Rosa's DopplerBench

Percentage Accurate: 72.3% → 98.3%

Time: 11.9s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. associate-/l* 72.5%

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

   2. distribute-lft-neg-out 72.5%

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

   3. distribute-rgt-neg-in 72.5%

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

   4. associate-/r* 81.7%

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

   5. distribute-neg-frac2 81.7%

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

3. Simplified 81.7%

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

   1. distribute-frac-neg2 81.7%

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

   2. distribute-rgt-neg-out 81.7%

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

   3. associate-/r* 72.5%

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

   4. distribute-lft-neg-out 72.5%

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

   5. associate-/l* 71.8%

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

   6. times-frac 97.5%

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

   7. frac-2neg 97.5%

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

   8. associate-*r/ 98.3%

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

   9. add-sqr-sqrt 46.5%

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

   10. sqrt-unprod 44.2%

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

   11. sqr-neg 44.2%

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

   12. sqrt-unprod 18.9%

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

   13. add-sqr-sqrt 35.5%

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

   14. add-sqr-sqrt 19.0%

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

   15. sqrt-unprod 56.5%

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

   16. sqr-neg 56.5%

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

   17. sqrt-prod 47.6%

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

   18. add-sqr-sqrt 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 2: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.4 \cdot 10^{+188}:\\ \;\;\;\;\frac{\left(-v\right) \cdot \frac{t1}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq -1.2 \cdot 10^{-132}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;u \leq 9.8 \cdot 10^{+145}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.4e+188)
   (/ (* (- v) (/ t1 u)) (+ t1 u))
   (if (<= u -1.2e-132)
     (* (- t1) (/ (/ v (+ t1 u)) (+ t1 u)))
     (if (<= u 7.2e-151)
       (/ v (- (- t1) (* u 2.0)))
       (if (<= u 9.8e+145)
         (* (- v) (/ t1 (* (+ t1 u) (+ t1 u))))
         (/ (/ t1 (/ u v)) (- (- u) t1)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.4e+188) {
		tmp = (-v * (t1 / u)) / (t1 + u);
	} else if (u <= -1.2e-132) {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	} else if (u <= 7.2e-151) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (u <= 9.8e+145) {
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = (t1 / (u / v)) / (-u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.4d+188)) then
        tmp = (-v * (t1 / u)) / (t1 + u)
    else if (u <= (-1.2d-132)) then
        tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
    else if (u <= 7.2d-151) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (u <= 9.8d+145) then
        tmp = -v * (t1 / ((t1 + u) * (t1 + u)))
    else
        tmp = (t1 / (u / v)) / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.4e+188) {
		tmp = (-v * (t1 / u)) / (t1 + u);
	} else if (u <= -1.2e-132) {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	} else if (u <= 7.2e-151) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (u <= 9.8e+145) {
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	} else {
		tmp = (t1 / (u / v)) / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.4e+188:
		tmp = (-v * (t1 / u)) / (t1 + u)
	elif u <= -1.2e-132:
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
	elif u <= 7.2e-151:
		tmp = v / (-t1 - (u * 2.0))
	elif u <= 9.8e+145:
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)))
	else:
		tmp = (t1 / (u / v)) / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.4e+188)
		tmp = Float64(Float64(Float64(-v) * Float64(t1 / u)) / Float64(t1 + u));
	elseif (u <= -1.2e-132)
		tmp = Float64(Float64(-t1) * Float64(Float64(v / Float64(t1 + u)) / Float64(t1 + u)));
	elseif (u <= 7.2e-151)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (u <= 9.8e+145)
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.4e+188)
		tmp = (-v * (t1 / u)) / (t1 + u);
	elseif (u <= -1.2e-132)
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	elseif (u <= 7.2e-151)
		tmp = v / (-t1 - (u * 2.0));
	elseif (u <= 9.8e+145)
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	else
		tmp = (t1 / (u / v)) / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.4e+188], N[(N[((-v) * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.2e-132], N[((-t1) * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.2e-151], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 9.8e+145], N[((-v) * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -5.4e188

