Rosa's DopplerBench

Percentage Accurate: 73.1% → 98.1%

Time: 12.2s

Alternatives: 18

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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. associate-*l/ 80.2%

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

   2. *-commutative 80.2%

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

3. Simplified 80.2%

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

   1. associate-*r/ 77.0%

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

   2. times-frac 97.2%

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

   3. *-commutative 97.2%

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

   4. frac-2neg 97.2%

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

   5. +-commutative 97.2%

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

   6. distribute-neg-in 97.2%

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

   7. sub-neg 97.2%

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

   8. associate-*r/ 99.0%

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

   9. add-sqr-sqrt 50.4%

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

   10. sqrt-unprod 51.7%

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

   11. sqr-neg 51.7%

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

   12. sqrt-unprod 19.9%

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

   13. add-sqr-sqrt 39.2%

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

   14. sub-neg 39.2%

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

   15. +-commutative 39.2%

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

   16. add-sqr-sqrt 19.3%

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

   17. sqrt-unprod 51.2%

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

   18. sqr-neg 51.2%

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

   19. sqrt-unprod 38.5%

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

   20. add-sqr-sqrt 19.1%

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

   21. sqrt-unprod 43.0%

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

   22. sqr-neg 43.0%

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

   23. sqrt-unprod 26.1%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+164}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq -5.6 \cdot 10^{-192}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.18 \cdot 10^{-169}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ t1 (* (+ t1 u) (- (- u) t1))))))
   (if (<= t1 -2.7e+164)
     (/ v (- t1))
     (if (<= t1 -5.6e-192)
       t_1
       (if (<= t1 1.18e-169)
         (/ (* t1 (/ v (- u))) (+ t1 u))
         (if (<= t1 4.8e+131) t_1 (/ v (- (* u (- 2.0)) t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	double tmp;
	if (t1 <= -2.7e+164) {
		tmp = v / -t1;
	} else if (t1 <= -5.6e-192) {
		tmp = t_1;
	} else if (t1 <= 1.18e-169) {
		tmp = (t1 * (v / -u)) / (t1 + u);
	} else if (t1 <= 4.8e+131) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v * (t1 / ((t1 + u) * (-u - t1)))
    if (t1 <= (-2.7d+164)) then
        tmp = v / -t1
    else if (t1 <= (-5.6d-192)) then
        tmp = t_1
    else if (t1 <= 1.18d-169) then
        tmp = (t1 * (v / -u)) / (t1 + u)
    else if (t1 <= 4.8d+131) then
        tmp = t_1
    else
        tmp = v / ((u * -2.0d0) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	double tmp;
	if (t1 <= -2.7e+164) {
		tmp = v / -t1;
	} else if (t1 <= -5.6e-192) {
		tmp = t_1;
	} else if (t1 <= 1.18e-169) {
		tmp = (t1 * (v / -u)) / (t1 + u);
	} else if (t1 <= 4.8e+131) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (t1 / ((t1 + u) * (-u - t1)))
	tmp = 0
	if t1 <= -2.7e+164:
		tmp = v / -t1
	elif t1 <= -5.6e-192:
		tmp = t_1
	elif t1 <= 1.18e-169:
		tmp = (t1 * (v / -u)) / (t1 + u)
	elif t1 <= 4.8e+131:
		tmp = t_1
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) - t1))))
	tmp = 0.0
	if (t1 <= -2.7e+164)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= -5.6e-192)
		tmp = t_1;
	elseif (t1 <= 1.18e-169)
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / Float64(t1 + u));
	elseif (t1 <= 4.8e+131)
		tmp = t_1;
	else
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (t1 / ((t1 + u) * (-u - t1)));
	tmp = 0.0;
	if (t1 <= -2.7e+164)
		tmp = v / -t1;
	elseif (t1 <= -5.6e-192)
		tmp = t_1;
	elseif (t1 <= 1.18e-169)
		tmp = (t1 * (v / -u)) / (t1 + u);
	elseif (t1 <= 4.8e+131)
		tmp = t_1;
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.7e+164], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, -5.6e-192], t$95$1, If[LessEqual[t1, 1.18e-169], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.8e+131], t$95$1, N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.70000000000000006e164

   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. associate-*l/ 40.6%

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

      2. *-commutative 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in t1 around 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%

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

   if -2.70000000000000006e164 < t1 < -5.60000000000000007e-192 or 1.18e-169 < t1 < 4.7999999999999999e131

   1. Initial program 89.7%

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

      1. associate-*l/ 96.3%

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

      2. *-commutative 96.3%

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

   3. Simplified 96.3%

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

   if -5.60000000000000007e-192 < t1 < 1.18e-169

   1. Initial program 87.8%

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

      1. associate-*l/ 80.0%

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

      2. *-commutative 80.0%

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

   3. Simplified 80.0%

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

      1. associate-*r/ 87.8%

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

      2. times-frac 94.2%

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

      3. *-commutative 94.2%

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

      4. frac-2neg 94.2%

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

      5. +-commutative 94.2%

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

      6. distribute-neg-in 94.2%

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

      7. sub-neg 94.2%

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

      8. associate-*r/ 96.1%

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

      9. add-sqr-sqrt 32.8%

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

      10. sqrt-unprod 53.6%

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

      11. sqr-neg 53.6%

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

      12. sqrt-unprod 38.1%

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

      13. add-sqr-sqrt 52.9%

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

      14. sub-neg 52.9%

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

      15. +-commutative 52.9%

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

      16. add-sqr-sqrt 14.7%

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

      17. sqrt-unprod 53.2%

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

      18. sqr-neg 53.2%

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

      19. sqrt-unprod 43.0%

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

      20. add-sqr-sqrt 19.3%

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

      21. sqrt-unprod 56.3%

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

      22. sqr-neg 56.3%

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

      23. sqrt-unprod 38.8%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in t1 around 0 90.0%

