neg-frac2 97.7%

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

      4. +-commutative 97.7%

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

      5. distribute-neg-in 97.7%

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

      6. unsub-neg 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t1 around inf 54.5%

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

      1. associate-*r/ 54.5%

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

      2. neg-mul-1 54.5%

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

   7. Simplified 54.5%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+215}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.2% 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 74.0%

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

   1. times-frac 97.9%

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

   2. distribute-frac-neg 97.9%

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

   3. distribute-neg-frac2 97.9%

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

   4. +-commutative 97.9%

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

   5. distribute-neg-in 97.9%

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

   6. unsub-neg 97.9%

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

3. Simplified 97.9%

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

   1. frac-2neg 97.9%

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

   2. frac-2neg 97.9%

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

   3. frac-times 74.0%

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

   4. sub-neg 74.0%

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

   5. distribute-neg-in 74.0%

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

   6. +-commutative 74.0%

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

   7. remove-double-neg 74.0%

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

   8. frac-times 97.9%

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

   9. associate-*r/ 98.5%

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

   10. add-sqr-sqrt 50.3%

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

   11. sqrt-unprod 42.9%

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

   12. sqr-neg 42.9%

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

   13. sqrt-unprod 20.1%

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

   14. add-sqr-sqrt 38.4%

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

   15. add-sqr-sqrt 17.9%

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

   16. sqrt-unprod 58.1%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in t1 around inf 57.7%

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

   1. mul-1-neg 57.7%

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

9. Simplified 57.7%

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

10. Final simplification 57.7%

    \[\leadsto \frac{v}{\left(-u\right) - t1} \]
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 74.0%

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

   1. times-frac 97.9%

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

   2. distribute-frac-neg 97.9%

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

   3. distribute-neg-frac2 97.9%

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

   4. +-commutative 97.9%

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

   5. distribute-neg-in 97.9%

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

   6. unsub-neg 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in t1 around inf 52.2%

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

   1. associate-*r/ 52.2%

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

   2. neg-mul-1 52.2%

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

7. Simplified 52.2%

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

   1. distribute-frac-neg 52.2%

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

   2. div-inv 52.1%

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

   3. distribute-rgt-neg-in 52.1%

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

   4. frac-2neg 52.1%

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

   5. metadata-eval 52.1%

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

   6. add-sqr-sqrt 26.1%

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

   7. sqrt-unprod 26.5%

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

   8. sqr-neg 26.5%

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

   9. sqrt-unprod 9.2%

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

   10. add-sqr-sqrt 15.3%

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

9. Applied egg-rr 15.3%

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

    1. distribute-rgt-neg-out 15.3%

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

    2. *-commutative 15.3%

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

    3. associate-*l/ 15.3%

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

    4. mul-1-neg 15.3%

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

    5. distribute-neg-frac 15.3%

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

    6. remove-double-neg 15.3%

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

11. Simplified 15.3%

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

Reproduce

herbie shell --seed 2024130 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))