Rosa's DopplerBench

Percentage Accurate: 73.0% → 98.1%

Time: 15.5s

Alternatives: 17

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: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (/ (/ (* t1 (/ v (+ t1 u))) -1.0) (+ t1 u)))

C

double code(double u, double v, double t1) {
	return ((t1 * (v / (t1 + u))) / -1.0) / (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))) / (-1.0d0)) / (t1 + u)
end function

Java

public static double code(double u, double v, double t1) {
	return ((t1 * (v / (t1 + u))) / -1.0) / (t1 + u);
}

Python

def code(u, v, t1):
	return ((t1 * (v / (t1 + u))) / -1.0) / (t1 + u)

Julia

function code(u, v, t1)
	return Float64(Float64(Float64(t1 * Float64(v / Float64(t1 + u))) / -1.0) / Float64(t1 + u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = ((t1 * (v / (t1 + u))) / -1.0) / (t1 + u);
end

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. times-frac 97.2%

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

   2. distribute-frac-neg 97.2%

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

   3. distribute-neg-frac2 97.2%

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

   4. +-commutative 97.2%

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

   5. distribute-neg-in 97.2%

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

   6. unsub-neg 97.2%

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

3. Simplified 97.2%

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

   1. associate-*l/ 98.4%

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

   2. sub-neg 98.4%

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

   3. distribute-neg-in 98.4%

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

   4. +-commutative 98.4%

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

   5. neg-mul-1 98.4%

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

   6. associate-/r* 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

   \[\leadsto \frac{\frac{t1 \cdot \frac{v}{t1 + u}}{-1}}{t1 + u} \]
8. Add Preprocessing

Alternative 2: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{if}\;t1 \leq -3.3 \cdot 10^{+173}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{-176}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{-229}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+167}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))))
   (if (<= t1 -3.3e+173)
     (/ v (- (- t1) (* u 2.0)))
     (if (<= t1 -5e-176)
       t_1
       (if (<= t1 2.9e-229)
         (/ (* t1 (/ v (- u))) (+ t1 u))
         (if (<= t1 2.1e+167) t_1 (/ v (- t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	double tmp;
	if (t1 <= -3.3e+173) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= -5e-176) {
		tmp = t_1;
	} else if (t1 <= 2.9e-229) {
		tmp = (t1 * (v / -u)) / (t1 + u);
	} else if (t1 <= 2.1e+167) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = t1 * ((v / (t1 + u)) / (-u - t1))
    if (t1 <= (-3.3d+173)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (t1 <= (-5d-176)) then
        tmp = t_1
    else if (t1 <= 2.9d-229) then
        tmp = (t1 * (v / -u)) / (t1 + u)
    else if (t1 <= 2.1d+167) then
        tmp = t_1
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	double tmp;
	if (t1 <= -3.3e+173) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= -5e-176) {
		tmp = t_1;
	} else if (t1 <= 2.9e-229) {
		tmp = (t1 * (v / -u)) / (t1 + u);
	} else if (t1 <= 2.1e+167) {
		tmp = t_1;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * ((v / (t1 + u)) / (-u - t1))
	tmp = 0
	if t1 <= -3.3e+173:
		tmp = v / (-t1 - (u * 2.0))
	elif t1 <= -5e-176:
		tmp = t_1
	elif t1 <= 2.9e-229:
		tmp = (t1 * (v / -u)) / (t1 + u)
	elif t1 <= 2.1e+167:
		tmp = t_1
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)))
	tmp = 0.0
	if (t1 <= -3.3e+173)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (t1 <= -5e-176)
		tmp = t_1;
	elseif (t1 <= 2.9e-229)
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / Float64(t1 + u));
	elseif (t1 <= 2.1e+167)
		tmp = t_1;
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	tmp = 0.0;
	if (t1 <= -3.3e+173)
		tmp = v / (-t1 - (u * 2.0));
	elseif (t1 <= -5e-176)
		tmp = t_1;
	elseif (t1 <= 2.9e-229)
		tmp = (t1 * (v / -u)) / (t1 + u);
	elseif (t1 <= 2.1e+167)
		tmp = t_1;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.3e+173], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -5e-176], t$95$1, If[LessEqual[t1, 2.9e-229], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.1e+167], t$95$1, N[(v / (-t1)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -3.29999999999999996e173

   1. Initial program 41.6%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 95.4%

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

      4. *-un-lft-identity 95.4%

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

      5. +-commutative 95.4%

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

      6. distribute-neg-in 95.4%

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

      7. sub-neg 95.4%

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

      8. frac-2neg 95.4%

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

      9. sub-neg 95.4%

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

      10. distribute-neg-in 95.4%

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

      11. +-commutative 95.4%

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

      12. remove-double-neg 95.4%

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

      13. add-sqr-sqrt 94.9%

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

      14. sqrt-unprod 1.9%

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

      15. sqr-neg 1.9%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 42.2%

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

      18. sub-neg 42.2%

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

      19. distribute-neg-in 42.2%

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

      20. +-commutative 42.2%

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

      21. add-sqr-sqrt 42.2%

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

      22. sqrt-unprod 43.1%

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

   6. Applied egg-rr 95.4%

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

   7. Taylor expanded in t1 around inf 95.5%

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

      1. *-commutative 95.5%

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

   9. Simplified 95.5%

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

   if -3.29999999999999996e173 < t1 < -5e-176 or 2.9e-229 < t1 < 2.0999999999999999e167

