Rosa's DopplerBench

Percentage Accurate: 73.2% → 94.3%

Time: 9.6s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% 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: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.28 \cdot 10^{+70}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.28e+70)
   (/ (* t1 (/ v u)) (- u))
   (/ (- v) (+ t1 (* u (+ 2.0 (/ u t1)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.28e+70) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -v / (t1 + (u * (2.0 + (u / t1))));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.28d+70)) then
        tmp = (t1 * (v / u)) / -u
    else
        tmp = -v / (t1 + (u * (2.0d0 + (u / t1))))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.28e+70) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -v / (t1 + (u * (2.0 + (u / t1))));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.28e+70:
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = -v / (t1 + (u * (2.0 + (u / t1))))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.28e+70)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * Float64(2.0 + Float64(u / t1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.28e+70)
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = -v / (t1 + (u * (2.0 + (u / t1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.28e+70], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[((-v) / N[(t1 + N[(u * N[(2.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.27999999999999994e70

   1. Initial program 85.8%

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

      1. associate-/l* 83.6%

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

      2. neg-mul-1 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-*r/ 83.6%

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

      5. associate-/l* 83.5%

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

      6. neg-mul-1 83.5%

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

      7. associate-/r* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in t1 around 0 83.5%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 83.5%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]

      2. neg-mul-1 83.5%

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

      3. unpow2 83.5%

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

   6. Simplified 83.5%

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

      1. associate-*r/ 84.3%

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

      2. times-frac 84.9%

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

      3. add-sqr-sqrt 53.6%

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

      4. sqrt-unprod 68.7%

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

      5. sqr-neg 68.7%

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

      6. sqrt-unprod 28.0%

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

      7. add-sqr-sqrt 64.7%

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

      8. times-frac 64.6%

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

      9. associate-/l/ 64.4%

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

      10. frac-2neg 64.4%

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

      11. distribute-neg-frac 64.4%

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

      12. distribute-rgt-neg-in 64.4%

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

      13. add-sqr-sqrt 36.5%

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

      14. sqrt-unprod 59.9%

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

      15. sqr-neg 59.9%

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

      16. sqrt-unprod 33.5%

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

      17. add-sqr-sqrt 96.0%

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

   8. Applied egg-rr 96.1%

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

   if -1.27999999999999994e70 < u

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. times-frac 95.7%

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

      3. neg-mul-1 95.7%

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

      4. associate-/l* 95.3%

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

      5. associate-*r/ 95.3%

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

      6. associate-/l* 95.3%

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

      7. associate-/l/ 95.3%

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

      8. neg-mul-1 95.3%

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

      9. *-lft-identity 95.3%

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

      10. metadata-eval 95.3%

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

      11. times-frac 95.3%

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

      12. neg-mul-1 95.3%

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

      13. remove-double-neg 95.3%

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

      14. neg-mul-1 95.3%

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

      15. sub0-neg 95.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 95.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 95.3%

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

      18. div-sub 95.3%

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

      19. distribute-frac-neg 95.3%

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

      20. *-inverses 95.3%

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

      21. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in v around 0 97.5%

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

      1. mul-1-neg 97.5%

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

   6. Simplified 97.5%

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

      1. distribute-rgt-in 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. associate-+l+ 97.5%

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

   8. Applied egg-rr 97.5%

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

   9. Taylor expanded in u around 0 93.2%

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

       1. +-commutative 93.2%

          \[\leadsto -\frac{v}{t1 + \color{blue}{\left(2 \cdot u + \frac{{u}^{2}}{t1}\right)}} \]

       2. *-commutative 93.2%

          \[\leadsto -\frac{v}{t1 + \left(\color{blue}{u \cdot 2} + \frac{{u}^{2}}{t1}\right)} \]

       3. unpow2 93.2%

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

       4. associate-*r/ 97.5%

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

       5. distribute-lft-out 97.5%

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

   11. Simplified 97.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.28 \cdot 10^{+70}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}\\ \end{array} \]

Alternative 2: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 1950000000000:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.45e+19)
   (/ (* t1 (/ v u)) (- u))
   (if (<= u 3.3e-40)
     (/ (- v) t1)
     (if (<= u 1.4e-24)
       (/ (- v) (/ (* u u) t1))
       (if (<= u 1950000000000.0)
         (/ (- v) (+ t1 (* u 2.0)))
         (* t1 (/ (/ (- v) u) (+ u t1))))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.45e+19) {
		tmp = (t1 * (v / u)) / -u;
	} else if (u <= 3.3e-40) {
		tmp = -v / t1;
	} else if (u <= 1.4e-24) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 1950000000000.0) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t1 * ((-v / u) / (u + t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.45d+19)) then
        tmp = (t1 * (v / u)) / -u
    else if (u <= 3.3d-40) then
        tmp = -v / t1
    else if (u <= 1.4d-24) then
        tmp = -v / ((u * u) / t1)
    else if (u <= 1950000000000.0d0) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = t1 * ((-v / u) / (u + t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.45e+19) {
		tmp = (t1 * (v / u)) / -u;
	} else if (u <= 3.3e-40) {
		tmp = -v / t1;
	} else if (u <= 1.4e-24) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 1950000000000.0) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t1 * ((-v / u) / (u + t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.45e+19:
		tmp = (t1 * (v / u)) / -u
	elif u <= 3.3e-40:
		tmp = -v / t1
	elif u <= 1.4e-24:
		tmp = -v / ((u * u) / t1)
	elif u <= 1950000000000.0:
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = t1 * ((-v / u) / (u + t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.45e+19)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	elseif (u <= 3.3e-40)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 1.4e-24)
		tmp = Float64(Float64(-v) / Float64(Float64(u * u) / t1));
	elseif (u <= 1950000000000.0)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(Float64(Float64(-v) / u) / Float64(u + t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.45e+19)
		tmp = (t1 * (v / u)) / -u;
	elseif (u <= 3.3e-40)
		tmp = -v / t1;
	elseif (u <= 1.4e-24)
		tmp = -v / ((u * u) / t1);
	elseif (u <= 1950000000000.0)
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = t1 * ((-v / u) / (u + t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.45e+19], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, 3.3e-40], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 1.4e-24], N[((-v) / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1950000000000.0], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[((-v) / u), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -1.45e19

   1. Initial program 85.1%

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

      1. associate-/l* 83.4%

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

      2. neg-mul-1 83.4%

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

      3. *-commutative 83.4%

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

      4. associate-*r/ 83.4%

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

      5. associate-/l* 83.4%

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

      6. neg-mul-1 83.4%

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

      7. associate-/r* 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in t1 around 0 81.9%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 81.9%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]