   1. Initial program 68.7%

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

      1. associate-/l* 69.2%

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

      2. distribute-lft-neg-out 69.2%

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

      3. distribute-rgt-neg-in 69.2%

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

      4. associate-/r* 84.8%

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

      5. distribute-neg-frac2 84.8%

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

   3. Simplified 84.8%

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

      1. distribute-frac-neg2 84.8%

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

      2. distribute-rgt-neg-out 84.8%

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

      3. associate-/r* 69.2%

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

      4. distribute-lft-neg-out 69.2%

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

      5. associate-/l* 68.7%

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

      6. times-frac 99.8%

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

      7. frac-2neg 99.8%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 43.1%

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

      10. sqrt-unprod 71.2%

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

      11. sqr-neg 71.2%

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

      12. sqrt-unprod 41.0%

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

      13. add-sqr-sqrt 68.7%

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

      14. add-sqr-sqrt 65.4%

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

      15. sqrt-unprod 69.2%

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

      16. sqr-neg 69.2%

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

      17. sqrt-prod 3.3%

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

      18. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around 0 99.9%

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

   if -5.4e188 < u < -1.20000000000000008e-132

   1. Initial program 77.0%

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

      1. associate-/l* 80.6%

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

      2. distribute-lft-neg-out 80.6%

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

      3. distribute-rgt-neg-in 80.6%

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

      4. associate-/r* 89.4%

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

      5. distribute-neg-frac2 89.4%

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

   3. Simplified 89.4%

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

   if -1.20000000000000008e-132 < u < 7.20000000000000064e-151

   1. Initial program 50.9%

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

      1. associate-/l* 54.5%

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

      2. distribute-lft-neg-out 54.5%

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

      3. distribute-rgt-neg-in 54.5%

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

      4. associate-/r* 63.4%

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

      5. distribute-neg-frac2 63.4%

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

   3. Simplified 63.4%

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

      1. associate-*r/ 98.5%

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

      2. +-commutative 98.5%

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

      3. distribute-neg-in 98.5%

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

      4. sub-neg 98.5%

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

      5. associate-*l/ 98.5%

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

      6. clear-num 98.4%

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

      7. frac-2neg 98.4%

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

      8. frac-times 100.0%

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

      9. *-un-lft-identity 100.0%

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

      10. frac-2neg 100.0%

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

      11. sub-neg 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. +-commutative 100.0%

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

      14. remove-double-neg 100.0%

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

      15. add-sqr-sqrt 45.7%

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

      16. sqrt-unprod 23.5%

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

      17. sqr-neg 23.5%

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

      18. sqrt-unprod 6.7%

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

      19. add-sqr-sqrt 11.5%

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

      20. add-sqr-sqrt 4.8%

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

      21. sqrt-unprod 41.7%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 94.1%

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

      1. *-commutative 94.1%

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

   9. Simplified 94.1%

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

   if 7.20000000000000064e-151 < u < 9.80000000000000006e145

   1. Initial program 86.3%

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

      1. associate-*l/ 88.0%

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

      2. *-commutative 88.0%

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

   3. Simplified 88.0%

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

   if 9.80000000000000006e145 < u

   1. Initial program 75.5%

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

      1. associate-/l* 76.1%

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

      2. distribute-lft-neg-out 76.1%

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

      3. distribute-rgt-neg-in 76.1%

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

      4. associate-/r* 90.4%

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

      5. distribute-neg-frac2 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in t1 around 0 90.4%

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

      1. associate-*r/ 99.9%

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

      2. frac-2neg 99.9%

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

      3. clear-num 100.0%

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

      4. un-div-inv 100.0%

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

      5. remove-double-neg 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.4 \cdot 10^{+188}:\\ \;\;\;\;\frac{\left(-v\right) \cdot \frac{t1}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq -1.2 \cdot 10^{-132}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;u \leq 9.8 \cdot 10^{+145}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 4 \cdot 10^{+107}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.4e+132)
   (/ v (- (- t1) (* u 2.0)))
   (if (<= t1 4e+107) (* (- t1) (/ (/ v (+ t1 u)) (+ t1 u))) (/ v (- t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.4e+132) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 4e+107) {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-4.4d+132)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (t1 <= 4d+107) then
        tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.4e+132) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 4e+107) {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4.4e+132:
		tmp = v / (-t1 - (u * 2.0))
	elif t1 <= 4e+107:
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.4e+132)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (t1 <= 4e+107)
		tmp = Float64(Float64(-t1) * Float64(Float64(v / Float64(t1 + u)) / Float64(t1 + u)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4.4e+132)
		tmp = v / (-t1 - (u * 2.0));
	elseif (t1 <= 4e+107)
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.4e+132], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4e+107], N[((-t1) * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.39999999999999977e132

   1. Initial program 38.4%

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

      1. associate-/l* 44.6%

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

      2. distribute-lft-neg-out 44.6%

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

      3. distribute-rgt-neg-in 44.6%

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

      4. associate-/r* 66.9%

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

      5. distribute-neg-frac2 66.9%

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

   3. Simplified 66.9%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-*l/ 100.0%

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

      6. clear-num 100.0%

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

      7. frac-2neg 100.0%

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

      8. frac-times 94.8%

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

      9. *-un-lft-identity 94.8%

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

      10. frac-2neg 94.8%

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

      11. sub-neg 94.8%

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

      12. distribute-neg-in 94.8%

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

      13. +-commutative 94.8%

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

      14. remove-double-neg 94.8%

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

      15. add-sqr-sqrt 94.6%

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

      16. sqrt-unprod 18.0%

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

      17. sqr-neg 18.0%

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

      18. sqrt-unprod 0.0%

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

      19. add-sqr-sqrt 33.5%

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

      20. add-sqr-sqrt 30.5%

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

      21. sqrt-unprod 34.3%

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

   6. Applied egg-rr 94.8%

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

   7. Taylor expanded in u around 0 89.9%

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

      1. *-commutative 89.9%

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

   9. Simplified 89.9%

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

   if -4.39999999999999977e132 < t1 < 3.9999999999999999e107

   1. Initial program 84.9%

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

      1. associate-/l* 85.8%

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

      2. distribute-lft-neg-out 85.8%

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

      3. distribute-rgt-neg-in 85.8%

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

      4. associate-/r* 90.2%

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

      5. distribute-neg-frac2 90.2%

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

   3. Simplified 90.2%

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

   if 3.9999999999999999e107 < t1

   1. Initial program 46.7%

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

      1. associate-/l* 41.7%

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

      2. distribute-lft-neg-out 41.7%

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

      3. distribute-rgt-neg-in 41.7%

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

      4. associate-/r* 59.8%

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

      5. distribute-neg-frac2 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in t1 around inf 86.7%