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

      1. mul-1-neg 90.0%

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

      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%

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

   if 4.7999999999999999e131 < t1

   1. Initial program 30.2%

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

      1. associate-*l/ 35.2%

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

      2. *-commutative 35.2%

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

   3. Simplified 35.2%

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

      1. associate-*r/ 30.2%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. clear-num 99.9%

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

      10. frac-2neg 99.9%

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

      11. frac-times 90.1%

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

      12. *-un-lft-identity 90.1%

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

      13. +-commutative 90.1%

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

      14. distribute-neg-in 90.1%

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

      15. sub-neg 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in u around 0 89.8%

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

      1. *-commutative 89.8%

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

   9. Simplified 89.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+164}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq -5.6 \cdot 10^{-192}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;t1 \leq 1.18 \cdot 10^{-169}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{+131}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 8.4 \cdot 10^{-91} \lor \neg \left(t1 \leq 3.6 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.9e+38)
   (/ v (- u t1))
   (if (<= t1 4.1e-125)
     (* (/ t1 (- u)) (/ v u))
     (if (or (<= t1 8.4e-91) (not (<= t1 3.6e-46)))
       (/ v (- (* u (- 2.0)) t1))
       (* t1 (/ (/ v (- u)) (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = v / (u - t1);
	} else if (t1 <= 4.1e-125) {
		tmp = (t1 / -u) * (v / u);
	} else if ((t1 <= 8.4e-91) || !(t1 <= 3.6e-46)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t1 * ((v / -u) / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5.9d+38)) then
        tmp = v / (u - t1)
    else if (t1 <= 4.1d-125) then
        tmp = (t1 / -u) * (v / u)
    else if ((t1 <= 8.4d-91) .or. (.not. (t1 <= 3.6d-46))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = t1 * ((v / -u) / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = v / (u - t1);
	} else if (t1 <= 4.1e-125) {
		tmp = (t1 / -u) * (v / u);
	} else if ((t1 <= 8.4e-91) || !(t1 <= 3.6e-46)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t1 * ((v / -u) / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.9e+38:
		tmp = v / (u - t1)
	elif t1 <= 4.1e-125:
		tmp = (t1 / -u) * (v / u)
	elif (t1 <= 8.4e-91) or not (t1 <= 3.6e-46):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t1 * ((v / -u) / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.9e+38)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 4.1e-125)
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	elseif ((t1 <= 8.4e-91) || !(t1 <= 3.6e-46))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	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 <= -5.9e+38)
		tmp = v / (u - t1);
	elseif (t1 <= 4.1e-125)
		tmp = (t1 / -u) * (v / u);
	elseif ((t1 <= 8.4e-91) || ~((t1 <= 3.6e-46)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t1 * ((v / -u) / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.9e+38], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.1e-125], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 8.4e-91], N[Not[LessEqual[t1, 3.6e-46]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -5.89999999999999981e38

   1. Initial program 55.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 88.6%

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

      1. *-commutative 88.6%

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

      2. clear-num 88.1%

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

      3. frac-times 70.5%

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

      4. *-un-lft-identity 70.5%

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

      5. sub-neg 70.5%

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

      6. add-sqr-sqrt 40.5%

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

      7. add-sqr-sqrt 40.5%

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

      8. sqrt-unprod 68.3%

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

      9. sqr-neg 68.3%

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

      10. sqrt-unprod 30.1%

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

      11. add-sqr-sqrt 70.6%

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

      12. sqrt-unprod 55.8%

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

      13. sqr-neg 55.8%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 38.6%

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

      16. +-commutative 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. associate-/l/ 35.0%

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

      2. associate-/r/ 35.0%

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

      3. /-rgt-identity 35.0%

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

      4. times-frac 35.0%

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

      5. *-rgt-identity 35.0%

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

      6. associate-*r/ 35.0%

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

      7. associate-*l/ 35.0%

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

      8. associate-/l* 38.6%

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

   9. Simplified 38.6%

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

       1. frac-2neg 38.6%

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

       2. associate-*r/ 35.0%

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

       3. distribute-neg-frac 35.0%

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

       4. add-sqr-sqrt 12.9%

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

       5. sqrt-unprod 47.7%

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

       6. sqr-neg 47.7%

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

       7. sqrt-unprod 50.2%

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

       8. add-sqr-sqrt 88.6%

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

       9. associate-*r/ 51.8%

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

       10. +-commutative 51.8%

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

       11. distribute-neg-in 51.8%

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

       12. add-sqr-sqrt 33.7%

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

       13. sqrt-unprod 56.0%

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

       14. sqr-neg 56.0%

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

       15. sqrt-unprod 18.3%

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

       16. add-sqr-sqrt 51.9%

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

   11. Applied egg-rr 51.9%

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

       1. *-lft-identity 51.9%

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

       2. *-commutative 51.9%

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

       3. times-frac 88.7%

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

       4. /-rgt-identity 88.7%

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

       5. *-inverses 88.7%

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

       6. *-rgt-identity 88.7%

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

       7. sub-neg 88.7%

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

   13. Simplified 88.7%

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

   if -5.89999999999999981e38 < t1 < 4.0999999999999997e-125

   1. Initial program 91.0%

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

      1. times-frac 94.7%

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

      2. distribute-frac-neg 94.7%

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

      3. distribute-neg-frac2 94.7%

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

      4. +-commutative 94.7%

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

      5. distribute-neg-in 94.7%

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

      6. unsub-neg 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in t1 around 0 79.3%

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

      1. associate-*r/ 79.3%

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

      2. mul-1-neg 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in t1 around 0 80.4%