   1. Initial program 82.9%

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

      1. associate-/l* 87.1%

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

      2. distribute-lft-neg-out 87.1%

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

      3. distribute-rgt-neg-in 87.1%

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

      4. associate-/r* 95.3%

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

      5. distribute-neg-frac2 95.3%

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

   3. Simplified 95.3%

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

   if -5e-176 < t1 < 2.9e-229

   1. Initial program 75.2%

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

      1. times-frac 86.5%

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

      2. distribute-frac-neg 86.5%

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

      3. distribute-neg-frac2 86.5%

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

      4. +-commutative 86.5%

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

      5. distribute-neg-in 86.5%

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

      6. unsub-neg 86.5%

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

   3. Simplified 86.5%

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

      1. frac-2neg 86.5%

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

      2. associate-*l/ 93.8%

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

      3. sub-neg 93.8%

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

      4. distribute-neg-in 93.8%

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

      5. +-commutative 93.8%

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

      6. remove-double-neg 93.8%

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

      7. associate-*l/ 86.5%

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

      8. frac-2neg 86.5%

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

      9. associate-*r/ 88.3%

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

      10. add-sqr-sqrt 56.1%

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

      11. sqrt-unprod 49.3%

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

      12. sqr-neg 49.3%

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

      13. sqrt-unprod 13.2%

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

      14. add-sqr-sqrt 48.7%

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

      15. add-sqr-sqrt 28.2%

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

      16. sqrt-unprod 57.3%

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

      17. sqr-neg 57.3%

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

      18. sqrt-prod 35.1%

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

   6. Applied egg-rr 88.3%

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

   7. Taylor expanded in t1 around 0 86.9%

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

      1. mul-1-neg 86.9%

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

      2. associate-/l* 91.2%

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

      3. distribute-rgt-neg-in 91.2%

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

      4. distribute-neg-frac2 91.2%

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

   9. Simplified 91.2%

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

   if 2.0999999999999999e167 < t1

   1. Initial program 38.1%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{+173}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq -5 \cdot 10^{-176}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 2.9 \cdot 10^{-229}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+167}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq -1.45 \cdot 10^{-158}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.15 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{\frac{t1}{\frac{u}{v}}}{-1}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{+166}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.55e+154)
   (/ v (- (- t1) (* u 2.0)))
   (if (<= t1 -1.45e-158)
     (* (- v) (/ t1 (* (+ t1 u) (+ t1 u))))
     (if (<= t1 2.15e-211)
       (/ (/ (/ t1 (/ u v)) -1.0) (+ t1 u))
       (if (<= t1 1.45e+166)
         (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))
         (/ v (- t1)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.55e+154) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= -1.45e-158) {
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	} else if (t1 <= 2.15e-211) {
		tmp = ((t1 / (u / v)) / -1.0) / (t1 + u);
	} else if (t1 <= 1.45e+166) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.55d+154)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (t1 <= (-1.45d-158)) then
        tmp = -v * (t1 / ((t1 + u) * (t1 + u)))
    else if (t1 <= 2.15d-211) then
        tmp = ((t1 / (u / v)) / (-1.0d0)) / (t1 + u)
    else if (t1 <= 1.45d+166) then
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.55e+154) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= -1.45e-158) {
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	} else if (t1 <= 2.15e-211) {
		tmp = ((t1 / (u / v)) / -1.0) / (t1 + u);
	} else if (t1 <= 1.45e+166) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.55e+154:
		tmp = v / (-t1 - (u * 2.0))
	elif t1 <= -1.45e-158:
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)))
	elif t1 <= 2.15e-211:
		tmp = ((t1 / (u / v)) / -1.0) / (t1 + u)
	elif t1 <= 1.45e+166:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.55e+154)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (t1 <= -1.45e-158)
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(Float64(t1 + u) * Float64(t1 + u))));
	elseif (t1 <= 2.15e-211)
		tmp = Float64(Float64(Float64(t1 / Float64(u / v)) / -1.0) / Float64(t1 + u));
	elseif (t1 <= 1.45e+166)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.55e+154)
		tmp = v / (-t1 - (u * 2.0));
	elseif (t1 <= -1.45e-158)
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	elseif (t1 <= 2.15e-211)
		tmp = ((t1 / (u / v)) / -1.0) / (t1 + u);
	elseif (t1 <= 1.45e+166)
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.55e+154], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.45e-158], N[((-v) * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.15e-211], N[(N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.45e+166], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t1 < -2.55e154

   1. Initial program 41.3%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. frac-2neg 99.8%

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

      3. frac-times 96.3%

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

      4. *-un-lft-identity 96.3%

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

      5. +-commutative 96.3%

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

      6. distribute-neg-in 96.3%

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

      7. sub-neg 96.3%

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

      8. frac-2neg 96.3%

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

      9. sub-neg 96.3%

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

      10. distribute-neg-in 96.3%

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

      11. +-commutative 96.3%

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

      12. remove-double-neg 96.3%

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

      13. add-sqr-sqrt 95.8%

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

      14. sqrt-unprod 2.0%

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

      15. sqr-neg 2.0%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 42.0%

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

      18. sub-neg 42.0%

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

      19. distribute-neg-in 42.0%

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

      20. +-commutative 42.0%

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

      21. add-sqr-sqrt 38.2%

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

      22. sqrt-unprod 42.9%

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in t1 around inf 89.2%

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

      1. *-commutative 89.2%

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

   9. Simplified 89.2%

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

   if -2.55e154 < t1 < -1.4499999999999999e-158

   1. Initial program 89.4%

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

      1. associate-*l/ 98.0%

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

      2. *-commutative 98.0%

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

   3. Simplified 98.0%

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

   if -1.4499999999999999e-158 < t1 < 2.15e-211

   1. Initial program 79.1%

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

      1. times-frac 88.9%

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

      2. distribute-frac-neg 88.9%

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

      3. distribute-neg-frac2 88.9%

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

      4. +-commutative 88.9%

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

      5. distribute-neg-in 88.9%

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

      6. unsub-neg 88.9%

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

   3. Simplified 88.9%

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

      1. associate-*l/ 95.1%

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

      2. sub-neg 95.1%

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

      3. distribute-neg-in 95.1%

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

      4. +-commutative 95.1%

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

      5. neg-mul-1 95.1%

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

      6. associate-/r* 95.1%

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

   6. Applied egg-rr 95.1%

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

      1. clear-num 95.0%

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

      2. un-div-inv 95.1%

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

   8. Applied egg-rr 95.1%

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

   9. Taylor expanded in t1 around 0 91.6%

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

   if 2.15e-211 < t1 < 1.4500000000000001e166

   1. Initial program 77.4%

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

      1. associate-/l* 81.8%

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

      2. distribute-lft-neg-out 81.8%

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

      3. distribute-rgt-neg-in 81.8%

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

      4. associate-/r* 96.0%

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

      5. distribute-neg-frac2 96.0%

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

   3. Simplified 96.0%

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

   if 1.4500000000000001e166 < t1

   1. Initial program 38.1%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq -1.45 \cdot 10^{-158}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.15 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{\frac{t1}{\frac{u}{v}}}{-1}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{+166}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -440000000 \lor \neg \left(t1 \leq 6.5 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t1}{\frac{u}{v}}}{-1}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -440000000.0) (not (<= t1 6.5e-30)))
   (/ v (- (- t1) (* u 2.0)))
   (/ (/ (/ t1 (/ u v)) -1.0) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -440000000.0) || !(t1 <= 6.5e-30)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = ((t1 / (u / v)) / -1.0) / (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 <= (-440000000.0d0)) .or. (.not. (t1 <= 6.5d-30))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = ((t1 / (u / v)) / (-1.0d0)) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -440000000.0) || !(t1 <= 6.5e-30)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = ((t1 / (u / v)) / -1.0) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -440000000.0) or not (t1 <= 6.5e-30):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = ((t1 / (u / v)) / -1.0) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -440000000.0) || !(t1 <= 6.5e-30))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(Float64(t1 / Float64(u / v)) / -1.0) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -440000000.0) || ~((t1 <= 6.5e-30)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = ((t1 / (u / v)) / -1.0) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -440000000.0], N[Not[LessEqual[t1, 6.5e-30]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.4e8 or 6.5000000000000005e-30 < t1

   1. Initial program 61.5%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 95.8%

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

      4. *-un-lft-identity 95.8%

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

      5. +-commutative 95.8%

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

      6. distribute-neg-in 95.8%

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

      7. sub-neg 95.8%

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

      8. frac-2neg 95.8%

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

      9. sub-neg 95.8%

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

      10. distribute-neg-in 95.8%

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

      11. +-commutative 95.8%

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

      12. remove-double-neg 95.8%

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

      13. add-sqr-sqrt 51.5%

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

      14. sqrt-unprod 37.0%

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

      15. sqr-neg 37.0%

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

      16. sqrt-unprod 16.0%

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

      17. add-sqr-sqrt 31.7%

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

      18. sub-neg 31.7%

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

      19. distribute-neg-in 31.7%

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

      20. +-commutative 31.7%

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

      21. add-sqr-sqrt 15.8%

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

      22. sqrt-unprod 43.4%

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in t1 around inf 85.4%

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

      1. *-commutative 85.4%

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

   9. Simplified 85.4%

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

   if -4.4e8 < t1 < 6.5000000000000005e-30

   1. Initial program 84.5%

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

      1. times-frac 95.2%

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

      2. distribute-frac-neg 95.2%

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

      3. distribute-neg-frac2 95.2%

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

      4. +-commutative 95.2%

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

      5. distribute-neg-in 95.2%

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

      6. unsub-neg 95.2%

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

   3. Simplified 95.2%

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

      1. associate-*l/ 97.2%

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

      2. sub-neg 97.2%

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

      3. distribute-neg-in 97.2%

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

      4. +-commutative 97.2%

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

      5. neg-mul-1 97.2%

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

      6. associate-/r* 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. clear-num 97.1%

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

      2. un-div-inv 97.2%

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

   8. Applied egg-rr 97.2%

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

   9. Taylor expanded in t1 around 0 83.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -440000000 \lor \neg \left(t1 \leq 6.5 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t1}{\frac{u}{v}}}{-1}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -88000000000 \lor \neg \left(t1 \leq 1.15 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -88000000000.0) (not (<= t1 1.15e-30)))
   (/ v (- (- t1) (* u 2.0)))
   (* t1 (/ (/ v (- u)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -88000000000.0) || !(t1 <= 1.15e-30)) {
		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 <= (-88000000000.0d0)) .or. (.not. (t1 <= 1.15d-30))) 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 <= -88000000000.0) || !(t1 <= 1.15e-30)) {
		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 <= -88000000000.0) or not (t1 <= 1.15e-30):
		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 <= -88000000000.0) || !(t1 <= 1.15e-30))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -88000000000.0) || ~((t1 <= 1.15e-30)))
		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, -88000000000.0], N[Not[LessEqual[t1, 1.15e-30]], $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 < -8.8e10 or 1.14999999999999992e-30 < t1