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

      1. associate-*r/ 86.7%

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

      2. neg-mul-1 86.7%

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

   7. Simplified 86.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 4 \cdot 10^{+107}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(t1 \cdot \frac{-1}{u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.8e-40)
   (/ (/ t1 (/ u v)) (- (- u) t1))
   (if (<= u 1.55e-13)
     (/ v (- (- t1) (* u 2.0)))
     (* (/ v (+ t1 u)) (* t1 (/ -1.0 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.8e-40) {
		tmp = (t1 / (u / v)) / (-u - t1);
	} else if (u <= 1.55e-13) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / (t1 + u)) * (t1 * (-1.0 / 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 <= (-5.8d-40)) then
        tmp = (t1 / (u / v)) / (-u - t1)
    else if (u <= 1.55d-13) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (v / (t1 + u)) * (t1 * ((-1.0d0) / u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.8e-40) {
		tmp = (t1 / (u / v)) / (-u - t1);
	} else if (u <= 1.55e-13) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / (t1 + u)) * (t1 * (-1.0 / u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.8e-40:
		tmp = (t1 / (u / v)) / (-u - t1)
	elif u <= 1.55e-13:
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (v / (t1 + u)) * (t1 * (-1.0 / u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.8e-40)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(Float64(-u) - t1));
	elseif (u <= 1.55e-13)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 * Float64(-1.0 / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.8e-40)
		tmp = (t1 / (u / v)) / (-u - t1);
	elseif (u <= 1.55e-13)
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (v / (t1 + u)) * (t1 * (-1.0 / u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.8e-40], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.55e-13], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.7999999999999998e-40

   1. Initial program 74.9%

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

      1. associate-/l* 74.2%

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

      2. distribute-lft-neg-out 74.2%

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

      3. distribute-rgt-neg-in 74.2%

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

      4. associate-/r* 87.2%

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

      5. distribute-neg-frac2 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in t1 around 0 76.6%

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

      1. associate-*r/ 80.2%

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

      2. frac-2neg 80.2%

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

      3. clear-num 80.2%

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

      4. un-div-inv 80.2%

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

      5. remove-double-neg 80.2%

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

   7. Applied egg-rr 80.2%

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

   if -5.7999999999999998e-40 < u < 1.55e-13

   1. Initial program 63.2%

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

      1. associate-/l* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 73.4%

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

      5. distribute-neg-frac2 73.4%

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

   3. Simplified 73.4%

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

      1. associate-*r/ 96.7%

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

      2. +-commutative 96.7%

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

      3. distribute-neg-in 96.7%

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

      4. sub-neg 96.7%

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

      5. associate-*l/ 97.3%

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

      6. clear-num 97.3%

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

      7. frac-2neg 97.3%

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

      8. frac-times 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. frac-2neg 99.9%

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

      11. sub-neg 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. +-commutative 99.9%

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

      14. remove-double-neg 99.9%

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

      15. add-sqr-sqrt 48.3%

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

      16. sqrt-unprod 29.2%

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

      17. sqr-neg 29.2%

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

      18. sqrt-unprod 9.0%

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

      19. add-sqr-sqrt 15.9%

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

      20. add-sqr-sqrt 7.5%

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

      21. sqrt-unprod 47.2%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around 0 80.4%

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

      1. *-commutative 80.4%

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

   9. Simplified 80.4%

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

   if 1.55e-13 < u

   1. Initial program 81.3%

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

      1. associate-/l* 79.7%

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

      2. distribute-lft-neg-out 79.7%

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

      3. distribute-rgt-neg-in 79.7%

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

      4. associate-/r* 88.5%

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

      5. distribute-neg-frac2 88.5%

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

   3. Simplified 88.5%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-neg-in 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-*l/ 98.2%