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

   if 4.0999999999999997e-125 < t1 < 8.3999999999999997e-91 or 3.6e-46 < t1

   1. Initial program 68.2%

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

      1. associate-*l/ 74.1%

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

      2. *-commutative 74.1%

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

   3. Simplified 74.1%

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

      1. associate-*r/ 68.2%

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

      2. times-frac 98.6%

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

      3. *-commutative 98.6%

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

      4. frac-2neg 98.6%

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

      5. remove-double-neg 98.6%

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

      6. +-commutative 98.6%

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

      7. distribute-neg-in 98.6%

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

      8. sub-neg 98.6%

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

      9. clear-num 98.6%

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

      10. frac-2neg 98.6%

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

      11. frac-times 96.1%

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

      12. *-un-lft-identity 96.1%

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

      13. +-commutative 96.1%

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

      14. distribute-neg-in 96.1%

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

      15. sub-neg 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in u around 0 86.7%

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

      1. *-commutative 86.7%

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

   9. Simplified 86.7%

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

   if 8.3999999999999997e-91 < t1 < 3.6e-46

   1. Initial program 99.7%

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

      1. associate-*l/ 100.0%

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

      2. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 99.7%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.7%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 59.8%

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

      11. sqr-neg 59.8%

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

      12. sqrt-unprod 59.8%

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

      13. add-sqr-sqrt 59.8%

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

      14. sub-neg 59.8%

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

      15. +-commutative 59.8%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 67.9%

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

      18. sqr-neg 67.9%

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

      19. sqrt-unprod 67.9%

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

      20. add-sqr-sqrt 25.4%

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

      21. sqrt-unprod 91.9%

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

      22. sqr-neg 91.9%

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

      23. sqrt-unprod 66.4%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in t1 around 0 91.8%

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

      1. mul-1-neg 91.8%

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

      2. associate-/l* 91.9%

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

      3. distribute-rgt-neg-in 91.9%

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

      4. distribute-neg-frac2 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in v around 0 91.8%

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

       1. mul-1-neg 91.8%

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

       2. associate-/r* 91.8%

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

       3. associate-*r/ 91.9%

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

       4. associate-*r/ 91.8%

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

       5. distribute-rgt-neg-in 91.8%

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

       6. distribute-neg-frac 91.8%

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

       7. distribute-neg-frac2 91.8%

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

   12. Simplified 91.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 8.4 \cdot 10^{-91} \lor \neg \left(t1 \leq 3.6 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 3.95 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 10^{-90} \lor \neg \left(t1 \leq 2.7 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.9e+38)
   (/ v (- u t1))
   (if (<= t1 3.95e-125)
     (* (/ t1 (- u)) (/ v u))
     (if (or (<= t1 1e-90) (not (<= t1 2.7e-46)))
       (/ v (- (* u (- 2.0)) t1))
       (* (/ t1 (- (- u) t1)) (/ v u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = v / (u - t1);
	} else if (t1 <= 3.95e-125) {
		tmp = (t1 / -u) * (v / u);
	} else if ((t1 <= 1e-90) || !(t1 <= 2.7e-46)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / (-u - t1)) * (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 <= (-5.9d+38)) then
        tmp = v / (u - t1)
    else if (t1 <= 3.95d-125) then
        tmp = (t1 / -u) * (v / u)
    else if ((t1 <= 1d-90) .or. (.not. (t1 <= 2.7d-46))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = (t1 / (-u - t1)) * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = v / (u - t1);
	} else if (t1 <= 3.95e-125) {
		tmp = (t1 / -u) * (v / u);
	} else if ((t1 <= 1e-90) || !(t1 <= 2.7e-46)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / (-u - t1)) * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.9e+38:
		tmp = v / (u - t1)
	elif t1 <= 3.95e-125:
		tmp = (t1 / -u) * (v / u)
	elif (t1 <= 1e-90) or not (t1 <= 2.7e-46):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (t1 / (-u - t1)) * (v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.9e+38)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= 3.95e-125)
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	elseif ((t1 <= 1e-90) || !(t1 <= 2.7e-46))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(Float64(t1 / Float64(Float64(-u) - t1)) * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5.9e+38)
		tmp = v / (u - t1);
	elseif (t1 <= 3.95e-125)
		tmp = (t1 / -u) * (v / u);
	elseif ((t1 <= 1e-90) || ~((t1 <= 2.7e-46)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (t1 / (-u - t1)) * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.9e+38], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.95e-125], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 1e-90], N[Not[LessEqual[t1, 2.7e-46]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[((-u) - t1), $MachinePrecision]), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -5.89999999999999981e38

   1. Initial program 55.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 88.6%

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

      1. *-commutative 88.6%

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

      2. clear-num 88.1%

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

      3. frac-times 70.5%

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

      4. *-un-lft-identity 70.5%

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

      5. sub-neg 70.5%

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

      6. add-sqr-sqrt 40.5%

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

      7. add-sqr-sqrt 40.5%

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

      8. sqrt-unprod 68.3%

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

      9. sqr-neg 68.3%

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

      10. sqrt-unprod 30.1%

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

      11. add-sqr-sqrt 70.6%

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

      12. sqrt-unprod 55.8%

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

      13. sqr-neg 55.8%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 38.6%

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

      16. +-commutative 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. associate-/l/ 35.0%