      2. neg-mul-1 81.9%

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

      3. unpow2 81.9%

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

   6. Simplified 81.9%

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

      1. associate-*r/ 82.5%

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

      2. times-frac 81.7%

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

      3. add-sqr-sqrt 51.3%

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

      4. sqrt-unprod 65.3%

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

      5. sqr-neg 65.3%

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

      6. sqrt-unprod 24.0%

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

      7. add-sqr-sqrt 55.2%

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

      8. times-frac 55.2%

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

      9. associate-/l/ 55.0%

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

      10. frac-2neg 55.0%

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

      11. distribute-neg-frac 55.0%

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

      12. distribute-rgt-neg-in 55.0%

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

      13. add-sqr-sqrt 31.0%

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

      14. sqrt-unprod 55.2%

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

      15. sqr-neg 55.2%

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

      16. sqrt-unprod 32.1%

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

      17. add-sqr-sqrt 91.7%

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

   8. Applied egg-rr 91.7%

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

   if -1.45e19 < u < 3.29999999999999993e-40

   1. Initial program 74.5%

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

      1. associate-/l* 66.4%

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

      2. neg-mul-1 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*r/ 65.9%

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

      5. associate-/l* 66.0%

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

      6. neg-mul-1 66.0%

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

      7. associate-/r* 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in t1 around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   6. Simplified 81.3%

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

   if 3.29999999999999993e-40 < u < 1.4000000000000001e-24

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

      2. times-frac 81.0%

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

      3. neg-mul-1 81.0%

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

      4. associate-/l* 81.3%

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

      5. associate-*r/ 81.0%

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

      6. associate-/l* 81.0%

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

      7. associate-/l/ 81.0%

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

      8. neg-mul-1 81.0%

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

      9. *-lft-identity 81.0%

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

      10. metadata-eval 81.0%

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

      11. times-frac 81.0%

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

      12. neg-mul-1 81.0%

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

      13. remove-double-neg 81.0%

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

      14. neg-mul-1 81.0%

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

      15. sub0-neg 81.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 81.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 81.0%

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

      18. div-sub 81.0%

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

      19. distribute-frac-neg 81.0%

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

      20. *-inverses 81.0%

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

      21. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in v around 0 99.4%

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

      1. mul-1-neg 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in t1 around 0 99.7%

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

      1. unpow2 99.7%

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

   9. Simplified 99.7%

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

   if 1.4000000000000001e-24 < u < 1.95e12

   1. Initial program 52.4%

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

      1. *-commutative 52.4%

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

      2. times-frac 99.7%

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

      3. neg-mul-1 99.7%

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

      4. associate-/l* 99.7%

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

      5. associate-*r/ 99.7%

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

      6. associate-/l* 99.7%

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

      7. associate-/l/ 99.7%

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

      8. neg-mul-1 99.7%

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

      9. *-lft-identity 99.7%

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

      10. metadata-eval 99.7%

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

      11. times-frac 99.7%

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

      12. neg-mul-1 99.7%

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

      13. remove-double-neg 99.7%

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

      14. neg-mul-1 99.7%

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

      15. sub0-neg 99.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.7%

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

      18. div-sub 99.7%

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

      19. distribute-frac-neg 99.7%

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

      20. *-inverses 99.7%

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

      21. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in v around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in t1 around inf 83.4%

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

      1. *-commutative 83.4%

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

   9. Simplified 83.4%

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

   if 1.95e12 < u

   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* 87.1%

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

      2. neg-mul-1 87.1%

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

      3. *-commutative 87.1%

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

      4. associate-*r/ 87.0%

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

      5. associate-/l* 87.4%

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

      6. neg-mul-1 87.4%

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

      7. associate-/r* 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in t1 around 0 83.9%

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

      1. associate-*r/ 83.9%

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

      2. neg-mul-1 83.9%

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

   6. Simplified 83.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 1950000000000:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u + t1}\\ \end{array} \]

Alternative 3: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{if}\;u \leq -2.05 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* t1 (/ v u)) (- u))))
   (if (<= u -2.05e+18)
     t_1
     (if (<= u 1.1e-40)
       (/ (- v) t1)
       (if (<= u 1.25e+45)
         (/ (- v) (/ (* u u) t1))
         (if (<= u 1.8e+88) (/ (/ v t1) (- -1.0 (/ u t1))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 * (v / u)) / -u;
	double tmp;
	if (u <= -2.05e+18) {
		tmp = t_1;
	} else if (u <= 1.1e-40) {
		tmp = -v / t1;
	} else if (u <= 1.25e+45) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 1.8e+88) {
		tmp = (v / t1) / (-1.0 - (u / t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t1 * (v / u)) / -u
    if (u <= (-2.05d+18)) then
        tmp = t_1
    else if (u <= 1.1d-40) then
        tmp = -v / t1
    else if (u <= 1.25d+45) then
        tmp = -v / ((u * u) / t1)
    else if (u <= 1.8d+88) then
        tmp = (v / t1) / ((-1.0d0) - (u / t1))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 * (v / u)) / -u;
	double tmp;
	if (u <= -2.05e+18) {
		tmp = t_1;
	} else if (u <= 1.1e-40) {
		tmp = -v / t1;
	} else if (u <= 1.25e+45) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 1.8e+88) {
		tmp = (v / t1) / (-1.0 - (u / t1));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 * (v / u)) / -u
	tmp = 0
	if u <= -2.05e+18:
		tmp = t_1
	elif u <= 1.1e-40:
		tmp = -v / t1
	elif u <= 1.25e+45:
		tmp = -v / ((u * u) / t1)
	elif u <= 1.8e+88:
		tmp = (v / t1) / (-1.0 - (u / t1))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 * Float64(v / u)) / Float64(-u))
	tmp = 0.0
	if (u <= -2.05e+18)
		tmp = t_1;
	elseif (u <= 1.1e-40)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 1.25e+45)
		tmp = Float64(Float64(-v) / Float64(Float64(u * u) / t1));
	elseif (u <= 1.8e+88)
		tmp = Float64(Float64(v / t1) / Float64(-1.0 - Float64(u / t1)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 * (v / u)) / -u;
	tmp = 0.0;
	if (u <= -2.05e+18)
		tmp = t_1;
	elseif (u <= 1.1e-40)
		tmp = -v / t1;
	elseif (u <= 1.25e+45)
		tmp = -v / ((u * u) / t1);
	elseif (u <= 1.8e+88)
		tmp = (v / t1) / (-1.0 - (u / t1));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]}, If[LessEqual[u, -2.05e+18], t$95$1, If[LessEqual[u, 1.1e-40], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 1.25e+45], N[((-v) / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.8e+88], N[(N[(v / t1), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -2.05e18 or 1.8000000000000001e88 < u

   1. Initial program 84.6%

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

      1. associate-/l* 84.8%

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

      2. neg-mul-1 84.8%

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

      3. *-commutative 84.8%

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

      4. associate-*r/ 84.8%

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

      5. associate-/l* 84.8%

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

      6. neg-mul-1 84.8%

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

      7. associate-/r* 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in t1 around 0 84.0%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 84.0%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]