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

      6. clear-num 98.1%

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

      7. frac-2neg 98.1%

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

      8. frac-times 93.3%

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

      9. *-un-lft-identity 93.3%

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

      10. frac-2neg 93.3%

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

      11. sub-neg 93.3%

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

      12. distribute-neg-in 93.3%

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

      13. +-commutative 93.3%

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

      14. remove-double-neg 93.3%

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

      15. add-sqr-sqrt 45.6%

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

      16. sqrt-unprod 53.7%

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

      17. sqr-neg 53.7%

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

      18. sqrt-unprod 27.9%

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

      19. add-sqr-sqrt 52.0%

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

      20. add-sqr-sqrt 8.8%

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

      21. sqrt-unprod 75.8%

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

   6. Applied egg-rr 93.3%

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

      1. neg-mul-1 93.3%

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

      2. times-frac 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-/r/ 98.2%

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

   10. Simplified 98.2%

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

   11. Taylor expanded in t1 around 0 81.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(t1 \cdot \frac{-1}{u}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-39} \lor \neg \left(t1 \leq 1.55 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(-u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.2e-39) (not (<= t1 1.55e-34)))
   (/ v (- (- t1) (* u 2.0)))
   (* t1 (/ v (* (+ t1 u) (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.2e-39) || !(t1 <= 1.55e-34)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * (v / ((t1 + u) * -u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.2d-39)) .or. (.not. (t1 <= 1.55d-34))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t1 * (v / ((t1 + u) * -u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.2e-39) || !(t1 <= 1.55e-34)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * (v / ((t1 + u) * -u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.2e-39) or not (t1 <= 1.55e-34):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t1 * (v / ((t1 + u) * -u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.2e-39) || !(t1 <= 1.55e-34))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(v / Float64(Float64(t1 + u) * Float64(-u))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.2e-39) || ~((t1 <= 1.55e-34)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t1 * (v / ((t1 + u) * -u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.2e-39], N[Not[LessEqual[t1, 1.55e-34]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(v / N[(N[(t1 + u), $MachinePrecision] * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.2e-39 or 1.5499999999999999e-34 < t1

   1. Initial program 61.7%

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

      1. associate-/l* 62.1%

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

      2. distribute-lft-neg-out 62.1%

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

      3. distribute-rgt-neg-in 62.1%

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

      4. associate-/r* 75.9%

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

      5. distribute-neg-frac2 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 92.3%

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

      9. *-un-lft-identity 92.3%

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

      10. frac-2neg 92.3%

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

      11. sub-neg 92.3%

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

      12. distribute-neg-in 92.3%

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

      13. +-commutative 92.3%

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

      14. remove-double-neg 92.3%

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

      15. add-sqr-sqrt 46.2%

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

      16. sqrt-unprod 36.3%

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

      17. sqr-neg 36.3%

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

      18. sqrt-unprod 16.0%

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

      19. add-sqr-sqrt 32.6%

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

      20. add-sqr-sqrt 16.7%

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

      21. sqrt-unprod 52.2%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in u around 0 80.5%

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

      1. *-commutative 80.5%

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

   9. Simplified 80.5%

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

   if -5.2e-39 < t1 < 1.5499999999999999e-34

   1. Initial program 85.3%

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

      1. associate-/l* 86.5%

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

      2. distribute-lft-neg-out 86.5%

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

      3. distribute-rgt-neg-in 86.5%

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

      4. associate-/r* 89.5%

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

      5. distribute-neg-frac2 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in t1 around 0 77.2%

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

   6. Taylor expanded in v around 0 76.1%

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

      1. neg-mul-1 76.1%

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

      2. distribute-neg-frac2 76.1%

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

   8. Simplified 76.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-39} \lor \neg \left(t1 \leq 1.55 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(-u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{-39} \lor \neg \left(t1 \leq 22000\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.4e-39) (not (<= t1 22000.0)))
   (/ v (- (- t1) (* u 2.0)))
   (* (- t1) (/ (/ v u) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.4e-39) || !(t1 <= 22000.0)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = -t1 * ((v / u) / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.4d-39)) .or. (.not. (t1 <= 22000.0d0))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = -t1 * ((v / u) / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.4e-39) || !(t1 <= 22000.0)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = -t1 * ((v / u) / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.4e-39) or not (t1 <= 22000.0):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = -t1 * ((v / u) / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.4e-39) || !(t1 <= 22000.0))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(-t1) * Float64(Float64(v / u) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.4e-39) || ~((t1 <= 22000.0)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = -t1 * ((v / u) / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.4e-39], N[Not[LessEqual[t1, 22000.0]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t1) * N[(N[(v / u), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.4000000000000001e-39 or 22000 < t1

   1. Initial program 60.5%

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

      1. associate-/l* 60.9%

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

      2. distribute-lft-neg-out 60.9%

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

      3. distribute-rgt-neg-in 60.9%

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

      4. associate-/r* 74.7%

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

      5. distribute-neg-frac2 74.7%

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

   3. Simplified 74.7%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 92.6%

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

      9. *-un-lft-identity 92.6%

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

      10. frac-2neg 92.6%

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

      11. sub-neg 92.6%

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

      12. distribute-neg-in 92.6%

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

      13. +-commutative 92.6%

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

      14. remove-double-neg 92.6%

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

      15. add-sqr-sqrt 48.5%

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

      16. sqrt-unprod 36.6%

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

      17. sqr-neg 36.6%

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

      18. sqrt-unprod 15.2%

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

      19. add-sqr-sqrt 32.6%

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

      20. add-sqr-sqrt 16.1%

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

      21. sqrt-unprod 50.5%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in u around 0 81.6%