   1. Initial program 61.5%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 95.8%

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

      4. *-un-lft-identity 95.8%

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

      5. +-commutative 95.8%

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

      6. distribute-neg-in 95.8%

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

      7. sub-neg 95.8%

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

      8. frac-2neg 95.8%

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

      9. sub-neg 95.8%

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

      10. distribute-neg-in 95.8%

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

      11. +-commutative 95.8%

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

      12. remove-double-neg 95.8%

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

      13. add-sqr-sqrt 51.5%

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

      14. sqrt-unprod 37.0%

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

      15. sqr-neg 37.0%

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

      16. sqrt-unprod 16.0%

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

      17. add-sqr-sqrt 31.7%

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

      18. sub-neg 31.7%

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

      19. distribute-neg-in 31.7%

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

      20. +-commutative 31.7%

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

      21. add-sqr-sqrt 15.8%

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

      22. sqrt-unprod 43.4%

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in t1 around inf 85.4%

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

      1. *-commutative 85.4%

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

   9. Simplified 85.4%

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

   if -8.8e10 < t1 < 1.14999999999999992e-30

   1. Initial program 84.5%

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

      1. associate-/l* 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* 90.7%

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

      5. distribute-neg-frac2 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t1 around 0 79.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -88000000000 \lor \neg \left(t1 \leq 1.15 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1600000000 \lor \neg \left(t1 \leq 1.75 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1600000000.0) (not (<= t1 1.75e-29)))
   (/ v (- (- t1) (* u 2.0)))
   (* (/ v (+ t1 u)) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1600000000.0) || !(t1 <= 1.75e-29)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / (t1 + 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 <= (-1600000000.0d0)) .or. (.not. (t1 <= 1.75d-29))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (v / (t1 + u)) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1600000000.0) || !(t1 <= 1.75e-29)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / (t1 + u)) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1600000000.0) or not (t1 <= 1.75e-29):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (v / (t1 + u)) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1600000000.0) || !(t1 <= 1.75e-29))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1600000000.0) || ~((t1 <= 1.75e-29)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (v / (t1 + u)) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1600000000.0], N[Not[LessEqual[t1, 1.75e-29]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6e9 or 1.7499999999999999e-29 < t1

   1. Initial program 61.5%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 95.8%

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

      4. *-un-lft-identity 95.8%

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

      5. +-commutative 95.8%

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

      6. distribute-neg-in 95.8%

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

      7. sub-neg 95.8%

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

      8. frac-2neg 95.8%

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

      9. sub-neg 95.8%

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

      10. distribute-neg-in 95.8%

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

      11. +-commutative 95.8%

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

      12. remove-double-neg 95.8%

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

      13. add-sqr-sqrt 51.5%

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

      14. sqrt-unprod 37.0%

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

      15. sqr-neg 37.0%

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

      16. sqrt-unprod 16.0%

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

      17. add-sqr-sqrt 31.7%

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

      18. sub-neg 31.7%

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

      19. distribute-neg-in 31.7%

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

      20. +-commutative 31.7%

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

      21. add-sqr-sqrt 15.8%

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

      22. sqrt-unprod 43.4%

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in t1 around inf 85.4%

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

      1. *-commutative 85.4%

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

   9. Simplified 85.4%

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

   if -1.6e9 < t1 < 1.7499999999999999e-29

   1. Initial program 84.5%

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

      1. times-frac 95.2%

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

      2. distribute-frac-neg 95.2%

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

      3. distribute-neg-frac2 95.2%

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

      4. +-commutative 95.2%

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

      5. distribute-neg-in 95.2%

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

      6. unsub-neg 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in t1 around 0 81.4%

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

      1. mul-1-neg 81.4%

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

      2. distribute-neg-frac2 81.4%

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

   7. Simplified 81.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1600000000 \lor \neg \left(t1 \leq 1.75 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7800000000 \lor \neg \left(t1 \leq 3.2 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7800000000.0) (not (<= t1 3.2e-28)))
   (/ v (- (- t1) (* u 2.0)))
   (/ (* t1 (/ v (- u))) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7800000000.0) || !(t1 <= 3.2e-28)) {
		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 <= (-7800000000.0d0)) .or. (.not. (t1 <= 3.2d-28))) 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 <= -7800000000.0) || !(t1 <= 3.2e-28)) {
		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 <= -7800000000.0) or not (t1 <= 3.2e-28):
		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 <= -7800000000.0) || !(t1 <= 3.2e-28))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -7800000000.0) || ~((t1 <= 3.2e-28)))
		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, -7800000000.0], N[Not[LessEqual[t1, 3.2e-28]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.8e9 or 3.19999999999999982e-28 < t1

   1. Initial program 61.5%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 95.8%

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

      4. *-un-lft-identity 95.8%

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

      5. +-commutative 95.8%

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

      6. distribute-neg-in 95.8%

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

      7. sub-neg 95.8%

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

      8. frac-2neg 95.8%

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

      9. sub-neg 95.8%

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

      10. distribute-neg-in 95.8%

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

      11. +-commutative 95.8%

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

      12. remove-double-neg 95.8%

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

      13. add-sqr-sqrt 51.5%

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

      14. sqrt-unprod 37.0%

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

      15. sqr-neg 37.0%

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

      16. sqrt-unprod 16.0%

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

      17. add-sqr-sqrt 31.7%

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

      18. sub-neg 31.7%

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

      19. distribute-neg-in 31.7%

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

      20. +-commutative 31.7%

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

      21. add-sqr-sqrt 15.8%

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

      22. sqrt-unprod 43.4%

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in t1 around inf 85.4%

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

      1. *-commutative 85.4%

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

   9. Simplified 85.4%

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

   if -7.8e9 < t1 < 3.19999999999999982e-28

   1. Initial program 84.5%

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

      1. times-frac 95.2%

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

      2. distribute-frac-neg 95.2%

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

      3. distribute-neg-frac2 95.2%

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

      4. +-commutative 95.2%

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

      5. distribute-neg-in 95.2%

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

      6. unsub-neg 95.2%

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

   3. Simplified 95.2%

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

      1. frac-2neg 95.2%

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

      2. associate-*l/ 97.2%

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

      3. sub-neg 97.2%

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

      4. distribute-neg-in 97.2%

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

      5. +-commutative 97.2%

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

      6. remove-double-neg 97.2%

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

      7. associate-*l/ 95.2%

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

      8. frac-2neg 95.2%

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

      9. associate-*r/ 95.2%

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

      10. add-sqr-sqrt 54.8%

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

      11. sqrt-unprod 61.9%

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

      12. sqr-neg 61.9%

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

      13. sqrt-unprod 15.9%

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

      14. add-sqr-sqrt 43.3%

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

      15. add-sqr-sqrt 22.3%

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

      16. sqrt-unprod 61.7%

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

      17. sqr-neg 61.7%

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

      18. sqrt-prod 42.6%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in t1 around 0 81.1%

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

      1. mul-1-neg 81.1%

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

      2. associate-/l* 83.3%

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

      3. distribute-rgt-neg-in 83.3%

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

      4. distribute-neg-frac2 83.3%

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

   9. Simplified 83.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7800000000 \lor \neg \left(t1 \leq 3.2 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.04 \cdot 10^{+68} \lor \neg \left(u \leq 2.3 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.04e+68) (not (<= u 2.3e+165)))
   (* (/ v u) (/ t1 (+ t1 u)))
   (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.04e+68) || !(u <= 2.3e+165)) {
		tmp = (v / u) * (t1 / (t1 + u));
	} 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 <= (-1.04d+68)) .or. (.not. (u <= 2.3d+165))) then
        tmp = (v / u) * (t1 / (t1 + u))
    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 <= -1.04e+68) || !(u <= 2.3e+165)) {
		tmp = (v / u) * (t1 / (t1 + u));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.04e+68) or not (u <= 2.3e+165):
		tmp = (v / u) * (t1 / (t1 + u))
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.04e+68) || !(u <= 2.3e+165))
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(t1 + u)));
	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 <= -1.04e+68) || ~((u <= 2.3e+165)))
		tmp = (v / u) * (t1 / (t1 + u));
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.04e+68], N[Not[LessEqual[u, 2.3e+165]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.04e68 or 2.30000000000000016e165 < u