      2. neg-mul-1 84.0%

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

      3. unpow2 84.0%

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

   6. Simplified 84.0%

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

      1. associate-*r/ 82.1%

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

      2. times-frac 84.8%

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

      3. add-sqr-sqrt 47.5%

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

      4. sqrt-unprod 66.8%

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

      5. sqr-neg 66.8%

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

      6. sqrt-unprod 28.1%

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

      7. add-sqr-sqrt 60.4%

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

      8. times-frac 60.4%

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

      9. associate-/l/ 60.2%

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

      10. frac-2neg 60.2%

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

      11. distribute-neg-frac 60.2%

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

      12. distribute-rgt-neg-in 60.2%

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

      13. add-sqr-sqrt 32.2%

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

      14. sqrt-unprod 64.0%

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

      15. sqr-neg 64.0%

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

      16. sqrt-unprod 38.2%

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

      17. add-sqr-sqrt 89.2%

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

   8. Applied egg-rr 91.4%

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

   if -2.05e18 < u < 1.10000000000000004e-40

   1. Initial program 74.5%

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

      1. associate-/l* 66.4%

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

      2. neg-mul-1 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*r/ 65.9%

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

      5. associate-/l* 66.0%

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

      6. neg-mul-1 66.0%

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

      7. associate-/r* 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in t1 around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   6. Simplified 81.3%

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

   if 1.10000000000000004e-40 < u < 1.25e45

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. times-frac 94.0%

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

      3. neg-mul-1 94.0%

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

      4. associate-/l* 94.1%

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

      5. associate-*r/ 93.8%

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

      6. associate-/l* 93.8%

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

      7. associate-/l/ 93.8%

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

      8. neg-mul-1 93.8%

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

      9. *-lft-identity 93.8%

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

      10. metadata-eval 93.8%

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

      11. times-frac 93.8%

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

      12. neg-mul-1 93.8%

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

      13. remove-double-neg 93.8%

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

      14. neg-mul-1 93.8%

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

      15. sub0-neg 93.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 93.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 93.8%

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

      18. div-sub 93.8%

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

      19. distribute-frac-neg 93.8%

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

      20. *-inverses 93.8%

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

      21. metadata-eval 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t1 around 0 70.4%

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

      1. unpow2 70.4%

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

   9. Simplified 70.4%

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

   if 1.25e45 < u < 1.8000000000000001e88

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

      2. times-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 100.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around inf 83.9%

      \[\leadsto \frac{\color{blue}{\frac{v}{t1}}}{-1 - \frac{u}{t1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 4: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{if}\;u \leq -1.1 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (- v) (* u u)))))
   (if (<= u -1.1e+19)
     t_1
     (if (<= u 3.5e-42)
       (/ (- v) t1)
       (if (<= u 8.5e+47)
         (/ (- v) (/ (* u u) t1))
         (if (<= u 8.5e+62) (/ (- v) (+ t1 (* u 2.0))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (-v / (u * u));
	double tmp;
	if (u <= -1.1e+19) {
		tmp = t_1;
	} else if (u <= 3.5e-42) {
		tmp = -v / t1;
	} else if (u <= 8.5e+47) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 8.5e+62) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t1 * (-v / (u * u))
    if (u <= (-1.1d+19)) then
        tmp = t_1
    else if (u <= 3.5d-42) then
        tmp = -v / t1
    else if (u <= 8.5d+47) then
        tmp = -v / ((u * u) / t1)
    else if (u <= 8.5d+62) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * (-v / (u * u));
	double tmp;
	if (u <= -1.1e+19) {
		tmp = t_1;
	} else if (u <= 3.5e-42) {
		tmp = -v / t1;
	} else if (u <= 8.5e+47) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 8.5e+62) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * (-v / (u * u))
	tmp = 0
	if u <= -1.1e+19:
		tmp = t_1
	elif u <= 3.5e-42:
		tmp = -v / t1
	elif u <= 8.5e+47:
		tmp = -v / ((u * u) / t1)
	elif u <= 8.5e+62:
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(Float64(-v) / Float64(u * u)))
	tmp = 0.0
	if (u <= -1.1e+19)
		tmp = t_1;
	elseif (u <= 3.5e-42)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 8.5e+47)
		tmp = Float64(Float64(-v) / Float64(Float64(u * u) / t1));
	elseif (u <= 8.5e+62)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * (-v / (u * u));
	tmp = 0.0;
	if (u <= -1.1e+19)
		tmp = t_1;
	elseif (u <= 3.5e-42)
		tmp = -v / t1;
	elseif (u <= 8.5e+47)
		tmp = -v / ((u * u) / t1);
	elseif (u <= 8.5e+62)
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.1e+19], t$95$1, If[LessEqual[u, 3.5e-42], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 8.5e+47], N[((-v) / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.5e+62], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.1e19 or 8.4999999999999997e62 < u

   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* 85.4%

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

      2. neg-mul-1 85.4%

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

      3. *-commutative 85.4%

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

      4. associate-*r/ 85.4%

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

      5. associate-/l* 85.4%

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

      6. neg-mul-1 85.4%

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

      7. associate-/r* 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in t1 around 0 82.7%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 82.7%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]

      2. neg-mul-1 82.7%

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

      3. unpow2 82.7%

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

   6. Simplified 82.7%

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

   if -1.1e19 < u < 3.5000000000000002e-42

   1. Initial program 74.5%

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

      1. associate-/l* 66.4%

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

      2. neg-mul-1 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*r/ 65.9%

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

      5. associate-/l* 66.0%

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

      6. neg-mul-1 66.0%

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

      7. associate-/r* 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in t1 around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   6. Simplified 81.3%

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

   if 3.5000000000000002e-42 < u < 8.5000000000000008e47

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. times-frac 94.0%

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

      3. neg-mul-1 94.0%

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

      4. associate-/l* 94.1%

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

      5. associate-*r/ 93.8%

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

      6. associate-/l* 93.8%

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

      7. associate-/l/ 93.8%

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

      8. neg-mul-1 93.8%

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

      9. *-lft-identity 93.8%

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

      10. metadata-eval 93.8%

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

      11. times-frac 93.8%

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

      12. neg-mul-1 93.8%

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

      13. remove-double-neg 93.8%

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

      14. neg-mul-1 93.8%

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

      15. sub0-neg 93.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 93.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 93.8%

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

      18. div-sub 93.8%

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

      19. distribute-frac-neg 93.8%

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

      20. *-inverses 93.8%

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

      21. metadata-eval 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t1 around 0 70.4%

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

      1. unpow2 70.4%

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

   9. Simplified 70.4%

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

   if 8.5000000000000008e47 < u < 8.4999999999999997e62

   1. Initial program 75.1%

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

      1. *-commutative 75.1%

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

      2. times-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 100.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t1 around inf 88.2%