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

      1. *-commutative 81.6%

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

   9. Simplified 81.6%

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

   if -5.4000000000000001e-39 < t1 < 22000

   1. Initial program 85.4%

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

      1. associate-/l* 86.5%

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

      2. distribute-lft-neg-out 86.5%

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

      3. distribute-rgt-neg-in 86.5%

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

      4. associate-/r* 90.1%

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

      5. distribute-neg-frac2 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in t1 around 0 76.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \cdot 10^{-39} \lor \neg \left(t1 \leq 22000\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{-40} \lor \neg \left(u \leq 2.2 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.2e-40) (not (<= u 2.2e-13)))
   (/ (/ t1 (/ u v)) (- (- u) t1))
   (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.2e-40) || !(u <= 2.2e-13)) {
		tmp = (t1 / (u / v)) / (-u - t1);
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-7.2d-40)) .or. (.not. (u <= 2.2d-13))) then
        tmp = (t1 / (u / v)) / (-u - t1)
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.2e-40) || !(u <= 2.2e-13)) {
		tmp = (t1 / (u / v)) / (-u - t1);
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.2e-40) or not (u <= 2.2e-13):
		tmp = (t1 / (u / v)) / (-u - t1)
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.2e-40) || !(u <= 2.2e-13))
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(Float64(-u) - t1));
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.2e-40) || ~((u <= 2.2e-13)))
		tmp = (t1 / (u / v)) / (-u - t1);
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.2e-40], N[Not[LessEqual[u, 2.2e-13]], $MachinePrecision]], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.2e-40 or 2.19999999999999997e-13 < u

   1. Initial program 78.1%

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

      1. associate-/l* 77.0%

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

      2. distribute-lft-neg-out 77.0%

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

      3. distribute-rgt-neg-in 77.0%

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

      4. associate-/r* 87.9%

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

      5. distribute-neg-frac2 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in t1 around 0 77.4%

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

      1. associate-*r/ 79.6%

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

      2. frac-2neg 79.6%

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

      3. clear-num 79.6%

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

      4. un-div-inv 79.6%

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

      5. remove-double-neg 79.6%

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

   7. Applied egg-rr 79.6%

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

   if -7.2e-40 < u < 2.19999999999999997e-13

   1. Initial program 63.2%

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

      1. associate-/l* 66.4%

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

      2. distribute-lft-neg-out 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. associate-/r* 73.4%

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

      5. distribute-neg-frac2 73.4%

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

   3. Simplified 73.4%

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

      1. associate-*r/ 96.7%

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

      2. +-commutative 96.7%

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

      3. distribute-neg-in 96.7%

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

      4. sub-neg 96.7%

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

      5. associate-*l/ 97.3%

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

      6. clear-num 97.3%

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

      7. frac-2neg 97.3%

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

      8. frac-times 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. frac-2neg 99.9%

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

      11. sub-neg 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. +-commutative 99.9%

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

      14. remove-double-neg 99.9%

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

      15. add-sqr-sqrt 48.3%

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

      16. sqrt-unprod 29.2%

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

      17. sqr-neg 29.2%

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

      18. sqrt-unprod 9.0%

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

      19. add-sqr-sqrt 15.9%

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

      20. add-sqr-sqrt 7.5%

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

      21. sqrt-unprod 47.2%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in u around 0 80.4%

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

      1. *-commutative 80.4%

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

   9. Simplified 80.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{-40} \lor \neg \left(u \leq 2.2 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+79} \lor \neg \left(u \leq 7.2 \cdot 10^{+136}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(u - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.5e+79) (not (<= u 7.2e+136)))
   (* v (/ t1 (* u (- u t1))))
   (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.5e+79) || !(u <= 7.2e+136)) {
		tmp = v * (t1 / (u * (u - t1)));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.5d+79)) .or. (.not. (u <= 7.2d+136))) then
        tmp = v * (t1 / (u * (u - t1)))
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.5e+79) || !(u <= 7.2e+136)) {
		tmp = v * (t1 / (u * (u - t1)));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.5e+79) or not (u <= 7.2e+136):
		tmp = v * (t1 / (u * (u - t1)))
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.5e+79) || !(u <= 7.2e+136))
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(u - t1))));
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.5e+79) || ~((u <= 7.2e+136)))
		tmp = v * (t1 / (u * (u - t1)));
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.5e+79], N[Not[LessEqual[u, 7.2e+136]], $MachinePrecision]], N[(v * N[(t1 / N[(u * N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.4999999999999998e79 or 7.20000000000000011e136 < u

   1. Initial program 73.9%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 89.8%

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

      5. distribute-neg-frac2 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in t1 around 0 86.4%