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

      2. associate-/r/ 35.0%

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

      3. /-rgt-identity 35.0%

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

      4. times-frac 35.0%

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

      5. *-rgt-identity 35.0%

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

      6. associate-*r/ 35.0%

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

      7. associate-*l/ 35.0%

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

      8. associate-/l* 38.6%

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

   9. Simplified 38.6%

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

       1. frac-2neg 38.6%

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

       2. associate-*r/ 35.0%

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

       3. distribute-neg-frac 35.0%

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

       4. add-sqr-sqrt 12.9%

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

       5. sqrt-unprod 47.7%

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

       6. sqr-neg 47.7%

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

       7. sqrt-unprod 50.2%

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

       8. add-sqr-sqrt 88.6%

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

       9. associate-*r/ 51.8%

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

       10. +-commutative 51.8%

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

       11. distribute-neg-in 51.8%

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

       12. add-sqr-sqrt 33.7%

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

       13. sqrt-unprod 56.0%

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

       14. sqr-neg 56.0%

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

       15. sqrt-unprod 18.3%

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

       16. add-sqr-sqrt 51.9%

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

   11. Applied egg-rr 51.9%

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

       1. *-lft-identity 51.9%

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

       2. *-commutative 51.9%

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

       3. times-frac 88.7%

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

       4. /-rgt-identity 88.7%

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

       5. *-inverses 88.7%

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

       6. *-rgt-identity 88.7%

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

       7. sub-neg 88.7%

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

   13. Simplified 88.7%

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

   if -5.89999999999999981e38 < t1 < 3.94999999999999994e-125

   1. Initial program 91.0%

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

      1. times-frac 94.7%

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

      2. distribute-frac-neg 94.7%

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

      3. distribute-neg-frac2 94.7%

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

      4. +-commutative 94.7%

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

      5. distribute-neg-in 94.7%

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

      6. unsub-neg 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in t1 around 0 79.3%

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

      1. associate-*r/ 79.3%

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

      2. mul-1-neg 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in t1 around 0 80.4%

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

   if 3.94999999999999994e-125 < t1 < 9.99999999999999995e-91 or 2.7e-46 < t1

   1. Initial program 68.2%

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

      1. associate-*l/ 74.1%

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

      2. *-commutative 74.1%

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

   3. Simplified 74.1%

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

      1. associate-*r/ 68.2%

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

      2. times-frac 98.6%

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

      3. *-commutative 98.6%

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

      4. frac-2neg 98.6%

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

      5. remove-double-neg 98.6%

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

      6. +-commutative 98.6%

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

      7. distribute-neg-in 98.6%

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

      8. sub-neg 98.6%

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

      9. clear-num 98.6%

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

      10. frac-2neg 98.6%

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

      11. frac-times 96.1%

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

      12. *-un-lft-identity 96.1%

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

      13. +-commutative 96.1%

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

      14. distribute-neg-in 96.1%

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

      15. sub-neg 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in u around 0 86.7%

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

      1. *-commutative 86.7%

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

   9. Simplified 86.7%

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

   if 9.99999999999999995e-91 < t1 < 2.7e-46

   1. Initial program 99.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 91.9%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{u}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 3.95 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 10^{-90} \lor \neg \left(t1 \leq 2.7 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -7.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{t1 \cdot v}{u \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;t1 \leq -6 \cdot 10^{-94} \lor \neg \left(t1 \leq 5 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.9e+38)
   (/ v (- u t1))
   (if (<= t1 -7.4e-41)
     (/ (* t1 v) (* u (- t1 u)))
     (if (or (<= t1 -6e-94) (not (<= t1 5e-46)))
       (/ v (- (* u (- 2.0)) t1))
       (/ (/ (* t1 v) (- u)) (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = v / (u - t1);
	} else if (t1 <= -7.4e-41) {
		tmp = (t1 * v) / (u * (t1 - u));
	} else if ((t1 <= -6e-94) || !(t1 <= 5e-46)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = ((t1 * v) / -u) / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5.9d+38)) then
        tmp = v / (u - t1)
    else if (t1 <= (-7.4d-41)) then
        tmp = (t1 * v) / (u * (t1 - u))
    else if ((t1 <= (-6d-94)) .or. (.not. (t1 <= 5d-46))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = ((t1 * v) / -u) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = v / (u - t1);
	} else if (t1 <= -7.4e-41) {
		tmp = (t1 * v) / (u * (t1 - u));
	} else if ((t1 <= -6e-94) || !(t1 <= 5e-46)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = ((t1 * v) / -u) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.9e+38:
		tmp = v / (u - t1)
	elif t1 <= -7.4e-41:
		tmp = (t1 * v) / (u * (t1 - u))
	elif (t1 <= -6e-94) or not (t1 <= 5e-46):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = ((t1 * v) / -u) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.9e+38)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -7.4e-41)
		tmp = Float64(Float64(t1 * v) / Float64(u * Float64(t1 - u)));
	elseif ((t1 <= -6e-94) || !(t1 <= 5e-46))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(Float64(Float64(t1 * v) / Float64(-u)) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5.9e+38)
		tmp = v / (u - t1);
	elseif (t1 <= -7.4e-41)
		tmp = (t1 * v) / (u * (t1 - u));
	elseif ((t1 <= -6e-94) || ~((t1 <= 5e-46)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = ((t1 * v) / -u) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.9e+38], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -7.4e-41], N[(N[(t1 * v), $MachinePrecision] / N[(u * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -6e-94], N[Not[LessEqual[t1, 5e-46]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 * v), $MachinePrecision] / (-u)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -5.89999999999999981e38

   1. Initial program 55.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 88.6%

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

      1. *-commutative 88.6%

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

      2. clear-num 88.1%

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

      3. frac-times 70.5%

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

      4. *-un-lft-identity 70.5%

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

      5. sub-neg 70.5%

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

      6. add-sqr-sqrt 40.5%

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

      7. add-sqr-sqrt 40.5%

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

      8. sqrt-unprod 68.3%

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

      9. sqr-neg 68.3%

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

      10. sqrt-unprod 30.1%

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

      11. add-sqr-sqrt 70.6%

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

      12. sqrt-unprod 55.8%

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

      13. sqr-neg 55.8%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 38.6%

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

      16. +-commutative 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. associate-/l/ 35.0%