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

      1. *-commutative 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.1 \cdot 10^{+19}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{elif}\;u \leq 3.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 5: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{if}\;u \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.16 \cdot 10^{+45}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+63}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (/ (- v) u) u))))
   (if (<= u -1.1e+21)
     t_1
     (if (<= u 8e-41)
       (/ (- v) t1)
       (if (<= u 1.16e+45)
         (/ (- v) (/ (* u u) t1))
         (if (<= u 1.05e+63) (/ (- v) (+ t1 (* u 2.0))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * ((-v / u) / u);
	double tmp;
	if (u <= -1.1e+21) {
		tmp = t_1;
	} else if (u <= 8e-41) {
		tmp = -v / t1;
	} else if (u <= 1.16e+45) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 1.05e+63) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t1 * ((-v / u) / u)
    if (u <= (-1.1d+21)) then
        tmp = t_1
    else if (u <= 8d-41) then
        tmp = -v / t1
    else if (u <= 1.16d+45) then
        tmp = -v / ((u * u) / t1)
    else if (u <= 1.05d+63) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * ((-v / u) / u);
	double tmp;
	if (u <= -1.1e+21) {
		tmp = t_1;
	} else if (u <= 8e-41) {
		tmp = -v / t1;
	} else if (u <= 1.16e+45) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 1.05e+63) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * ((-v / u) / u)
	tmp = 0
	if u <= -1.1e+21:
		tmp = t_1
	elif u <= 8e-41:
		tmp = -v / t1
	elif u <= 1.16e+45:
		tmp = -v / ((u * u) / t1)
	elif u <= 1.05e+63:
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(Float64(Float64(-v) / u) / u))
	tmp = 0.0
	if (u <= -1.1e+21)
		tmp = t_1;
	elseif (u <= 8e-41)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 1.16e+45)
		tmp = Float64(Float64(-v) / Float64(Float64(u * u) / t1));
	elseif (u <= 1.05e+63)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * ((-v / u) / u);
	tmp = 0.0;
	if (u <= -1.1e+21)
		tmp = t_1;
	elseif (u <= 8e-41)
		tmp = -v / t1;
	elseif (u <= 1.16e+45)
		tmp = -v / ((u * u) / t1);
	elseif (u <= 1.05e+63)
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.1e+21], t$95$1, If[LessEqual[u, 8e-41], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 1.16e+45], N[((-v) / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.05e+63], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.1e21 or 1.0500000000000001e63 < u

   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* 85.4%

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

      2. neg-mul-1 85.4%

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

      3. *-commutative 85.4%

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

      4. associate-*r/ 85.4%

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

      5. associate-/l* 85.4%

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

      6. neg-mul-1 85.4%

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

      7. associate-/r* 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in t1 around 0 82.7%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 82.7%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]

      2. neg-mul-1 82.7%

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

      3. unpow2 82.7%

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

   6. Simplified 82.7%

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

   7. Taylor expanded in v around 0 82.7%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   8. Step-by-step derivation

      1. unpow2 82.7%

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

      2. associate-/r* 87.5%

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

      3. associate-*r/ 87.5%

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

      4. neg-mul-1 87.5%

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

      5. distribute-neg-frac 87.5%

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

   9. Simplified 87.5%

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

   if -1.1e21 < u < 8.00000000000000005e-41

   1. Initial program 74.5%

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

      1. associate-/l* 66.4%

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

      2. neg-mul-1 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*r/ 65.9%

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

      5. associate-/l* 66.0%

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

      6. neg-mul-1 66.0%

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

      7. associate-/r* 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in t1 around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   6. Simplified 81.3%

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

   if 8.00000000000000005e-41 < u < 1.1600000000000001e45

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. times-frac 94.0%

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

      3. neg-mul-1 94.0%

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

      4. associate-/l* 94.1%

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

      5. associate-*r/ 93.8%

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

      6. associate-/l* 93.8%

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

      7. associate-/l/ 93.8%

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

      8. neg-mul-1 93.8%

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

      9. *-lft-identity 93.8%

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

      10. metadata-eval 93.8%

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

      11. times-frac 93.8%

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

      12. neg-mul-1 93.8%

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

      13. remove-double-neg 93.8%

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

      14. neg-mul-1 93.8%

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

      15. sub0-neg 93.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 93.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 93.8%

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

      18. div-sub 93.8%

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

      19. distribute-frac-neg 93.8%

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

      20. *-inverses 93.8%

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

      21. metadata-eval 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t1 around 0 70.4%

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

      1. unpow2 70.4%

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

   9. Simplified 70.4%

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

   if 1.1600000000000001e45 < u < 1.0500000000000001e63

   1. Initial program 75.1%

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

      1. *-commutative 75.1%

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

      2. times-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 100.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t1 around inf 88.2%

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

      1. *-commutative 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.16 \cdot 10^{+45}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+63}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \end{array} \]

Alternative 6: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{if}\;u \leq -1.4 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* t1 (/ v u)) (- u))))
   (if (<= u -1.4e+18)
     t_1
     (if (<= u 1.4e-42)
       (/ (- v) t1)
       (if (<= u 1.35e+48)
         (/ (- v) (/ (* u u) t1))
         (if (<= u 7.6e+66) (/ (- v) (+ t1 (* u 2.0))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 * (v / u)) / -u;
	double tmp;
	if (u <= -1.4e+18) {
		tmp = t_1;
	} else if (u <= 1.4e-42) {
		tmp = -v / t1;
	} else if (u <= 1.35e+48) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 7.6e+66) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t1 * (v / u)) / -u
    if (u <= (-1.4d+18)) then
        tmp = t_1
    else if (u <= 1.4d-42) then
        tmp = -v / t1
    else if (u <= 1.35d+48) then
        tmp = -v / ((u * u) / t1)
    else if (u <= 7.6d+66) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 * (v / u)) / -u;
	double tmp;
	if (u <= -1.4e+18) {
		tmp = t_1;
	} else if (u <= 1.4e-42) {
		tmp = -v / t1;
	} else if (u <= 1.35e+48) {
		tmp = -v / ((u * u) / t1);
	} else if (u <= 7.6e+66) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 * (v / u)) / -u
	tmp = 0
	if u <= -1.4e+18:
		tmp = t_1
	elif u <= 1.4e-42:
		tmp = -v / t1
	elif u <= 1.35e+48:
		tmp = -v / ((u * u) / t1)
	elif u <= 7.6e+66:
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 * Float64(v / u)) / Float64(-u))
	tmp = 0.0
	if (u <= -1.4e+18)
		tmp = t_1;
	elseif (u <= 1.4e-42)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 1.35e+48)
		tmp = Float64(Float64(-v) / Float64(Float64(u * u) / t1));
	elseif (u <= 7.6e+66)
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 * (v / u)) / -u;
	tmp = 0.0;
	if (u <= -1.4e+18)
		tmp = t_1;
	elseif (u <= 1.4e-42)
		tmp = -v / t1;
	elseif (u <= 1.35e+48)
		tmp = -v / ((u * u) / t1);
	elseif (u <= 7.6e+66)
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]}, If[LessEqual[u, -1.4e+18], t$95$1, If[LessEqual[u, 1.4e-42], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 1.35e+48], N[((-v) / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.6e+66], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -1.4e18 or 7.6000000000000004e66 < u