   1. Initial program 83.2%

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

      1. times-frac 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. distribute-neg-frac2 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-neg-in 97.3%

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

      6. unsub-neg 97.3%

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

   3. Simplified 97.3%

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

      1. frac-2neg 97.3%

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

      2. associate-*l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. associate-*l/ 97.3%

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

      8. frac-2neg 97.3%

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

      9. associate-*r/ 97.3%

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

      10. add-sqr-sqrt 58.3%

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

      11. sqrt-unprod 71.1%

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

      12. sqr-neg 71.1%

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

      13. sqrt-unprod 25.9%

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

      14. add-sqr-sqrt 71.3%

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

      15. add-sqr-sqrt 43.4%

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

      16. sqrt-unprod 71.6%

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

      17. sqr-neg 71.6%

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

      18. sqrt-prod 33.4%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. associate-/l* 94.4%

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

      3. distribute-rgt-neg-in 94.4%

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

      4. distribute-neg-frac2 94.4%

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

   9. Simplified 94.4%

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

       1. associate-/l* 92.2%

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

       2. associate-/l/ 82.6%

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

       3. add-sqr-sqrt 53.3%

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

       4. sqrt-unprod 82.6%

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

       5. sqr-neg 82.6%

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

       6. sqrt-unprod 29.2%

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

       7. add-sqr-sqrt 71.5%

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

   11. Applied egg-rr 71.5%

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

       1. associate-*r/ 71.1%

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

       2. times-frac 71.5%

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

       3. *-commutative 71.5%

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

   13. Simplified 71.5%

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

   if -1.04e68 < u < 2.30000000000000016e165

   1. Initial program 70.0%

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

      1. times-frac 97.2%

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

      2. distribute-frac-neg 97.2%

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

      3. distribute-neg-frac2 97.2%

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

      4. +-commutative 97.2%

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

      5. distribute-neg-in 97.2%

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

      6. unsub-neg 97.2%

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

   3. Simplified 97.2%

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

      1. clear-num 97.1%

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

      2. frac-2neg 97.1%

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

      3. frac-times 97.1%

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

      4. *-un-lft-identity 97.1%

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

      5. +-commutative 97.1%

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

      6. distribute-neg-in 97.1%

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

      7. sub-neg 97.1%

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

      8. frac-2neg 97.1%

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

      9. sub-neg 97.1%

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

      10. distribute-neg-in 97.1%

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

      11. +-commutative 97.1%

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

      12. remove-double-neg 97.1%

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

      13. add-sqr-sqrt 51.3%

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

      14. sqrt-unprod 40.6%

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

      15. sqr-neg 40.6%

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

      16. sqrt-unprod 10.8%

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

      17. add-sqr-sqrt 21.1%

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

      18. sub-neg 21.1%

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

      19. distribute-neg-in 21.1%

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

      20. +-commutative 21.1%

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

      21. add-sqr-sqrt 7.1%

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

      22. sqrt-unprod 44.0%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in t1 around inf 64.3%

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

      1. *-commutative 64.3%

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

   9. Simplified 64.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.04 \cdot 10^{+68} \lor \neg \left(u \leq 2.3 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+100}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.1e+100)
   (* t1 (/ v (* u (+ t1 u))))
   (if (<= u 4e+166) (/ v (- (- t1) (* u 2.0))) (* (/ v u) (/ t1 (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.1e+100) {
		tmp = t1 * (v / (u * (t1 + u)));
	} else if (u <= 4e+166) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / u) * (t1 / (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 <= (-3.1d+100)) then
        tmp = t1 * (v / (u * (t1 + u)))
    else if (u <= 4d+166) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (v / u) * (t1 / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.1e+100) {
		tmp = t1 * (v / (u * (t1 + u)));
	} else if (u <= 4e+166) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v / u) * (t1 / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.1e+100:
		tmp = t1 * (v / (u * (t1 + u)))
	elif u <= 4e+166:
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (v / u) * (t1 / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.1e+100)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(t1 + u))));
	elseif (u <= 4e+166)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.1e+100)
		tmp = t1 * (v / (u * (t1 + u)));
	elseif (u <= 4e+166)
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (v / u) * (t1 / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.1e+100], N[(t1 * N[(v / N[(u * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4e+166], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.10000000000000007e100

   1. Initial program 85.7%

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

      1. times-frac 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. distribute-neg-frac2 99.2%

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

      4. +-commutative 99.2%

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

      5. distribute-neg-in 99.2%

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

      6. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

      1. frac-2neg 99.2%

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

      2. associate-*l/ 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. +-commutative 99.8%

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

      6. remove-double-neg 99.8%

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

      7. associate-*l/ 99.2%

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

      8. frac-2neg 99.2%

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

      9. associate-*r/ 99.1%

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

      10. add-sqr-sqrt 68.0%

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

      11. sqrt-unprod 75.7%

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

      12. sqr-neg 75.7%

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

      13. sqrt-unprod 21.3%

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

      14. add-sqr-sqrt 72.0%

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

      15. add-sqr-sqrt 72.0%

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

      16. sqrt-unprod 72.2%

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

      17. sqr-neg 72.2%

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

      18. sqrt-prod 0.0%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in t1 around 0 89.8%

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

      1. mul-1-neg 89.8%

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

      2. associate-/l* 97.9%

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

      3. distribute-rgt-neg-in 97.9%

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

      4. distribute-neg-frac2 97.9%

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

   9. Simplified 97.9%

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

       1. associate-/l* 94.0%

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

       2. *-commutative 94.0%

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

       3. associate-/l/ 86.2%

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

       4. add-sqr-sqrt 86.0%

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

       5. sqrt-unprod 86.2%

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

       6. sqr-neg 86.2%

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

       7. sqrt-unprod 0.0%

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

       8. add-sqr-sqrt 72.2%

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

   11. Applied egg-rr 72.2%

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

   if -3.10000000000000007e100 < u < 3.99999999999999976e166

   1. Initial program 70.6%

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

      1. times-frac 96.7%

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

      2. distribute-frac-neg 96.7%

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

      3. distribute-neg-frac2 96.7%

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

      4. +-commutative 96.7%

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

      5. distribute-neg-in 96.7%

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

      6. unsub-neg 96.7%

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

   3. Simplified 96.7%

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

      1. clear-num 96.7%

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

      2. frac-2neg 96.7%

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

      3. frac-times 96.7%

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

      4. *-un-lft-identity 96.7%

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

      5. +-commutative 96.7%

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

      6. distribute-neg-in 96.7%

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

      7. sub-neg 96.7%

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

      8. frac-2neg 96.7%

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

      9. sub-neg 96.7%

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

      10. distribute-neg-in 96.7%

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

      11. +-commutative 96.7%

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

      12. remove-double-neg 96.7%

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

      13. add-sqr-sqrt 49.8%

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

      14. sqrt-unprod 40.8%

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

      15. sqr-neg 40.8%

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

      16. sqrt-unprod 11.6%

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

      17. add-sqr-sqrt 21.5%

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

      18. sub-neg 21.5%

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

      19. distribute-neg-in 21.5%

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

      20. +-commutative 21.5%

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

      21. add-sqr-sqrt 8.1%

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

      22. sqrt-unprod 43.5%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in t1 around inf 63.0%

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

      1. *-commutative 63.0%

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

   9. Simplified 63.0%

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

   if 3.99999999999999976e166 < u

   1. Initial program 78.9%

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

      1. times-frac 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-neg-frac2 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. unsub-neg 96.9%

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

   3. Simplified 96.9%

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

      1. frac-2neg 96.9%

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

      2. associate-*l/ 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. associate-*l/ 96.9%

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

      8. frac-2neg 96.9%

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

      9. associate-*r/ 97.0%

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

      10. add-sqr-sqrt 53.3%

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

      11. sqrt-unprod 69.9%

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

      12. sqr-neg 69.9%

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

      13. sqrt-unprod 31.8%

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

      14. add-sqr-sqrt 79.1%

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

      15. add-sqr-sqrt 3.3%

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

      16. sqrt-unprod 79.4%

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

      17. sqr-neg 79.4%

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

      18. sqrt-prod 87.7%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in t1 around 0 84.8%