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

      2. associate-/r/ 35.0%

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

      3. /-rgt-identity 35.0%

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

      4. times-frac 35.0%

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

      5. *-rgt-identity 35.0%

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

      6. associate-*r/ 35.0%

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

      7. associate-*l/ 35.0%

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

      8. associate-/l* 38.6%

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

   9. Simplified 38.6%

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

       1. frac-2neg 38.6%

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

       2. associate-*r/ 35.0%

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

       3. distribute-neg-frac 35.0%

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

       4. add-sqr-sqrt 12.9%

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

       5. sqrt-unprod 47.7%

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

       6. sqr-neg 47.7%

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

       7. sqrt-unprod 50.2%

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

       8. add-sqr-sqrt 88.6%

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

       9. associate-*r/ 51.8%

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

       10. +-commutative 51.8%

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

       11. distribute-neg-in 51.8%

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

       12. add-sqr-sqrt 33.7%

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

       13. sqrt-unprod 56.0%

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

       14. sqr-neg 56.0%

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

       15. sqrt-unprod 18.3%

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

       16. add-sqr-sqrt 51.9%

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

   11. Applied egg-rr 51.9%

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

       1. *-lft-identity 51.9%

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

       2. *-commutative 51.9%

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

       3. times-frac 88.7%

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

       4. /-rgt-identity 88.7%

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

       5. *-inverses 88.7%

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

       6. *-rgt-identity 88.7%

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

       7. sub-neg 88.7%

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

   13. Simplified 88.7%

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

   if -5.89999999999999981e38 < t1 < -7.4000000000000004e-41

   1. Initial program 99.7%

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

      1. times-frac 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t1 around 0 67.4%

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

      1. associate-*r/ 67.4%

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

      2. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

      1. *-commutative 67.4%

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

      2. frac-2neg 67.4%

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

      3. frac-times 67.6%

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

      4. add-sqr-sqrt 67.5%

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

      5. sqrt-unprod 67.6%

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

      6. sqr-neg 67.6%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 52.1%

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

      9. add-sqr-sqrt 31.9%

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

      10. sqrt-unprod 55.1%

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

      11. sqr-neg 55.1%

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

      12. sqrt-unprod 23.2%

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

      13. add-sqr-sqrt 67.6%

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

      14. distribute-neg-in 67.6%

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

      15. add-sqr-sqrt 67.6%

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

      16. sqrt-unprod 67.6%

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

      17. sqr-neg 67.6%

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

      18. sqrt-unprod 0.0%

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

      19. add-sqr-sqrt 74.6%

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

      20. sub-neg 74.6%

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

   9. Applied egg-rr 74.6%

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

   if -7.4000000000000004e-41 < t1 < -6.0000000000000003e-94 or 4.99999999999999992e-46 < t1

   1. Initial program 68.5%

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

      1. associate-*l/ 73.6%

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

      2. *-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. associate-*r/ 68.5%

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

      2. times-frac 99.5%

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

      3. *-commutative 99.5%

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

      4. frac-2neg 99.5%

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

      5. remove-double-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-neg-in 99.5%

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

      8. sub-neg 99.5%

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

      9. clear-num 98.6%

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

      10. frac-2neg 98.6%

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

      11. frac-times 94.9%

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

      12. *-un-lft-identity 94.9%

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

      13. +-commutative 94.9%

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

      14. distribute-neg-in 94.9%

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

      15. sub-neg 94.9%

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in u around 0 85.7%

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

      1. *-commutative 85.7%

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

   9. Simplified 85.7%

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

   if -6.0000000000000003e-94 < t1 < 4.99999999999999992e-46

   1. Initial program 90.9%

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

      1. associate-*l/ 90.0%

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

      2. *-commutative 90.0%

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

   3. Simplified 90.0%

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

      1. associate-*r/ 90.9%

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

      2. times-frac 93.9%

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

      3. *-commutative 93.9%

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

      4. frac-2neg 93.9%

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

      5. +-commutative 93.9%

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

      6. distribute-neg-in 93.9%

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

      7. sub-neg 93.9%

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

      8. associate-*r/ 98.1%

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

      9. add-sqr-sqrt 41.2%

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

      10. sqrt-unprod 60.8%

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

      11. sqr-neg 60.8%

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

      12. sqrt-unprod 30.3%

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

      13. add-sqr-sqrt 47.3%

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

      14. sub-neg 47.3%

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

      15. +-commutative 47.3%

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

      16. add-sqr-sqrt 17.0%

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

      17. sqrt-unprod 54.7%

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

      18. sqr-neg 54.7%

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

      19. sqrt-unprod 39.7%

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

      20. add-sqr-sqrt 20.6%

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

      21. sqrt-unprod 51.9%

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

      22. sqr-neg 51.9%

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

      23. sqrt-unprod 32.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in t1 around 0 85.0%