   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* 85.4%

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

      2. neg-mul-1 85.4%

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

      3. *-commutative 85.4%

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

      4. associate-*r/ 85.4%

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

      5. associate-/l* 85.4%

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

      6. neg-mul-1 85.4%

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

      7. associate-/r* 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in t1 around 0 82.7%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 82.7%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]

      2. neg-mul-1 82.7%

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

      3. unpow2 82.7%

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

   6. Simplified 82.7%

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

      1. associate-*r/ 80.9%

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

      2. times-frac 83.5%

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

      3. add-sqr-sqrt 46.6%

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

      4. sqrt-unprod 66.2%

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

      5. sqr-neg 66.2%

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

      6. sqrt-unprod 27.9%

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

      7. add-sqr-sqrt 59.0%

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

      8. times-frac 59.0%

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

      9. associate-/l/ 58.9%

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

      10. frac-2neg 58.9%

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

      11. distribute-neg-frac 58.9%

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

      12. distribute-rgt-neg-in 58.9%

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

      13. add-sqr-sqrt 31.0%

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

      14. sqrt-unprod 62.5%

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

      15. sqr-neg 62.5%

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

      16. sqrt-unprod 37.7%

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

      17. add-sqr-sqrt 87.7%

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

   8. Applied egg-rr 89.8%

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

   if -1.4e18 < u < 1.39999999999999999e-42

   1. Initial program 74.5%

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

      1. associate-/l* 66.4%

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

      2. neg-mul-1 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-*r/ 65.9%

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

      5. associate-/l* 66.0%

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

      6. neg-mul-1 66.0%

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

      7. associate-/r* 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in t1 around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   6. Simplified 81.3%

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

   if 1.39999999999999999e-42 < u < 1.35000000000000002e48

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. times-frac 94.0%

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

      3. neg-mul-1 94.0%

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

      4. associate-/l* 94.1%

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

      5. associate-*r/ 93.8%

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

      6. associate-/l* 93.8%

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

      7. associate-/l/ 93.8%

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

      8. neg-mul-1 93.8%

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

      9. *-lft-identity 93.8%

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

      10. metadata-eval 93.8%

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

      11. times-frac 93.8%

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

      12. neg-mul-1 93.8%

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

      13. remove-double-neg 93.8%

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

      14. neg-mul-1 93.8%

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

      15. sub0-neg 93.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 93.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 93.8%

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

      18. div-sub 93.8%

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

      19. distribute-frac-neg 93.8%

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

      20. *-inverses 93.8%

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

      21. metadata-eval 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t1 around 0 70.4%

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

      1. unpow2 70.4%

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

   9. Simplified 70.4%

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

   if 1.35000000000000002e48 < u < 7.6000000000000004e66

   1. Initial program 75.1%

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

      1. *-commutative 75.1%

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

      2. times-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 100.0%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t1 around inf 88.2%

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

      1. *-commutative 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 7.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 7: 94.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+70}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(\frac{u}{t1} + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.65e+70)
   (/ (* t1 (/ v u)) (- u))
   (/ (- v) (* (+ u t1) (+ (/ u t1) 1.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.65e+70) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -v / ((u + t1) * ((u / t1) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.65d+70)) then
        tmp = (t1 * (v / u)) / -u
    else
        tmp = -v / ((u + t1) * ((u / t1) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.65e+70) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -v / ((u + t1) * ((u / t1) + 1.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.65e+70:
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = -v / ((u + t1) * ((u / t1) + 1.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.65e+70)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	else
		tmp = Float64(Float64(-v) / Float64(Float64(u + t1) * Float64(Float64(u / t1) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.65e+70)
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = -v / ((u + t1) * ((u / t1) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.65e+70], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[((-v) / N[(N[(u + t1), $MachinePrecision] * N[(N[(u / t1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.65e70

   1. Initial program 85.8%

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

      1. associate-/l* 83.6%

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

      2. neg-mul-1 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-*r/ 83.6%

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

      5. associate-/l* 83.5%

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

      6. neg-mul-1 83.5%

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

      7. associate-/r* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in t1 around 0 83.5%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 83.5%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]

      2. neg-mul-1 83.5%

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

      3. unpow2 83.5%

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

   6. Simplified 83.5%

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

      1. associate-*r/ 84.3%

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

      2. times-frac 84.9%

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

      3. add-sqr-sqrt 53.6%

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

      4. sqrt-unprod 68.7%

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

      5. sqr-neg 68.7%

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

      6. sqrt-unprod 28.0%

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

      7. add-sqr-sqrt 64.7%

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

      8. times-frac 64.6%

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

      9. associate-/l/ 64.4%

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

      10. frac-2neg 64.4%

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

      11. distribute-neg-frac 64.4%

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

      12. distribute-rgt-neg-in 64.4%

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

      13. add-sqr-sqrt 36.5%

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

      14. sqrt-unprod 59.9%

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

      15. sqr-neg 59.9%

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

      16. sqrt-unprod 33.5%

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

      17. add-sqr-sqrt 96.0%

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

   8. Applied egg-rr 96.1%

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

   if -2.65e70 < u

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. times-frac 95.7%

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

      3. neg-mul-1 95.7%

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

      4. associate-/l* 95.3%

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

      5. associate-*r/ 95.3%

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

      6. associate-/l* 95.3%

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

      7. associate-/l/ 95.3%

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

      8. neg-mul-1 95.3%

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

      9. *-lft-identity 95.3%

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

      10. metadata-eval 95.3%

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

      11. times-frac 95.3%

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

      12. neg-mul-1 95.3%

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

      13. remove-double-neg 95.3%

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

      14. neg-mul-1 95.3%

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

      15. sub0-neg 95.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 95.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 95.3%

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

      18. div-sub 95.3%

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

      19. distribute-frac-neg 95.3%

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

      20. *-inverses 95.3%

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

      21. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in v around 0 97.5%

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

      1. mul-1-neg 97.5%

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

   6. Simplified 97.5%

      \[\leadsto \color{blue}{-\frac{v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+70}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\left(u + t1\right) \cdot \left(\frac{u}{t1} + 1\right)}\\ \end{array} \]

Alternative 8: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-55} \lor \neg \left(t1 \leq 8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.2e-55) (not (<= t1 8e-9)))
   (/ (- v) (+ t1 (* u 2.0)))
   (/ (- v) (/ (* u u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.2e-55) || !(t1 <= 8e-9)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = -v / ((u * 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-55)) .or. (.not. (t1 <= 8d-9))) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = -v / ((u * 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-55) || !(t1 <= 8e-9)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = -v / ((u * u) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.2e-55) or not (t1 <= 8e-9):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = -v / ((u * u) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.2e-55) || !(t1 <= 8e-9))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(-v) / Float64(Float64(u * u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.2e-55) || ~((t1 <= 8e-9)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = -v / ((u * u) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.2e-55], N[Not[LessEqual[t1, 8e-9]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.19999999999999996e-55 or 8.0000000000000005e-9 < t1

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 97.9%

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

      1. mul-1-neg 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in t1 around inf 83.4%