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

      1. mul-1-neg 84.8%

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

      2. associate-/l* 93.9%

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

      3. distribute-rgt-neg-in 93.9%

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

      4. distribute-neg-frac2 93.9%

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

   9. Simplified 93.9%

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

       1. associate-/l* 93.8%

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

       2. associate-/l/ 79.4%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 79.4%

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

       5. sqr-neg 79.4%

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

       6. sqrt-unprod 79.4%

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

       7. add-sqr-sqrt 79.4%

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

   11. Applied egg-rr 79.4%

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

       1. associate-*r/ 78.9%

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

       2. times-frac 79.6%

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

       3. *-commutative 79.6%

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

   13. Simplified 79.6%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+100}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+111} \lor \neg \left(u \leq 2.9 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.4e+111) (not (<= u 2.9e+165)))
   (/ (* v (/ t1 u)) t1)
   (/ v (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+111) || !(u <= 2.9e+165)) {
		tmp = (v * (t1 / u)) / t1;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.4d+111)) .or. (.not. (u <= 2.9d+165))) then
        tmp = (v * (t1 / u)) / t1
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+111) || !(u <= 2.9e+165)) {
		tmp = (v * (t1 / u)) / t1;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.4e+111) or not (u <= 2.9e+165):
		tmp = (v * (t1 / u)) / t1
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.4e+111) || !(u <= 2.9e+165))
		tmp = Float64(Float64(v * Float64(t1 / u)) / t1);
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.4e+111) || ~((u <= 2.9e+165)))
		tmp = (v * (t1 / u)) / t1;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.4e+111], N[Not[LessEqual[u, 2.9e+165]], $MachinePrecision]], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.4e111 or 2.90000000000000006e165 < u

   1. Initial program 82.6%

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

      1. associate-/l* 83.1%

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

      2. distribute-lft-neg-out 83.1%

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

      3. distribute-rgt-neg-in 83.1%

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

      4. associate-/r* 95.0%

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

      5. distribute-neg-frac2 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in t1 around inf 47.0%

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

   6. Taylor expanded in t1 around 0 46.1%

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

      1. associate-*r/ 46.1%

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

      2. mul-1-neg 46.1%

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

   8. Simplified 46.1%

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

      1. associate-*r/ 45.7%

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

      2. *-commutative 45.7%

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

      3. distribute-rgt-neg-in 45.7%

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

      4. associate-/r* 52.2%

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

      5. remove-double-neg 52.2%

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

      6. frac-2neg 52.2%

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

      7. *-commutative 52.2%

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

      8. associate-/l* 68.6%

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

      9. add-sqr-sqrt 40.8%

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

      10. sqrt-unprod 77.0%

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

      11. sqr-neg 77.0%

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

      12. sqrt-unprod 27.8%

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

      13. add-sqr-sqrt 68.5%

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

   10. Applied egg-rr 68.5%

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

   if -1.4e111 < u < 2.90000000000000006e165

   1. Initial program 70.9%

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

      1. associate-/l* 74.0%

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

      2. distribute-lft-neg-out 74.0%

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

      3. distribute-rgt-neg-in 74.0%

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

      4. associate-/r* 83.0%

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

      5. distribute-neg-frac2 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in t1 around inf 50.3%

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

      1. associate-/l/ 43.6%

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

      2. associate-*r/ 43.5%

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

      3. *-commutative 43.5%

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

      4. +-commutative 43.5%

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

      5. distribute-neg-in 43.5%

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

      6. sub-neg 43.5%

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

      7. sub-neg 43.5%

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

      8. distribute-neg-in 43.5%

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

      9. +-commutative 43.5%

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

      10. add-sqr-sqrt 22.1%

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

      11. sqrt-unprod 34.4%

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

      12. sqr-neg 34.4%

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

      13. sqrt-unprod 12.2%

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

      14. add-sqr-sqrt 18.8%

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

   7. Applied egg-rr 18.8%

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

      1. frac-2neg 18.8%

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

      2. div-inv 18.8%

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

      3. distribute-rgt-neg-in 18.8%

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

      4. add-sqr-sqrt 10.4%

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

      5. sqrt-unprod 29.8%

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

      6. sqr-neg 29.8%

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

      7. sqrt-unprod 20.4%

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

      8. add-sqr-sqrt 43.2%

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

      9. *-commutative 43.2%

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

      10. distribute-rgt-neg-in 43.2%

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

      11. +-commutative 43.2%

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

      12. distribute-neg-in 43.2%

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

      13. add-sqr-sqrt 21.1%

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

      14. sqrt-unprod 44.0%

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

      15. sqr-neg 44.0%

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

      16. sqrt-unprod 22.3%

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

      17. add-sqr-sqrt 43.6%

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

   9. Applied egg-rr 43.6%

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

       1. associate-*r/ 43.9%

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

       2. *-rgt-identity 43.9%

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

       3. associate-/r* 51.1%

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

       4. associate-/l* 62.1%

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

       5. *-inverses 62.1%

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

       6. *-rgt-identity 62.1%

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

       7. sub-neg 62.1%

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

   11. Simplified 62.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+111} \lor \neg \left(u \leq 2.9 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.08 \cdot 10^{+111}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.08e+111)
   (/ v (- u))
   (if (<= u 1.35e+91) (/ v (- t1)) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.08e+111) {
		tmp = v / -u;
	} else if (u <= 1.35e+91) {
		tmp = v / -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 <= (-1.08d+111)) then
        tmp = v / -u
    else if (u <= 1.35d+91) then
        tmp = v / -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 <= -1.08e+111) {
		tmp = v / -u;
	} else if (u <= 1.35e+91) {
		tmp = v / -t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.08e+111:
		tmp = v / -u
	elif u <= 1.35e+91:
		tmp = v / -t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.08e+111)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 1.35e+91)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.08e+111)
		tmp = v / -u;
	elseif (u <= 1.35e+91)
		tmp = v / -t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.08e+111], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 1.35e+91], N[(v / (-t1)), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.08000000000000004e111

   1. Initial program 85.5%

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

      1. associate-/l* 85.9%

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

      2. distribute-lft-neg-out 85.9%

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

      3. distribute-rgt-neg-in 85.9%

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

      4. associate-/r* 93.9%

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

      5. distribute-neg-frac2 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in t1 around inf 38.8%

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

   6. Taylor expanded in t1 around 0 35.5%

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

      1. mul-1-neg 35.5%

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

      2. distribute-neg-frac2 35.5%

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

   8. Simplified 35.5%

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

   if -1.08000000000000004e111 < u < 1.35e91

   1. Initial program 70.7%

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

      1. times-frac 96.4%

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

      2. distribute-frac-neg 96.4%

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

      3. distribute-neg-frac2 96.4%

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

      4. +-commutative 96.4%

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

      5. distribute-neg-in 96.4%

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

      6. unsub-neg 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t1 around inf 63.7%

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

      1. associate-*r/ 63.7%

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

      2. neg-mul-1 63.7%

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

   7. Simplified 63.7%

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

   if 1.35e91 < u

   1. Initial program 76.2%

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

      1. associate-/l* 78.7%

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

      2. distribute-lft-neg-out 78.7%

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

      3. distribute-rgt-neg-in 78.7%

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

      4. associate-/r* 95.8%

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

      5. distribute-neg-frac2 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in t1 around inf 53.3%

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

      1. clear-num 51.6%

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

      2. un-div-inv 51.7%

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

      3. div-inv 51.7%

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

      4. add-sqr-sqrt 4.4%

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

      5. sqrt-unprod 58.6%

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

      6. sqr-neg 58.6%

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

      7. sqrt-unprod 46.3%

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

      8. add-sqr-sqrt 48.6%

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

      9. clear-num 48.6%

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

   7. Applied egg-rr 48.6%

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

      1. associate-/l/ 44.9%

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

      2. associate-/r/ 45.0%

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

      3. *-inverses 45.0%

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

      4. *-lft-identity 45.0%

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

   9. Simplified 45.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.08 \cdot 10^{+111}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+110} \lor \neg \left(u \leq 2.5 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9e+110) (not (<= u 2.5e+91))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9e+110) || !(u <= 2.5e+91)) {
		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 <= (-9d+110)) .or. (.not. (u <= 2.5d+91))) 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 <= -9e+110) || !(u <= 2.5e+91)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9e+110) or not (u <= 2.5e+91):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9e+110) || !(u <= 2.5e+91))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9e+110) || ~((u <= 2.5e+91)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9e+110], N[Not[LessEqual[u, 2.5e+91]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -9.0000000000000005e110 or 2.5000000000000001e91 < u