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

      1. associate-/l/ 74.4%

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

      2. associate-*r/ 73.9%

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

      3. *-commutative 73.9%

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

      4. add-sqr-sqrt 47.8%

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

      5. sqrt-unprod 73.9%

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

      6. sqr-neg 73.9%

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

      7. sqrt-unprod 26.1%

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

      8. add-sqr-sqrt 66.0%

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

   7. Applied egg-rr 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-/l* 66.6%

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

      3. +-commutative 66.6%

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

      4. distribute-lft-in 56.9%

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

      5. add-sqr-sqrt 22.6%

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

      6. sqrt-unprod 41.4%

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

      7. sqr-neg 41.4%

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

      8. sqrt-unprod 33.0%

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

      9. add-sqr-sqrt 60.3%

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

      10. distribute-lft-neg-out 60.3%

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

      11. *-commutative 60.3%

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

      12. sub-neg 60.3%

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

      13. distribute-rgt-out-- 66.6%

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

   9. Applied egg-rr 66.6%

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

   if -3.4999999999999998e79 < u < 7.20000000000000011e136

   1. Initial program 70.7%

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

      1. associate-/l* 71.5%

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

      2. distribute-lft-neg-out 71.5%

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

      3. distribute-rgt-neg-in 71.5%

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

      4. associate-/r* 77.6%

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

      5. distribute-neg-frac2 77.6%

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

   3. Simplified 77.6%

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

      1. associate-*r/ 97.8%

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

      2. +-commutative 97.8%

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

      3. distribute-neg-in 97.8%

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

      4. sub-neg 97.8%

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

      5. associate-*l/ 97.5%

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

      6. clear-num 97.4%

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

      7. frac-2neg 97.4%

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

      8. frac-times 99.2%

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

      9. *-un-lft-identity 99.2%

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

      10. frac-2neg 99.2%

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

      11. sub-neg 99.2%

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

      12. distribute-neg-in 99.2%

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

      13. +-commutative 99.2%

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

      14. remove-double-neg 99.2%

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

      15. add-sqr-sqrt 48.1%

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

      16. sqrt-unprod 35.3%

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

      17. sqr-neg 35.3%

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

      18. sqrt-unprod 10.9%

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

      19. add-sqr-sqrt 20.3%

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

      20. add-sqr-sqrt 8.6%

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

      21. sqrt-unprod 51.6%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in u around 0 69.4%

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

      1. *-commutative 69.4%

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

   9. Simplified 69.4%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+79} \lor \neg \left(u \leq 7.2 \cdot 10^{+136}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(u - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+114} \lor \neg \left(u \leq 1.35 \cdot 10^{+163}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.6e+114) (not (<= u 1.35e+163)))
   (* v (/ (/ t1 u) t1))
   (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.6e+114) || !(u <= 1.35e+163)) {
		tmp = v * ((t1 / u) / t1);
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.6d+114)) .or. (.not. (u <= 1.35d+163))) then
        tmp = v * ((t1 / u) / t1)
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.6e+114) || !(u <= 1.35e+163)) {
		tmp = v * ((t1 / u) / t1);
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.6e+114) or not (u <= 1.35e+163):
		tmp = v * ((t1 / u) / t1)
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.6e+114) || !(u <= 1.35e+163))
		tmp = Float64(v * Float64(Float64(t1 / u) / t1));
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.6e+114) || ~((u <= 1.35e+163)))
		tmp = v * ((t1 / u) / t1);
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.6e+114], N[Not[LessEqual[u, 1.35e+163]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.6000000000000001e114 or 1.35e163 < u

   1. Initial program 73.4%

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

      1. associate-/l* 73.8%

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

      2. distribute-lft-neg-out 73.8%

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

      3. distribute-rgt-neg-in 73.8%

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

      4. associate-/r* 89.4%

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

      5. distribute-neg-frac2 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in t1 around 0 88.0%

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

   6. Taylor expanded in u around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. associate-/r* 41.4%

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

      3. distribute-neg-frac2 41.4%

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

   8. Simplified 41.4%

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

      1. clear-num 41.4%

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

      2. un-div-inv 41.4%

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

      3. div-inv 41.4%

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

      4. add-sqr-sqrt 22.2%

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

      5. sqrt-unprod 49.5%

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

      6. sqr-neg 49.5%

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

      7. sqrt-unprod 19.0%

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

      8. add-sqr-sqrt 41.4%

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

      9. clear-num 41.4%

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

   10. Applied egg-rr 41.4%

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

       1. associate-/r* 46.3%

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

       2. associate-/r/ 60.1%

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

   12. Applied egg-rr 60.1%

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

   if -3.6000000000000001e114 < u < 1.35e163

   1. Initial program 71.2%

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

      1. associate-/l* 72.0%

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

      2. distribute-lft-neg-out 72.0%

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

      3. distribute-rgt-neg-in 72.0%

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

      4. associate-/r* 78.7%

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

      5. distribute-neg-frac2 78.7%

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

   3. Simplified 78.7%

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

      1. associate-*r/ 98.0%

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

      2. +-commutative 98.0%

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

      3. distribute-neg-in 98.0%

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

      4. sub-neg 98.0%

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

      5. associate-*l/ 96.6%

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

      6. clear-num 96.6%

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

      7. frac-2neg 96.6%

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

      8. frac-times 98.2%

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

      9. *-un-lft-identity 98.2%

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

      10. frac-2neg 98.2%

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

      11. sub-neg 98.2%

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

      12. distribute-neg-in 98.2%

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

      13. +-commutative 98.2%

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

      14. remove-double-neg 98.2%

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

      15. add-sqr-sqrt 47.5%

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

      16. sqrt-unprod 35.7%

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

      17. sqr-neg 35.7%

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

      18. sqrt-unprod 11.8%

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

      19. add-sqr-sqrt 22.8%

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

      20. add-sqr-sqrt 11.3%

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

      21. sqrt-unprod 51.9%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in u around 0 67.9%