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

      1. associate-*r/ 85.0%

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

      2. mul-1-neg 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

   9. Simplified 85.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -7.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{t1 \cdot v}{u \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;t1 \leq -6 \cdot 10^{-94} \lor \neg \left(t1 \leq 5 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{-32}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq -2.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;u \leq -1.15 \cdot 10^{-102}:\\ \;\;\;\;\frac{-v}{\left(t1 + u\right) \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 1.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.25e-32)
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (if (<= u -2.8e-92)
     (/ v (- (* u (- 2.0)) t1))
     (if (<= u -1.15e-102)
       (/ (- v) (* (+ t1 u) (/ u t1)))
       (if (<= u 1.3e-13) (/ v (- t1)) (/ (* t1 (/ v (- u))) (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.25e-32) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else if (u <= -2.8e-92) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -1.15e-102) {
		tmp = -v / ((t1 + u) * (u / t1));
	} else if (u <= 1.3e-13) {
		tmp = v / -t1;
	} 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 (u <= (-1.25d-32)) then
        tmp = (v / (t1 + u)) * (t1 / -u)
    else if (u <= (-2.8d-92)) then
        tmp = v / ((u * -2.0d0) - t1)
    else if (u <= (-1.15d-102)) then
        tmp = -v / ((t1 + u) * (u / t1))
    else if (u <= 1.3d-13) then
        tmp = v / -t1
    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 (u <= -1.25e-32) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else if (u <= -2.8e-92) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -1.15e-102) {
		tmp = -v / ((t1 + u) * (u / t1));
	} else if (u <= 1.3e-13) {
		tmp = v / -t1;
	} else {
		tmp = (t1 * (v / -u)) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.25e-32:
		tmp = (v / (t1 + u)) * (t1 / -u)
	elif u <= -2.8e-92:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -1.15e-102:
		tmp = -v / ((t1 + u) * (u / t1))
	elif u <= 1.3e-13:
		tmp = v / -t1
	else:
		tmp = (t1 * (v / -u)) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.25e-32)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	elseif (u <= -2.8e-92)
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	elseif (u <= -1.15e-102)
		tmp = Float64(Float64(-v) / Float64(Float64(t1 + u) * Float64(u / t1)));
	elseif (u <= 1.3e-13)
		tmp = Float64(v / Float64(-t1));
	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 (u <= -1.25e-32)
		tmp = (v / (t1 + u)) * (t1 / -u);
	elseif (u <= -2.8e-92)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -1.15e-102)
		tmp = -v / ((t1 + u) * (u / t1));
	elseif (u <= 1.3e-13)
		tmp = v / -t1;
	else
		tmp = (t1 * (v / -u)) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.25e-32], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -2.8e-92], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.15e-102], N[((-v) / N[(N[(t1 + u), $MachinePrecision] * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.3e-13], N[(v / (-t1)), $MachinePrecision], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -1.25e-32

   1. Initial program 86.2%

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

      1. times-frac 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around 0 84.3%

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

      1. associate-*r/ 84.3%

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

      2. mul-1-neg 84.3%

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

   7. Simplified 84.3%

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

   if -1.25e-32 < u < -2.8e-92

   1. Initial program 64.2%

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

      1. associate-*l/ 65.2%

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

      2. *-commutative 65.2%

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

   3. Simplified 65.2%

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

      1. associate-*r/ 64.2%

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

      2. times-frac 99.7%

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

      3. *-commutative 99.7%

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

      4. frac-2neg 99.7%

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

      5. remove-double-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. sub-neg 99.7%

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

      9. clear-num 99.7%

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

      10. frac-2neg 99.7%

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

      11. frac-times 99.7%

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

      12. *-un-lft-identity 99.7%

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

      13. +-commutative 99.7%

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

      14. distribute-neg-in 99.7%

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

      15. sub-neg 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in u around 0 84.6%

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

      1. *-commutative 84.6%

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

   9. Simplified 84.6%

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

   if -2.8e-92 < u < -1.14999999999999993e-102

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. times-frac 77.1%

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

      3. *-commutative 77.1%

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

      4. frac-2neg 77.1%

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

      5. remove-double-neg 77.1%

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

      6. +-commutative 77.1%

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

      7. distribute-neg-in 77.1%

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

      8. sub-neg 77.1%

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

      9. clear-num 76.3%

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

      10. frac-2neg 76.3%

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

      11. frac-times 99.6%

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

      12. *-un-lft-identity 99.6%

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

      13. +-commutative 99.6%

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

      14. distribute-neg-in 99.6%

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

      15. sub-neg 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in t1 around 0 99.6%

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

   if -1.14999999999999993e-102 < u < 1.3e-13

   1. Initial program 69.9%

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

      1. associate-*l/ 80.1%

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

      2. *-commutative 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in t1 around inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   7. Simplified 84.2%

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

   if 1.3e-13 < u

   1. Initial program 80.1%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

      1. associate-*r/ 80.1%

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

      2. times-frac 98.1%

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

      3. *-commutative 98.1%

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

      4. frac-2neg 98.1%

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

      5. +-commutative 98.1%

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

      6. distribute-neg-in 98.1%

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

      7. sub-neg 98.1%

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

      8. associate-*r/ 98.1%

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

      9. add-sqr-sqrt 43.6%

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

      10. sqrt-unprod 64.0%

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

      11. sqr-neg 64.0%

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

      12. sqrt-unprod 31.4%

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

      13. add-sqr-sqrt 53.0%

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

      14. sub-neg 53.0%

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

      15. +-commutative 53.0%

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

      16. add-sqr-sqrt 21.5%

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

      17. sqrt-unprod 55.8%

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

      18. sqr-neg 55.8%

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

      19. sqrt-unprod 39.7%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 48.9%

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

      22. sqr-neg 48.9%

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

      23. sqrt-unprod 54.2%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in t1 around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. associate-/l* 82.7%

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

      3. distribute-rgt-neg-in 82.7%

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

      4. distribute-neg-frac2 82.7%

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

   9. Simplified 82.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{-32}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq -2.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;u \leq -1.15 \cdot 10^{-102}:\\ \;\;\;\;\frac{-v}{\left(t1 + u\right) \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 1.3 \cdot 10^{-13}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.8 \cdot 10^{-125} \lor \neg \left(t1 \leq 8.4 \cdot 10^{-91}\right) \land t1 \leq 3.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.9e+38)
   (/ v (- u t1))
   (if (or (<= t1 1.8e-125) (and (not (<= t1 8.4e-91)) (<= t1 3.2e-46)))
     (* (/ t1 (- u)) (/ v u))
     (/ v (- (* u (- 2.0)) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = v / (u - t1);
	} else if ((t1 <= 1.8e-125) || (!(t1 <= 8.4e-91) && (t1 <= 3.2e-46))) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5.9d+38)) then
        tmp = v / (u - t1)
    else if ((t1 <= 1.8d-125) .or. (.not. (t1 <= 8.4d-91)) .and. (t1 <= 3.2d-46)) then
        tmp = (t1 / -u) * (v / u)
    else
        tmp = v / ((u * -2.0d0) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = v / (u - t1);
	} else if ((t1 <= 1.8e-125) || (!(t1 <= 8.4e-91) && (t1 <= 3.2e-46))) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.9e+38:
		tmp = v / (u - t1)
	elif (t1 <= 1.8e-125) or (not (t1 <= 8.4e-91) and (t1 <= 3.2e-46)):
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.9e+38)
		tmp = Float64(v / Float64(u - t1));
	elseif ((t1 <= 1.8e-125) || (!(t1 <= 8.4e-91) && (t1 <= 3.2e-46)))
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5.9e+38)
		tmp = v / (u - t1);
	elseif ((t1 <= 1.8e-125) || (~((t1 <= 8.4e-91)) && (t1 <= 3.2e-46)))
		tmp = (t1 / -u) * (v / u);
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.9e+38], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 1.8e-125], And[N[Not[LessEqual[t1, 8.4e-91]], $MachinePrecision], LessEqual[t1, 3.2e-46]]], N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.89999999999999981e38