   1. Initial program 80.8%

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

      1. associate-/l* 82.2%

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

      2. distribute-lft-neg-out 82.2%

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

      3. distribute-rgt-neg-in 82.2%

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

      4. associate-/r* 94.8%

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

      5. distribute-neg-frac2 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in t1 around inf 46.2%

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

   6. Taylor expanded in t1 around 0 39.7%

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

      1. mul-1-neg 39.7%

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

      2. distribute-neg-frac2 39.7%

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

   8. Simplified 39.7%

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

   if -9.0000000000000005e110 < u < 2.5000000000000001e91

   1. Initial program 70.7%

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

      1. times-frac 96.4%

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

      2. distribute-frac-neg 96.4%

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

      3. distribute-neg-frac2 96.4%

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

      4. +-commutative 96.4%

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

      5. distribute-neg-in 96.4%

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

      6. unsub-neg 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t1 around inf 63.7%

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

      1. associate-*r/ 63.7%

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

      2. neg-mul-1 63.7%

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

   7. Simplified 63.7%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+110} \lor \neg \left(u \leq 2.5 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 23.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+173} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3e+173) (not (<= t1 7.5e+35))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e+173) || !(t1 <= 7.5e+35)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3d+173)) .or. (.not. (t1 <= 7.5d+35))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3e+173) || !(t1 <= 7.5e+35)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3e+173) or not (t1 <= 7.5e+35):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3e+173) || !(t1 <= 7.5e+35))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3e+173) || ~((t1 <= 7.5e+35)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3e+173], N[Not[LessEqual[t1, 7.5e+35]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.9999999999999998e173 or 7.4999999999999999e35 < t1

   1. Initial program 44.8%

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

      1. associate-/l* 47.9%

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

      2. distribute-lft-neg-out 47.9%

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

      3. distribute-rgt-neg-in 47.9%

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

      4. associate-/r* 71.0%

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

      5. distribute-neg-frac2 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in t1 around inf 63.8%

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

      1. associate-*r/ 89.5%

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

      2. +-commutative 89.5%

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

      3. distribute-neg-in 89.5%

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

      4. sub-neg 89.5%

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

      5. associate-*l/ 89.5%

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

      6. clear-num 89.5%

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

      7. associate-*l/ 89.5%

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

      8. *-un-lft-identity 89.5%

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

      9. sub-neg 89.5%

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

      10. distribute-neg-in 89.5%

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

      11. +-commutative 89.5%

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

      12. add-sqr-sqrt 33.6%

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

      13. sqrt-unprod 38.5%

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

      14. sqr-neg 38.5%

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

      15. sqrt-unprod 21.6%

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

      16. add-sqr-sqrt 37.8%

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

   7. Applied egg-rr 37.8%

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

   8. Taylor expanded in t1 around inf 37.6%

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

   if -2.9999999999999998e173 < t1 < 7.4999999999999999e35

   1. Initial program 83.9%

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

      1. associate-/l* 86.0%

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

      2. distribute-lft-neg-out 86.0%

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

      3. distribute-rgt-neg-in 86.0%

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

      4. associate-/r* 91.6%

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

      5. distribute-neg-frac2 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in t1 around inf 44.6%

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

      1. associate-*r/ 46.2%

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

      2. +-commutative 46.2%

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

      3. distribute-neg-in 46.2%

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

      4. sub-neg 46.2%

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

      5. associate-*l/ 51.3%

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

      6. clear-num 51.8%

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

      7. associate-*l/ 51.8%

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

      8. *-un-lft-identity 51.8%

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

      9. sub-neg 51.8%

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

      10. distribute-neg-in 51.8%

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

      11. +-commutative 51.8%

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

      12. add-sqr-sqrt 31.9%

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

      13. sqrt-unprod 47.2%

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

      14. sqr-neg 47.2%

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

      15. sqrt-unprod 12.8%

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

      16. add-sqr-sqrt 25.6%

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

   7. Applied egg-rr 25.6%

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

   8. Taylor expanded in t1 around 0 20.5%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+173} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.7e+68) (/ (* v (/ t1 u)) t1) (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.7e+68) {
		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 <= (-5.7d+68)) 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 <= -5.7e+68) {
		tmp = (v * (t1 / u)) / t1;
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.7e+68:
		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 <= -5.7e+68)
		tmp = Float64(Float64(v * 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 <= -5.7e+68)
		tmp = (v * (t1 / u)) / t1;
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.7e+68], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.6999999999999996e68

   1. Initial program 85.7%

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

      1. associate-/l* 86.1%

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

      2. distribute-lft-neg-out 86.1%

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

      3. distribute-rgt-neg-in 86.1%

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

      4. associate-/r* 94.7%

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

      5. distribute-neg-frac2 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in t1 around inf 37.1%

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

   6. Taylor expanded in t1 around 0 33.9%

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

      1. associate-*r/ 33.9%

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

      2. mul-1-neg 33.9%

         \[\leadsto t1 \cdot \frac{\color{blue}{-v}}{t1 \cdot u} \]

   8. Simplified 33.9%

      \[\leadsto t1 \cdot \color{blue}{\frac{-v}{t1 \cdot u}} \]
   9. Step-by-step derivation

      1. associate-*r/ 35.2%

         \[\leadsto \color{blue}{\frac{t1 \cdot \left(-v\right)}{t1 \cdot u}} \]

      2. *-commutative 35.2%

         \[\leadsto \frac{t1 \cdot \left(-v\right)}{\color{blue}{u \cdot t1}} \]

      3. distribute-rgt-neg-in 35.2%

         \[\leadsto \frac{\color{blue}{-t1 \cdot v}}{u \cdot t1} \]

      4. associate-/r* 42.4%

         \[\leadsto \color{blue}{\frac{\frac{-t1 \cdot v}{u}}{t1}} \]

      5. remove-double-neg 42.4%

         \[\leadsto \frac{\frac{-t1 \cdot v}{\color{blue}{-\left(-u\right)}}}{t1} \]

      6. frac-2neg 42.4%

         \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{-u}}}{t1} \]

      7. *-commutative 42.4%

         \[\leadsto \frac{\frac{\color{blue}{v \cdot t1}}{-u}}{t1} \]

      8. associate-/l* 62.0%

         \[\leadsto \frac{\color{blue}{v \cdot \frac{t1}{-u}}}{t1} \]

      9. add-sqr-sqrt 62.0%

         \[\leadsto \frac{v \cdot \frac{t1}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}}}}{t1} \]

      10. sqrt-unprod 67.0%

          \[\leadsto \frac{v \cdot \frac{t1}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}}}{t1} \]

      11. sqr-neg 67.0%

          \[\leadsto \frac{v \cdot \frac{t1}{\sqrt{\color{blue}{u \cdot u}}}}{t1} \]

      12. sqrt-unprod 0.0%

          \[\leadsto \frac{v \cdot \frac{t1}{\color{blue}{\sqrt{u} \cdot \sqrt{u}}}}{t1} \]

      13. add-sqr-sqrt 61.8%

          \[\leadsto \frac{v \cdot \frac{t1}{\color{blue}{u}}}{t1} \]

   10. Applied egg-rr 61.8%

       \[\leadsto \color{blue}{\frac{v \cdot \frac{t1}{u}}{t1}} \]

   if -5.6999999999999996e68 < u

   1. Initial program 71.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 97.1%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 97.1%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 97.1%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 97.1%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 97.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 97.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 97.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(-u\right) - t1}{t1}}} \cdot \frac{v}{t1 + u} \]

      2. frac-2neg 97.1%

         \[\leadsto \frac{1}{\frac{\left(-u\right) - t1}{t1}} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

      3. frac-times 95.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(-v\right)}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)}} \]