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

      1. *-commutative 83.4%

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

   9. Simplified 83.4%

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

   if -1.19999999999999996e-55 < t1 < 8.0000000000000005e-9

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. times-frac 88.4%

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

      3. neg-mul-1 88.4%

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

      4. associate-/l* 87.8%

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

      5. associate-*r/ 87.8%

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

      6. associate-/l* 87.8%

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

      7. associate-/l/ 87.8%

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

      8. neg-mul-1 87.8%

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

      9. *-lft-identity 87.8%

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

      10. metadata-eval 87.8%

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

      11. times-frac 87.8%

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

      12. neg-mul-1 87.8%

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

      13. remove-double-neg 87.8%

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

      14. neg-mul-1 87.8%

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

      15. sub0-neg 87.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 87.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 87.8%

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

      18. div-sub 87.8%

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

      19. distribute-frac-neg 87.8%

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

      20. *-inverses 87.8%

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

      21. metadata-eval 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in v around 0 90.9%

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

      1. mul-1-neg 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in t1 around 0 71.4%

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

      1. unpow2 71.4%

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

   9. Simplified 71.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-55} \lor \neg \left(t1 \leq 8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\frac{u \cdot u}{t1}}\\ \end{array} \]

Alternative 9: 66.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.25 \cdot 10^{+38} \lor \neg \left(u \leq 6.2 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.25e+38) (not (<= u 6.2e+76)))
   (* (/ v u) (/ t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.25e+38) || !(u <= 6.2e+76)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.25d+38)) .or. (.not. (u <= 6.2d+76))) then
        tmp = (v / u) * (t1 / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.25e+38) || !(u <= 6.2e+76)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.25e+38) or not (u <= 6.2e+76):
		tmp = (v / u) * (t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.25e+38) || !(u <= 6.2e+76))
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.25e+38) || ~((u <= 6.2e+76)))
		tmp = (v / u) * (t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.25e+38], N[Not[LessEqual[u, 6.2e+76]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.25e38 or 6.20000000000000023e76 < u

   1. Initial program 85.1%

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

   2. Taylor expanded in t1 around 0 81.6%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{{u}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 81.6%

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

   4. Simplified 81.6%

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

      1. *-commutative 81.6%

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

      2. times-frac 85.0%

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

      3. add-sqr-sqrt 41.6%

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

      4. sqrt-unprod 66.1%

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

      5. sqr-neg 66.1%

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

      6. sqrt-unprod 33.4%

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

      7. add-sqr-sqrt 63.3%

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

   6. Applied egg-rr 63.3%

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

   if -3.25e38 < u < 6.20000000000000023e76

   1. Initial program 75.3%

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

      1. associate-/l* 68.2%

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

      2. neg-mul-1 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*r/ 67.8%

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

      5. associate-/l* 68.5%

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

      6. neg-mul-1 68.5%

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

      7. associate-/r* 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in t1 around inf 74.0%

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

      1. associate-*r/ 74.0%

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

      2. neg-mul-1 74.0%

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

   6. Simplified 74.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.25 \cdot 10^{+38} \lor \neg \left(u \leq 6.2 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10: 66.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+38} \lor \neg \left(u \leq 1.75 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3e+38) (not (<= u 1.75e+77)))
   (/ v (* u (/ u t1)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e+38) || !(u <= 1.75e+77)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3d+38)) .or. (.not. (u <= 1.75d+77))) then
        tmp = v / (u * (u / t1))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e+38) || !(u <= 1.75e+77)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3e+38) or not (u <= 1.75e+77):
		tmp = v / (u * (u / t1))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3e+38) || !(u <= 1.75e+77))
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3e+38) || ~((u <= 1.75e+77)))
		tmp = v / (u * (u / t1));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3e+38], N[Not[LessEqual[u, 1.75e+77]], $MachinePrecision]], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.0000000000000001e38 or 1.7500000000000001e77 < u

   1. Initial program 85.1%

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

      1. associate-/l* 85.3%

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

      2. neg-mul-1 85.3%

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

      3. *-commutative 85.3%

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

      4. associate-*r/ 85.2%

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

      5. associate-/l* 85.3%

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

      6. neg-mul-1 85.3%

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

      7. associate-/r* 90.5%

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

   3. Simplified 90.5%

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

      1. associate-/l/ 85.3%

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

      2. associate-*r/ 85.1%

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

      3. distribute-rgt-neg-in 85.1%

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

      4. distribute-lft-neg-out 85.1%

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

      5. associate-/r* 94.4%

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

   5. Applied egg-rr 93.0%

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

      1. *-rgt-identity 93.0%

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

      2. associate-/r/ 86.9%

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

      3. times-frac 79.2%

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

      4. /-rgt-identity 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in t1 around 0 72.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{t1}{{u}^{2}}\right)} \cdot v \]
   9. Step-by-step derivation

      1. associate-*r/ 72.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot t1}{{u}^{2}}} \cdot v \]

      2. mul-1-neg 72.9%

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

      3. unpow2 72.9%

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

   10. Simplified 72.9%

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

       1. *-commutative 72.9%

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

       2. clear-num 72.9%

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

       3. un-div-inv 72.9%

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

       4. add-sqr-sqrt 33.3%

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

       5. sqrt-unprod 59.8%

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

       6. sqr-neg 59.8%

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

       7. sqrt-unprod 33.6%

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

       8. add-sqr-sqrt 63.6%

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

       9. associate-*r/ 63.5%

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

   12. Applied egg-rr 63.5%

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

   if -3.0000000000000001e38 < u < 1.7500000000000001e77

   1. Initial program 75.3%

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

      1. associate-/l* 68.2%

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

      2. neg-mul-1 68.2%

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

      3. *-commutative 68.2%

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

      4. associate-*r/ 67.8%

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

      5. associate-/l* 68.5%

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

      6. neg-mul-1 68.5%

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

      7. associate-/r* 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in t1 around inf 74.0%

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

      1. associate-*r/ 74.0%

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

      2. neg-mul-1 74.0%

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

   6. Simplified 74.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+38} \lor \neg \left(u \leq 1.75 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / (u + t1)) / ((-1.0d0) - (u / t1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. *-commutative 78.7%

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

   2. times-frac 94.6%

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

   3. neg-mul-1 94.6%

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

   4. associate-/l* 94.3%

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

   5. associate-*r/ 94.3%

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

   6. associate-/l* 94.3%

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

   7. associate-/l/ 94.3%

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

   8. neg-mul-1 94.3%

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

   9. *-lft-identity 94.3%

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

   10. metadata-eval 94.3%

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

   11. times-frac 94.3%

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

   12. neg-mul-1 94.3%

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

   13. remove-double-neg 94.3%

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

   14. neg-mul-1 94.3%

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

   15. sub0-neg 94.3%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 94.3%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 94.3%