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

      1. *-commutative 67.9%

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

   9. Simplified 67.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+114} \lor \neg \left(u \leq 1.35 \cdot 10^{+163}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.4 \cdot 10^{+114} \lor \neg \left(u \leq 8 \cdot 10^{+146}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.4e+114) (not (<= u 8e+146)))
   (* v (/ (/ t1 u) t1))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.4e+114) || !(u <= 8e+146)) {
		tmp = v * ((t1 / u) / t1);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.4d+114)) .or. (.not. (u <= 8d+146))) then
        tmp = v * ((t1 / u) / t1)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.4e+114) || !(u <= 8e+146)) {
		tmp = v * ((t1 / u) / t1);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.4e+114) or not (u <= 8e+146):
		tmp = v * ((t1 / u) / t1)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.4e+114) || !(u <= 8e+146))
		tmp = Float64(v * Float64(Float64(t1 / u) / t1));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.4e+114) || ~((u <= 8e+146)))
		tmp = v * ((t1 / u) / t1);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.4e+114], N[Not[LessEqual[u, 8e+146]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.4000000000000001e114 or 7.99999999999999947e146 < u

   1. Initial program 74.1%

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

      1. associate-/l* 74.5%

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

      2. distribute-lft-neg-out 74.5%

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

      3. distribute-rgt-neg-in 74.5%

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

      4. associate-/r* 89.6%

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

      5. distribute-neg-frac2 89.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in t1 around 0 88.3%

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

   6. Taylor expanded in u around 0 42.0%

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

      1. mul-1-neg 42.0%

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

      2. associate-/r* 41.7%

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

      3. distribute-neg-frac2 41.7%

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

   8. Simplified 41.7%

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

      1. clear-num 41.7%

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

      2. un-div-inv 41.7%

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

      3. div-inv 41.7%

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

      4. add-sqr-sqrt 21.6%

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

      5. sqrt-unprod 50.8%

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

      6. sqr-neg 50.8%

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

      7. sqrt-unprod 19.9%

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

      8. add-sqr-sqrt 41.7%

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

      9. clear-num 41.7%

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

   10. Applied egg-rr 41.7%

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

       1. associate-/r* 46.5%

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

       2. associate-/r/ 59.9%

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

   12. Applied egg-rr 59.9%

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

   if -4.4000000000000001e114 < u < 7.99999999999999947e146

   1. Initial program 70.8%

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

      1. associate-/l* 71.6%

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

      2. distribute-lft-neg-out 71.6%

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

      3. distribute-rgt-neg-in 71.6%

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

      4. associate-/r* 78.5%

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

      5. distribute-neg-frac2 78.5%

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

   3. Simplified 78.5%

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

      1. distribute-frac-neg2 78.5%

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

      2. distribute-rgt-neg-out 78.5%

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

      3. associate-/r* 71.6%

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

      4. distribute-lft-neg-out 71.6%

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

      5. associate-/l* 70.8%

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

      6. times-frac 96.6%

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

      7. frac-2neg 96.6%

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

      8. associate-*r/ 97.8%

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

      9. add-sqr-sqrt 46.9%

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

      10. sqrt-unprod 35.5%

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

      11. sqr-neg 35.5%

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

      12. sqrt-unprod 11.9%

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

      13. add-sqr-sqrt 21.9%

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

      14. add-sqr-sqrt 10.9%

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

      15. sqrt-unprod 51.2%

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

      16. sqr-neg 51.2%

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

      17. sqrt-prod 50.1%

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

      18. add-sqr-sqrt 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in t1 around inf 67.7%

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

      1. mul-1-neg 67.7%

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

   9. Simplified 67.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.4 \cdot 10^{+114} \lor \neg \left(u \leq 8 \cdot 10^{+146}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+159} \lor \neg \left(u \leq 9 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.2e+159) (not (<= u 9e+111))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+159) || !(u <= 9e+111)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.2d+159)) .or. (.not. (u <= 9d+111))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+159) || !(u <= 9e+111)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.2e+159) or not (u <= 9e+111):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.2e+159) || !(u <= 9e+111))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.2e+159) || ~((u <= 9e+111)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.2e+159], N[Not[LessEqual[u, 9e+111]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.1999999999999999e159 or 9.00000000000000001e111 < u

   1. Initial program 73.4%

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

      1. associate-/l* 73.3%

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

      2. distribute-lft-neg-out 73.3%

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

      3. distribute-rgt-neg-in 73.3%

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

      4. associate-/r* 87.5%

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

      5. distribute-neg-frac2 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in t1 around 0 87.5%