      4. *-un-lft-identity 95.7%

         \[\leadsto \frac{\color{blue}{-v}}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      5. +-commutative 95.7%

         \[\leadsto \frac{-v}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\color{blue}{\left(u + t1\right)}\right)} \]

      6. distribute-neg-in 95.7%

         \[\leadsto \frac{-v}{\frac{\left(-u\right) - t1}{t1} \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

      7. sub-neg 95.7%

         \[\leadsto \frac{-v}{\frac{\left(-u\right) - t1}{t1} \cdot \color{blue}{\left(\left(-u\right) - t1\right)}} \]

      8. frac-2neg 95.7%

         \[\leadsto \frac{-v}{\color{blue}{\frac{-\left(\left(-u\right) - t1\right)}{-t1}} \cdot \left(\left(-u\right) - t1\right)} \]

      9. sub-neg 95.7%

         \[\leadsto \frac{-v}{\frac{-\color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}}{-t1} \cdot \left(\left(-u\right) - t1\right)} \]

      10. distribute-neg-in 95.7%

          \[\leadsto \frac{-v}{\frac{-\color{blue}{\left(-\left(u + t1\right)\right)}}{-t1} \cdot \left(\left(-u\right) - t1\right)} \]

      11. +-commutative 95.7%

          \[\leadsto \frac{-v}{\frac{-\left(-\color{blue}{\left(t1 + u\right)}\right)}{-t1} \cdot \left(\left(-u\right) - t1\right)} \]

      12. remove-double-neg 95.7%

          \[\leadsto \frac{-v}{\frac{\color{blue}{t1 + u}}{-t1} \cdot \left(\left(-u\right) - t1\right)} \]

      13. add-sqr-sqrt 51.6%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}} \cdot \left(\left(-u\right) - t1\right)} \]

      14. sqrt-unprod 45.3%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}} \cdot \left(\left(-u\right) - t1\right)} \]

      15. sqr-neg 45.3%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\sqrt{\color{blue}{t1 \cdot t1}}} \cdot \left(\left(-u\right) - t1\right)} \]

      16. sqrt-unprod 14.2%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}} \cdot \left(\left(-u\right) - t1\right)} \]

      17. add-sqr-sqrt 30.4%

          \[\leadsto \frac{-v}{\frac{t1 + u}{\color{blue}{t1}} \cdot \left(\left(-u\right) - t1\right)} \]

      18. sub-neg 30.4%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

      19. distribute-neg-in 30.4%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\left(-\left(u + t1\right)\right)}} \]

      20. +-commutative 30.4%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \left(-\color{blue}{\left(t1 + u\right)}\right)} \]

      21. add-sqr-sqrt 6.5%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\left(\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}\right)}} \]

      22. sqrt-unprod 49.6%

          \[\leadsto \frac{-v}{\frac{t1 + u}{t1} \cdot \color{blue}{\sqrt{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

   6. Applied egg-rr 95.7%

      \[\leadsto \color{blue}{\frac{-v}{\frac{t1 + u}{t1} \cdot \left(t1 + u\right)}} \]

   7. Taylor expanded in t1 around inf 62.9%

      \[\leadsto \frac{-v}{\color{blue}{t1 + 2 \cdot u}} \]
   8. Step-by-step derivation

      1. *-commutative 62.9%

         \[\leadsto \frac{-v}{t1 + \color{blue}{u \cdot 2}} \]

   9. Simplified 62.9%

      \[\leadsto \frac{-v}{\color{blue}{t1 + u \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{t1 + u} \cdot \frac{t1}{\left(-u\right) - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (* (/ v (+ t1 u)) (/ t1 (- (- u) t1))))

C

double code(double u, double v, double t1) {
	return (v / (t1 + u)) * (t1 / (-u - t1));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / (t1 + u)) * (t1 / (-u - t1))
end function

Java

public static double code(double u, double v, double t1) {
	return (v / (t1 + u)) * (t1 / (-u - t1));
}

Python

def code(u, v, t1):
	return (v / (t1 + u)) * (t1 / (-u - t1))

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(Float64(-u) - t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / (t1 + u)) * (t1 / (-u - t1));
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 97.2%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 97.2%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 97.2%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 97.2%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 97.2%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 97.2%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Final simplification 97.2%

   \[\leadsto \frac{v}{t1 + u} \cdot \frac{t1}{\left(-u\right) - t1} \]
6. Add Preprocessing

Alternative 16: 61.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- u t1)))

C

double code(double u, double v, double t1) {
	return v / (u - t1);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / (u - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / (u - t1);
}

Python

def code(u, v, t1):
	return v / (u - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(u - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (u - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 76.8%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. distribute-lft-neg-out 76.8%

      \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. distribute-rgt-neg-in 76.8%

      \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

   4. associate-/r* 86.6%

      \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

   5. distribute-neg-frac2 86.6%

      \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

3. Simplified 86.6%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t1 around inf 49.3%

   \[\leadsto t1 \cdot \frac{\color{blue}{\frac{v}{t1}}}{-\left(t1 + u\right)} \]
6. Step-by-step derivation

   1. associate-/l/ 44.4%

      \[\leadsto t1 \cdot \color{blue}{\frac{v}{\left(-\left(t1 + u\right)\right) \cdot t1}} \]

   2. associate-*r/ 44.2%

      \[\leadsto \color{blue}{\frac{t1 \cdot v}{\left(-\left(t1 + u\right)\right) \cdot t1}} \]

   3. *-commutative 44.2%

      \[\leadsto \frac{t1 \cdot v}{\color{blue}{t1 \cdot \left(-\left(t1 + u\right)\right)}} \]

   4. +-commutative 44.2%

      \[\leadsto \frac{t1 \cdot v}{t1 \cdot \left(-\color{blue}{\left(u + t1\right)}\right)} \]

   5. distribute-neg-in 44.2%

      \[\leadsto \frac{t1 \cdot v}{t1 \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

   6. sub-neg 44.2%

      \[\leadsto \frac{t1 \cdot v}{t1 \cdot \color{blue}{\left(\left(-u\right) - t1\right)}} \]

   7. sub-neg 44.2%

      \[\leadsto \frac{t1 \cdot v}{t1 \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

   8. distribute-neg-in 44.2%

      \[\leadsto \frac{t1 \cdot v}{t1 \cdot \color{blue}{\left(-\left(u + t1\right)\right)}} \]

   9. +-commutative 44.2%

      \[\leadsto \frac{t1 \cdot v}{t1 \cdot \left(-\color{blue}{\left(t1 + u\right)}\right)} \]

   10. add-sqr-sqrt 22.9%

       \[\leadsto \frac{t1 \cdot v}{t1 \cdot \color{blue}{\left(\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}\right)}} \]

   11. sqrt-unprod 46.5%

       \[\leadsto \frac{t1 \cdot v}{t1 \cdot \color{blue}{\sqrt{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}} \]

   12. sqr-neg 46.5%

       \[\leadsto \frac{t1 \cdot v}{t1 \cdot \sqrt{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}} \]

   13. sqrt-unprod 14.9%

       \[\leadsto \frac{t1 \cdot v}{t1 \cdot \color{blue}{\left(\sqrt{t1 + u} \cdot \sqrt{t1 + u}\right)}} \]

   14. add-sqr-sqrt 27.0%

       \[\leadsto \frac{t1 \cdot v}{t1 \cdot \color{blue}{\left(t1 + u\right)}} \]

7. Applied egg-rr 27.0%

   \[\leadsto \color{blue}{\frac{t1 \cdot v}{t1 \cdot \left(t1 + u\right)}} \]
8. Step-by-step derivation

   1. frac-2neg 27.0%

      \[\leadsto \color{blue}{\frac{-t1 \cdot v}{-t1 \cdot \left(t1 + u\right)}} \]

   2. div-inv 27.0%

      \[\leadsto \color{blue}{\left(-t1 \cdot v\right) \cdot \frac{1}{-t1 \cdot \left(t1 + u\right)}} \]

   3. distribute-rgt-neg-in 27.0%

      \[\leadsto \color{blue}{\left(t1 \cdot \left(-v\right)\right)} \cdot \frac{1}{-t1 \cdot \left(t1 + u\right)} \]