   1. Initial program 55.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 88.6%

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

      1. *-commutative 88.6%

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

      2. clear-num 88.1%

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

      3. frac-times 70.5%

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

      4. *-un-lft-identity 70.5%

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

      5. sub-neg 70.5%

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

      6. add-sqr-sqrt 40.5%

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

      7. add-sqr-sqrt 40.5%

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

      8. sqrt-unprod 68.3%

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

      9. sqr-neg 68.3%

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

      10. sqrt-unprod 30.1%

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

      11. add-sqr-sqrt 70.6%

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

      12. sqrt-unprod 55.8%

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

      13. sqr-neg 55.8%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 38.6%

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

      16. +-commutative 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. associate-/l/ 35.0%

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

      2. associate-/r/ 35.0%

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

      3. /-rgt-identity 35.0%

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

      4. times-frac 35.0%

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

      5. *-rgt-identity 35.0%

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

      6. associate-*r/ 35.0%

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

      7. associate-*l/ 35.0%

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

      8. associate-/l* 38.6%

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

   9. Simplified 38.6%

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

       1. frac-2neg 38.6%

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

       2. associate-*r/ 35.0%

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

       3. distribute-neg-frac 35.0%

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

       4. add-sqr-sqrt 12.9%

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

       5. sqrt-unprod 47.7%

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

       6. sqr-neg 47.7%

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

       7. sqrt-unprod 50.2%

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

       8. add-sqr-sqrt 88.6%

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

       9. associate-*r/ 51.8%

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

       10. +-commutative 51.8%

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

       11. distribute-neg-in 51.8%

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

       12. add-sqr-sqrt 33.7%

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

       13. sqrt-unprod 56.0%

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

       14. sqr-neg 56.0%

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

       15. sqrt-unprod 18.3%

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

       16. add-sqr-sqrt 51.9%

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

   11. Applied egg-rr 51.9%

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

       1. *-lft-identity 51.9%

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

       2. *-commutative 51.9%

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

       3. times-frac 88.7%

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

       4. /-rgt-identity 88.7%

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

       5. *-inverses 88.7%

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

       6. *-rgt-identity 88.7%

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

       7. sub-neg 88.7%

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

   13. Simplified 88.7%

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

   if -5.89999999999999981e38 < t1 < 1.8000000000000001e-125 or 8.3999999999999997e-91 < t1 < 3.1999999999999999e-46

   1. Initial program 91.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 80.5%

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

      1. associate-*r/ 80.5%

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

      2. mul-1-neg 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in t1 around 0 81.5%

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

   if 1.8000000000000001e-125 < t1 < 8.3999999999999997e-91 or 3.1999999999999999e-46 < t1

   1. Initial program 68.2%

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

      1. associate-*l/ 74.1%

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

      2. *-commutative 74.1%

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

   3. Simplified 74.1%

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

      1. associate-*r/ 68.2%

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

      2. times-frac 98.6%

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

      3. *-commutative 98.6%

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

      4. frac-2neg 98.6%

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

      5. remove-double-neg 98.6%

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

      6. +-commutative 98.6%

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

      7. distribute-neg-in 98.6%

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

      8. sub-neg 98.6%

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

      9. clear-num 98.6%

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

      10. frac-2neg 98.6%

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

      11. frac-times 96.1%

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

      12. *-un-lft-identity 96.1%

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

      13. +-commutative 96.1%

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

      14. distribute-neg-in 96.1%

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

      15. sub-neg 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in u around 0 86.7%