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

      1. associate-/l/ 73.3%

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

      2. associate-*r/ 73.4%

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

      3. *-commutative 73.4%

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

      4. add-sqr-sqrt 35.6%

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

      5. sqrt-unprod 70.7%

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

      6. sqr-neg 70.7%

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

      7. sqrt-unprod 35.1%

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

      8. add-sqr-sqrt 70.7%

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

   7. Applied egg-rr 70.7%

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

   8. Taylor expanded in t1 around inf 45.6%

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

   if -2.1999999999999999e159 < u < 9.00000000000000001e111

   1. Initial program 71.1%

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

      1. associate-/l* 72.2%

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

      2. distribute-lft-neg-out 72.2%

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

      3. distribute-rgt-neg-in 72.2%

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

      4. associate-/r* 79.4%

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

      5. distribute-neg-frac2 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+159} \lor \neg \left(u \leq 9 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.28 \cdot 10^{+113}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.8e+160) (/ v u) (if (<= u 1.28e+113) (/ v (- t1)) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e+160) {
		tmp = v / u;
	} else if (u <= 1.28e+113) {
		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 <= (-4.8d+160)) then
        tmp = v / u
    else if (u <= 1.28d+113) 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 <= -4.8e+160) {
		tmp = v / u;
	} else if (u <= 1.28e+113) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.8e+160:
		tmp = v / u
	elif u <= 1.28e+113:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.8e+160)
		tmp = Float64(v / u);
	elseif (u <= 1.28e+113)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.8e+160)
		tmp = v / u;
	elseif (u <= 1.28e+113)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.8e+160], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.28e+113], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.8000000000000003e160

   1. Initial program 67.7%

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

      1. associate-/l* 68.2%

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

      2. distribute-lft-neg-out 68.2%

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

      3. distribute-rgt-neg-in 68.2%

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

      4. associate-/r* 85.8%

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

      5. distribute-neg-frac2 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in t1 around 0 85.8%

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

      1. associate-/l/ 68.2%

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

      2. associate-*r/ 67.7%

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

      3. *-commutative 67.7%

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

      4. add-sqr-sqrt 64.6%

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

      5. sqrt-unprod 67.7%

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

      6. sqr-neg 67.7%

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

      7. sqrt-unprod 3.1%

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

      8. add-sqr-sqrt 67.7%

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

   7. Applied egg-rr 67.7%

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

   8. Taylor expanded in t1 around inf 46.3%

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

   if -4.8000000000000003e160 < u < 1.27999999999999994e113

   1. Initial program 71.1%

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

      1. associate-/l* 72.2%

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

      2. distribute-lft-neg-out 72.2%

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

      3. distribute-rgt-neg-in 72.2%

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

      4. associate-/r* 79.4%

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

      5. distribute-neg-frac2 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

   if 1.27999999999999994e113 < u

   1. Initial program 77.9%

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

      1. associate-/l* 77.4%

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

      2. distribute-lft-neg-out 77.4%

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

      3. distribute-rgt-neg-in 77.4%

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

      4. associate-/r* 88.8%

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

      5. distribute-neg-frac2 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in t1 around 0 88.8%

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

   6. Taylor expanded in t1 around inf 45.6%

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

      1. associate-*r/ 45.6%

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

      2. mul-1-neg 45.6%

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

   8. Simplified 45.6%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.28 \cdot 10^{+113}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. times-frac 97.5%

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

   2. distribute-frac-neg 97.5%

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

   3. distribute-neg-frac2 97.5%

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

   4. +-commutative 97.5%

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

   5. distribute-neg-in 97.5%

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

   6. unsub-neg 97.5%

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

3. Simplified 97.5%

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

5. Final simplification 97.5%

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

Alternative 14: 62.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

   1. associate-/l* 72.5%

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

   2. distribute-lft-neg-out 72.5%

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

   3. distribute-rgt-neg-in 72.5%

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

   4. associate-/r* 81.7%

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

   5. distribute-neg-frac2 81.7%

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

3. Simplified 81.7%

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

   1. distribute-frac-neg2 81.7%

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

   2. distribute-rgt-neg-out 81.7%

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

   3. associate-/r* 72.5%

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

   4. distribute-lft-neg-out 72.5%

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

   5. associate-/l* 71.8%

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

   6. times-frac 97.5%

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

   7. frac-2neg 97.5%

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

   8. associate-*r/ 98.3%

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

   9. add-sqr-sqrt 46.5%

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

   10. sqrt-unprod 44.2%

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

   11. sqr-neg 44.2%

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

   12. sqrt-unprod 18.9%

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

   13. add-sqr-sqrt 35.5%

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

   14. add-sqr-sqrt 19.0%

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

   15. sqrt-unprod 56.5%

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

   16. sqr-neg 56.5%

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

   17. sqrt-prod 47.6%

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

   18. add-sqr-sqrt 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in t1 around inf 60.2%

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

   1. mul-1-neg 60.2%

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

9. Simplified 60.2%

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

10. Final simplification 60.2%

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

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 71.8%

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

   1. associate-/l* 72.5%

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

   2. distribute-lft-neg-out 72.5%

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

   3. distribute-rgt-neg-in 72.5%

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

   4. associate-/r* 81.7%

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

   5. distribute-neg-frac2 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in t1 around 0 52.4%

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

   1. associate-/l/ 48.4%

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

   2. associate-*r/ 47.0%

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

   3. *-commutative 47.0%

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

   4. add-sqr-sqrt 24.4%

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

   5. sqrt-unprod 40.8%

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

   6. sqr-neg 40.8%

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

   7. sqrt-unprod 13.7%

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

   8. add-sqr-sqrt 29.9%

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

7. Applied egg-rr 29.9%

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

8. Taylor expanded in t1 around inf 16.6%

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

9. Final simplification 16.6%

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

Reproduce

herbie shell --seed 2024060 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))