   4. add-sqr-sqrt 13.0%

      \[\leadsto \left(t1 \cdot \color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)}\right) \cdot \frac{1}{-t1 \cdot \left(t1 + u\right)} \]

   5. sqrt-unprod 34.5%

      \[\leadsto \left(t1 \cdot \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}\right) \cdot \frac{1}{-t1 \cdot \left(t1 + u\right)} \]

   6. sqr-neg 34.5%

      \[\leadsto \left(t1 \cdot \sqrt{\color{blue}{v \cdot v}}\right) \cdot \frac{1}{-t1 \cdot \left(t1 + u\right)} \]

   7. sqrt-unprod 22.3%

      \[\leadsto \left(t1 \cdot \color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)}\right) \cdot \frac{1}{-t1 \cdot \left(t1 + u\right)} \]

   8. add-sqr-sqrt 44.0%

      \[\leadsto \left(t1 \cdot \color{blue}{v}\right) \cdot \frac{1}{-t1 \cdot \left(t1 + u\right)} \]

   9. *-commutative 44.0%

      \[\leadsto \color{blue}{\left(v \cdot t1\right)} \cdot \frac{1}{-t1 \cdot \left(t1 + u\right)} \]

   10. distribute-rgt-neg-in 44.0%

       \[\leadsto \left(v \cdot t1\right) \cdot \frac{1}{\color{blue}{t1 \cdot \left(-\left(t1 + u\right)\right)}} \]

   11. +-commutative 44.0%

       \[\leadsto \left(v \cdot t1\right) \cdot \frac{1}{t1 \cdot \left(-\color{blue}{\left(u + t1\right)}\right)} \]

   12. distribute-neg-in 44.0%

       \[\leadsto \left(v \cdot t1\right) \cdot \frac{1}{t1 \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

   13. add-sqr-sqrt 21.8%

       \[\leadsto \left(v \cdot t1\right) \cdot \frac{1}{t1 \cdot \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)\right)} \]

   14. sqrt-unprod 53.2%

       \[\leadsto \left(v \cdot t1\right) \cdot \frac{1}{t1 \cdot \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)\right)} \]

   15. sqr-neg 53.2%

       \[\leadsto \left(v \cdot t1\right) \cdot \frac{1}{t1 \cdot \left(\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)\right)} \]

   16. sqrt-unprod 22.3%

       \[\leadsto \left(v \cdot t1\right) \cdot \frac{1}{t1 \cdot \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)\right)} \]

   17. add-sqr-sqrt 44.2%

       \[\leadsto \left(v \cdot t1\right) \cdot \frac{1}{t1 \cdot \left(\color{blue}{u} + \left(-t1\right)\right)} \]

9. Applied egg-rr 44.2%

   \[\leadsto \color{blue}{\left(v \cdot t1\right) \cdot \frac{1}{t1 \cdot \left(u + \left(-t1\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ 44.4%

       \[\leadsto \color{blue}{\frac{\left(v \cdot t1\right) \cdot 1}{t1 \cdot \left(u + \left(-t1\right)\right)}} \]

    2. *-rgt-identity 44.4%

       \[\leadsto \frac{\color{blue}{v \cdot t1}}{t1 \cdot \left(u + \left(-t1\right)\right)} \]

    3. associate-/r* 48.8%

       \[\leadsto \color{blue}{\frac{\frac{v \cdot t1}{t1}}{u + \left(-t1\right)}} \]

    4. associate-/l* 56.9%

       \[\leadsto \frac{\color{blue}{v \cdot \frac{t1}{t1}}}{u + \left(-t1\right)} \]

    5. *-inverses 56.9%

       \[\leadsto \frac{v \cdot \color{blue}{1}}{u + \left(-t1\right)} \]

    6. *-rgt-identity 56.9%

       \[\leadsto \frac{\color{blue}{v}}{u + \left(-t1\right)} \]

    7. sub-neg 56.9%

       \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

11. Simplified 56.9%

    \[\leadsto \color{blue}{\frac{v}{u - t1}} \]

12. Final simplification 56.9%

    \[\leadsto \frac{v}{u - t1} \]
13. Add Preprocessing

Alternative 17: 14.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v t1))

C

double code(double u, double v, double t1) {
	return v / t1;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / t1
end function

Java

public static double code(double u, double v, double t1) {
	return v / t1;
}

Python

def code(u, v, t1):
	return v / t1

Julia

function code(u, v, t1)
	return Float64(v / t1)
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / t1;
end

Wolfram

code[u_, v_, t1_] := N[(v / t1), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 76.8%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. distribute-lft-neg-out 76.8%

      \[\leadsto \color{blue}{-t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. distribute-rgt-neg-in 76.8%

      \[\leadsto \color{blue}{t1 \cdot \left(-\frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right)} \]

   4. associate-/r* 86.6%

      \[\leadsto t1 \cdot \left(-\color{blue}{\frac{\frac{v}{t1 + u}}{t1 + u}}\right) \]

   5. distribute-neg-frac2 86.6%

      \[\leadsto t1 \cdot \color{blue}{\frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]

3. Simplified 86.6%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t1 around inf 49.3%

   \[\leadsto t1 \cdot \frac{\color{blue}{\frac{v}{t1}}}{-\left(t1 + u\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 56.6%

      \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1}}{-\left(t1 + u\right)}} \]

   2. +-commutative 56.6%

      \[\leadsto \frac{t1 \cdot \frac{v}{t1}}{-\color{blue}{\left(u + t1\right)}} \]

   3. distribute-neg-in 56.6%

      \[\leadsto \frac{t1 \cdot \frac{v}{t1}}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   4. sub-neg 56.6%

      \[\leadsto \frac{t1 \cdot \frac{v}{t1}}{\color{blue}{\left(-u\right) - t1}} \]

   5. associate-*l/ 60.6%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1}} \]

   6. clear-num 60.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(-u\right) - t1}{t1}}} \cdot \frac{v}{t1} \]

   7. associate-*l/ 60.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{v}{t1}}{\frac{\left(-u\right) - t1}{t1}}} \]

   8. *-un-lft-identity 60.9%

      \[\leadsto \frac{\color{blue}{\frac{v}{t1}}}{\frac{\left(-u\right) - t1}{t1}} \]

   9. sub-neg 60.9%

      \[\leadsto \frac{\frac{v}{t1}}{\frac{\color{blue}{\left(-u\right) + \left(-t1\right)}}{t1}} \]

   10. distribute-neg-in 60.9%

       \[\leadsto \frac{\frac{v}{t1}}{\frac{\color{blue}{-\left(u + t1\right)}}{t1}} \]

   11. +-commutative 60.9%

       \[\leadsto \frac{\frac{v}{t1}}{\frac{-\color{blue}{\left(t1 + u\right)}}{t1}} \]

   12. add-sqr-sqrt 32.3%

       \[\leadsto \frac{\frac{v}{t1}}{\frac{\color{blue}{\sqrt{-\left(t1 + u\right)} \cdot \sqrt{-\left(t1 + u\right)}}}{t1}} \]

   13. sqrt-unprod 45.1%

       \[\leadsto \frac{\frac{v}{t1}}{\frac{\color{blue}{\sqrt{\left(-\left(t1 + u\right)\right) \cdot \left(-\left(t1 + u\right)\right)}}}{t1}} \]

   14. sqr-neg 45.1%

       \[\leadsto \frac{\frac{v}{t1}}{\frac{\sqrt{\color{blue}{\left(t1 + u\right) \cdot \left(t1 + u\right)}}}{t1}} \]

   15. sqrt-unprod 15.0%

       \[\leadsto \frac{\frac{v}{t1}}{\frac{\color{blue}{\sqrt{t1 + u} \cdot \sqrt{t1 + u}}}{t1}} \]

   16. add-sqr-sqrt 28.5%

       \[\leadsto \frac{\frac{v}{t1}}{\frac{\color{blue}{t1 + u}}{t1}} \]

7. Applied egg-rr 28.5%

   \[\leadsto \color{blue}{\frac{\frac{v}{t1}}{\frac{t1 + u}{t1}}} \]

8. Taylor expanded in t1 around inf 11.9%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]

9. Final simplification 11.9%

   \[\leadsto \frac{v}{t1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))