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

   18. div-sub 94.3%

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

   19. distribute-frac-neg 94.3%

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

   20. *-inverses 94.3%

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

   21. metadata-eval 94.3%

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

3. Simplified 94.3%

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

4. Final simplification 94.3%

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

Alternative 12: 56.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+69}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+221}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6e+69) (/ v u) (if (<= u 1.35e+221) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6e+69) {
		tmp = v / u;
	} else if (u <= 1.35e+221) {
		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 <= (-6d+69)) then
        tmp = v / u
    else if (u <= 1.35d+221) 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 <= -6e+69) {
		tmp = v / u;
	} else if (u <= 1.35e+221) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6e+69:
		tmp = v / u
	elif u <= 1.35e+221:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6e+69)
		tmp = Float64(v / u);
	elseif (u <= 1.35e+221)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6e+69)
		tmp = v / u;
	elseif (u <= 1.35e+221)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6e+69], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.35e+221], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -5.99999999999999967e69 or 1.35e221 < u

   1. Initial program 86.2%

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

      1. associate-/l* 84.9%

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

      2. neg-mul-1 84.9%

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

      3. *-commutative 84.9%

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

      4. associate-*r/ 84.9%

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

      5. associate-/l* 84.8%

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

      6. neg-mul-1 84.8%

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

      7. associate-/r* 92.4%

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

   3. Simplified 92.4%

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

      1. associate-/l/ 84.8%

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

      2. associate-*r/ 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. distribute-lft-neg-out 86.2%

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

      5. associate-/r* 96.8%

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

   5. Applied egg-rr 97.4%

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

      1. *-rgt-identity 97.4%

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

      2. associate-/r/ 90.1%

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

      3. times-frac 82.6%

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

      4. /-rgt-identity 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in t1 around inf 38.8%

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

   9. Taylor expanded in t1 around 0 38.8%

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

   if -5.99999999999999967e69 < u < 1.35e221

   1. Initial program 76.3%

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

      1. associate-/l* 70.8%

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

      2. neg-mul-1 70.8%

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

      3. *-commutative 70.8%

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

      4. associate-*r/ 70.4%

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

      5. associate-/l* 71.0%

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

      6. neg-mul-1 71.0%

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

      7. associate-/r* 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in t1 around inf 67.0%

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

      1. associate-*r/ 67.0%

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

      2. neg-mul-1 67.0%

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

   6. Simplified 67.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+69}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+221}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 13: 56.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+70}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{+221}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.65e+70) (/ (- v) u) (if (<= u 3.3e+221) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.65e+70) {
		tmp = -v / u;
	} else if (u <= 3.3e+221) {
		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 <= (-2.65d+70)) then
        tmp = -v / u
    else if (u <= 3.3d+221) 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 <= -2.65e+70) {
		tmp = -v / u;
	} else if (u <= 3.3e+221) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.65e+70:
		tmp = -v / u
	elif u <= 3.3e+221:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.65e+70)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 3.3e+221)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.65e+70)
		tmp = -v / u;
	elseif (u <= 3.3e+221)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.65e+70], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 3.3e+221], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.65e70

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

      2. times-frac 88.6%

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

      3. neg-mul-1 88.6%

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

      4. associate-/l* 88.6%

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

      5. associate-*r/ 88.7%

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

      6. associate-/l* 88.7%

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

      7. associate-/l/ 88.7%

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

      8. neg-mul-1 88.7%

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

      9. *-lft-identity 88.7%

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

      10. metadata-eval 88.7%

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

      11. times-frac 88.7%

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

      12. neg-mul-1 88.7%

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

      13. remove-double-neg 88.7%

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

      14. neg-mul-1 88.7%

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

      15. sub0-neg 88.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 88.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 88.7%

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

      18. div-sub 88.7%

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

      19. distribute-frac-neg 88.7%

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

      20. *-inverses 88.7%

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

      21. metadata-eval 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in t1 around 0 85.0%

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

   5. Taylor expanded in u around 0 34.7%

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

      1. mul-1-neg 34.7%

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

      2. distribute-frac-neg 34.7%

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

   7. Simplified 34.7%

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

   if -2.65e70 < u < 3.29999999999999991e221

   1. Initial program 76.3%

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

      1. associate-/l* 70.8%

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

      2. neg-mul-1 70.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. *-commutative 70.8%

         \[\leadsto \frac{\color{blue}{t1 \cdot -1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      4. associate-*r/ 70.4%

         \[\leadsto \color{blue}{t1 \cdot \frac{-1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      5. associate-/l* 71.0%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      6. neg-mul-1 71.0%

         \[\leadsto t1 \cdot \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      7. associate-/r* 77.1%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{-v}{t1 + u}}{t1 + u}} \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{-v}{t1 + u}}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 67.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   5. Step-by-step derivation

      1. associate-*r/ 67.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 67.0%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   6. Simplified 67.0%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 3.29999999999999991e221 < u

   1. Initial program 86.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 87.2%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 87.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. *-commutative 87.2%

         \[\leadsto \frac{\color{blue}{t1 \cdot -1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      4. associate-*r/ 87.2%

         \[\leadsto \color{blue}{t1 \cdot \frac{-1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      5. associate-/l* 87.2%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      6. neg-mul-1 87.2%

         \[\leadsto t1 \cdot \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      7. associate-/r* 91.5%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{-v}{t1 + u}}{t1 + u}} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{-v}{t1 + u}}{t1 + u}} \]
   4. Step-by-step derivation

      1. associate-/l/ 87.2%

         \[\leadsto t1 \cdot \color{blue}{\frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. associate-*r/ 86.9%

         \[\leadsto \color{blue}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      3. distribute-rgt-neg-in 86.9%

         \[\leadsto \frac{\color{blue}{-t1 \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      4. distribute-lft-neg-out 86.9%

         \[\leadsto \frac{\color{blue}{\left(-t1\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      5. associate-/r* 95.5%

         \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{t1}{\frac{t1 - u}{v}}}{t1 + u}} \]
   6. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto \frac{\frac{t1}{\frac{t1 - u}{v}}}{\color{blue}{\left(t1 + u\right) \cdot 1}} \]

      2. associate-/r/ 99.9%

         \[\leadsto \frac{\color{blue}{\frac{t1}{t1 - u} \cdot v}}{\left(t1 + u\right) \cdot 1} \]

      3. times-frac 95.6%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 - u}}{t1 + u} \cdot \frac{v}{1}} \]

      4. /-rgt-identity 95.6%

         \[\leadsto \frac{\frac{t1}{t1 - u}}{t1 + u} \cdot \color{blue}{v} \]

   7. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 - u}}{t1 + u} \cdot v} \]

   8. Taylor expanded in t1 around inf 46.5%

      \[\leadsto \frac{\color{blue}{1}}{t1 + u} \cdot v \]

   9. Taylor expanded in t1 around 0 46.5%

      \[\leadsto \color{blue}{\frac{v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+70}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{+221}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 14: 60.6% 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 78.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 74.2%

      \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   2. neg-mul-1 74.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

   3. *-commutative 74.2%

      \[\leadsto \frac{\color{blue}{t1 \cdot -1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

   4. associate-*r/ 73.9%

      \[\leadsto \color{blue}{t1 \cdot \frac{-1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   5. associate-/l* 74.4%

      \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   6. neg-mul-1 74.4%

      \[\leadsto t1 \cdot \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   7. associate-/r* 80.8%

      \[\leadsto t1 \cdot \color{blue}{\frac{\frac{-v}{t1 + u}}{t1 + u}} \]

3. Simplified 80.8%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{-v}{t1 + u}}{t1 + u}} \]
4. Step-by-step derivation

   1. associate-/l/ 74.4%

      \[\leadsto t1 \cdot \color{blue}{\frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. associate-*r/ 78.7%

      \[\leadsto \color{blue}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. distribute-rgt-neg-in 78.7%

      \[\leadsto \frac{\color{blue}{-t1 \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   4. distribute-lft-neg-out 78.7%

      \[\leadsto \frac{\color{blue}{\left(-t1\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   5. associate-/r* 87.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

5. Applied egg-rr 56.3%

   \[\leadsto \color{blue}{\frac{\frac{t1}{\frac{t1 - u}{v}}}{t1 + u}} \]
6. Step-by-step derivation

   1. *-rgt-identity 56.3%

      \[\leadsto \frac{\frac{t1}{\frac{t1 - u}{v}}}{\color{blue}{\left(t1 + u\right) \cdot 1}} \]

   2. associate-/r/ 56.5%

      \[\leadsto \frac{\color{blue}{\frac{t1}{t1 - u} \cdot v}}{\left(t1 + u\right) \cdot 1} \]

   3. times-frac 54.3%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 - u}}{t1 + u} \cdot \frac{v}{1}} \]

   4. /-rgt-identity 54.3%

      \[\leadsto \frac{\frac{t1}{t1 - u}}{t1 + u} \cdot \color{blue}{v} \]

7. Simplified 54.3%

   \[\leadsto \color{blue}{\frac{\frac{t1}{t1 - u}}{t1 + u} \cdot v} \]

8. Taylor expanded in t1 around inf 23.2%

   \[\leadsto \frac{\color{blue}{1}}{t1 + u} \cdot v \]
9. Step-by-step derivation

   1. associate-*l/ 23.2%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{t1 + u}} \]

   2. *-un-lft-identity 23.2%

      \[\leadsto \frac{\color{blue}{v}}{t1 + u} \]

   3. frac-2neg 23.2%

      \[\leadsto \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

   4. add-sqr-sqrt 11.9%

      \[\leadsto \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{-\left(t1 + u\right)} \]

   5. sqrt-unprod 34.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{-\left(t1 + u\right)} \]

   6. sqr-neg 34.8%

      \[\leadsto \frac{\sqrt{\color{blue}{v \cdot v}}}{-\left(t1 + u\right)} \]

   7. sqrt-unprod 29.2%

      \[\leadsto \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{-\left(t1 + u\right)} \]

   8. add-sqr-sqrt 60.8%

      \[\leadsto \frac{\color{blue}{v}}{-\left(t1 + u\right)} \]

10. Applied egg-rr 60.8%

    \[\leadsto \color{blue}{\frac{v}{-\left(t1 + u\right)}} \]
11. Step-by-step derivation

    1. neg-sub0 60.8%

       \[\leadsto \frac{v}{\color{blue}{0 - \left(t1 + u\right)}} \]

    2. +-commutative 60.8%

       \[\leadsto \frac{v}{0 - \color{blue}{\left(u + t1\right)}} \]

    3. associate--r+ 60.8%

       \[\leadsto \frac{v}{\color{blue}{\left(0 - u\right) - t1}} \]

    4. neg-sub0 60.8%

       \[\leadsto \frac{v}{\color{blue}{\left(-u\right)} - t1} \]

12. Simplified 60.8%

    \[\leadsto \color{blue}{\frac{v}{\left(-u\right) - t1}} \]

13. Final simplification 60.8%

    \[\leadsto \frac{v}{\left(-u\right) - t1} \]

Alternative 15: 17.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v u))

C

double code(double u, double v, double t1) {
	return v / u;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / u
end function

Java

public static double code(double u, double v, double t1) {
	return v / u;
}

Python

def code(u, v, t1):
	return v / u

Julia

function code(u, v, t1)
	return Float64(v / u)
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / u;
end

Wolfram

code[u_, v_, t1_] := N[(v / u), $MachinePrecision]

Derivation

1. Initial program 78.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 74.2%

      \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   2. neg-mul-1 74.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

   3. *-commutative 74.2%

      \[\leadsto \frac{\color{blue}{t1 \cdot -1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

   4. associate-*r/ 73.9%

      \[\leadsto \color{blue}{t1 \cdot \frac{-1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   5. associate-/l* 74.4%

      \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   6. neg-mul-1 74.4%

      \[\leadsto t1 \cdot \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   7. associate-/r* 80.8%

      \[\leadsto t1 \cdot \color{blue}{\frac{\frac{-v}{t1 + u}}{t1 + u}} \]

3. Simplified 80.8%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{-v}{t1 + u}}{t1 + u}} \]
4. Step-by-step derivation

   1. associate-/l/ 74.4%

      \[\leadsto t1 \cdot \color{blue}{\frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. associate-*r/ 78.7%

      \[\leadsto \color{blue}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. distribute-rgt-neg-in 78.7%

      \[\leadsto \frac{\color{blue}{-t1 \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   4. distribute-lft-neg-out 78.7%

      \[\leadsto \frac{\color{blue}{\left(-t1\right) \cdot v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

   5. associate-/r* 87.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}} \]

5. Applied egg-rr 56.3%

   \[\leadsto \color{blue}{\frac{\frac{t1}{\frac{t1 - u}{v}}}{t1 + u}} \]
6. Step-by-step derivation

   1. *-rgt-identity 56.3%

      \[\leadsto \frac{\frac{t1}{\frac{t1 - u}{v}}}{\color{blue}{\left(t1 + u\right) \cdot 1}} \]

   2. associate-/r/ 56.5%

      \[\leadsto \frac{\color{blue}{\frac{t1}{t1 - u} \cdot v}}{\left(t1 + u\right) \cdot 1} \]

   3. times-frac 54.3%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 - u}}{t1 + u} \cdot \frac{v}{1}} \]

   4. /-rgt-identity 54.3%

      \[\leadsto \frac{\frac{t1}{t1 - u}}{t1 + u} \cdot \color{blue}{v} \]

7. Simplified 54.3%

   \[\leadsto \color{blue}{\frac{\frac{t1}{t1 - u}}{t1 + u} \cdot v} \]

8. Taylor expanded in t1 around inf 23.2%

   \[\leadsto \frac{\color{blue}{1}}{t1 + u} \cdot v \]

9. Taylor expanded in t1 around 0 16.0%

   \[\leadsto \color{blue}{\frac{v}{u}} \]

10. Final simplification 16.0%

    \[\leadsto \frac{v}{u} \]

Reproduce

herbie shell --seed 2023172 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))