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

      1. *-commutative 86.7%

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

   9. Simplified 86.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.8 \cdot 10^{-125} \lor \neg \left(t1 \leq 8.4 \cdot 10^{-91}\right) \land t1 \leq 3.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u - t1}\\ t_2 := \frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{-125}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 8.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 3.6 \cdot 10^{-46}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- u t1))) (t_2 (* (/ t1 (- u)) (/ v u))))
   (if (<= t1 -7.4e+38)
     t_1
     (if (<= t1 4.1e-125)
       t_2
       (if (<= t1 8.8e-91) (/ v (- (- u) t1)) (if (<= t1 3.6e-46) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (u - t1);
	double t_2 = (t1 / -u) * (v / u);
	double tmp;
	if (t1 <= -7.4e+38) {
		tmp = t_1;
	} else if (t1 <= 4.1e-125) {
		tmp = t_2;
	} else if (t1 <= 8.8e-91) {
		tmp = v / (-u - t1);
	} else if (t1 <= 3.6e-46) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = v / (u - t1)
    t_2 = (t1 / -u) * (v / u)
    if (t1 <= (-7.4d+38)) then
        tmp = t_1
    else if (t1 <= 4.1d-125) then
        tmp = t_2
    else if (t1 <= 8.8d-91) then
        tmp = v / (-u - t1)
    else if (t1 <= 3.6d-46) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (u - t1);
	double t_2 = (t1 / -u) * (v / u);
	double tmp;
	if (t1 <= -7.4e+38) {
		tmp = t_1;
	} else if (t1 <= 4.1e-125) {
		tmp = t_2;
	} else if (t1 <= 8.8e-91) {
		tmp = v / (-u - t1);
	} else if (t1 <= 3.6e-46) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (u - t1)
	t_2 = (t1 / -u) * (v / u)
	tmp = 0
	if t1 <= -7.4e+38:
		tmp = t_1
	elif t1 <= 4.1e-125:
		tmp = t_2
	elif t1 <= 8.8e-91:
		tmp = v / (-u - t1)
	elif t1 <= 3.6e-46:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(u - t1))
	t_2 = Float64(Float64(t1 / Float64(-u)) * Float64(v / u))
	tmp = 0.0
	if (t1 <= -7.4e+38)
		tmp = t_1;
	elseif (t1 <= 4.1e-125)
		tmp = t_2;
	elseif (t1 <= 8.8e-91)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= 3.6e-46)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (u - t1);
	t_2 = (t1 / -u) * (v / u);
	tmp = 0.0;
	if (t1 <= -7.4e+38)
		tmp = t_1;
	elseif (t1 <= 4.1e-125)
		tmp = t_2;
	elseif (t1 <= 8.8e-91)
		tmp = v / (-u - t1);
	elseif (t1 <= 3.6e-46)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t1 / (-u)), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.4e+38], t$95$1, If[LessEqual[t1, 4.1e-125], t$95$2, If[LessEqual[t1, 8.8e-91], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.6e-46], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -7.4000000000000002e38 or 3.6e-46 < t1

   1. Initial program 60.4%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 87.9%

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

      1. *-commutative 87.9%

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

      2. clear-num 87.4%

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

      3. frac-times 71.6%

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

      4. *-un-lft-identity 71.6%

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

      5. sub-neg 71.6%

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

      6. add-sqr-sqrt 37.6%

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

      7. add-sqr-sqrt 19.6%

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

      8. sqrt-unprod 33.0%

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

      9. sqr-neg 33.0%

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

      10. sqrt-unprod 14.5%

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

      11. add-sqr-sqrt 34.1%

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

      12. sqrt-unprod 41.5%

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

      13. sqr-neg 41.5%

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

      14. sqrt-unprod 14.5%

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

      15. add-sqr-sqrt 33.1%

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

      16. +-commutative 33.1%

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

   7. Applied egg-rr 33.1%

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

      1. associate-/l/ 30.7%

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

      2. associate-/r/ 30.7%

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

      3. /-rgt-identity 30.7%

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

      4. times-frac 30.7%

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

      5. *-rgt-identity 30.7%

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

      6. associate-*r/ 30.7%

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

      7. associate-*l/ 30.7%

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

      8. associate-/l* 33.1%

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

   9. Simplified 33.1%

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

       1. frac-2neg 33.1%

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

       2. associate-*r/ 30.7%

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

       3. distribute-neg-frac 30.7%

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

       4. add-sqr-sqrt 14.5%

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

       5. sqrt-unprod 47.8%

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

       6. sqr-neg 47.8%

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

       7. sqrt-unprod 47.2%

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

       8. add-sqr-sqrt 87.9%

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

       9. associate-*r/ 58.0%

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

       10. +-commutative 58.0%

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

       11. distribute-neg-in 58.0%

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

       12. add-sqr-sqrt 31.1%

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

       13. sqrt-unprod 59.7%

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

       14. sqr-neg 59.7%

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

       15. sqrt-unprod 26.9%

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

       16. add-sqr-sqrt 58.1%

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

   11. Applied egg-rr 58.1%

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

       1. *-lft-identity 58.1%

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

       2. *-commutative 58.1%

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

       3. times-frac 88.0%

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

       4. /-rgt-identity 88.0%

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

       5. *-inverses 88.0%

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

       6. *-rgt-identity 88.0%

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

       7. sub-neg 88.0%

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

   13. Simplified 88.0%

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

   if -7.4000000000000002e38 < t1 < 4.0999999999999997e-125 or 8.8000000000000003e-91 < t1 < 3.6e-46

   1. Initial program 91.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 80.5%

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

      1. associate-*r/ 80.5%

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

      2. mul-1-neg 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in t1 around 0 81.5%

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

   if 4.0999999999999997e-125 < t1 < 8.8000000000000003e-91

   1. Initial program 88.8%

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

      1. associate-*l/ 99.3%

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

      2. *-commutative 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r/ 88.8%

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

      2. times-frac 89.8%

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

      3. *-commutative 89.8%

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

      4. frac-2neg 89.8%

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

      5. +-commutative 89.8%

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

      6. distribute-neg-in 89.8%

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

      7. sub-neg 89.8%

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

      8. associate-*r/ 99.8%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 11.9%

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

      11. sqr-neg 11.9%

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

      12. sqrt-unprod 11.9%

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

      13. add-sqr-sqrt 11.9%

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

      14. sub-neg 11.9%

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

      15. +-commutative 11.9%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-unprod 88.9%

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

      18. sqr-neg 88.9%

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

      19. sqrt-unprod 88.6%

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

      20. add-sqr-sqrt 55.4%

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

      21. sqrt-unprod 99.5%

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

      22. sqr-neg 99.5%

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

      23. sqrt-unprod 44.3%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t1 around inf 79.5%

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

      1. mul-1-neg 79.5%

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

   9. Simplified 79.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 8.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 3.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{-u}\\ \mathbf{if}\;u \leq -6.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot t\_1\\ \mathbf{elif}\;u \leq -2 \cdot 10^{-91}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;u \leq -1.15 \cdot 10^{-102}:\\ \;\;\;\;\frac{-v}{\left(t1 + u\right) \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ t1 (- u))))
   (if (<= u -6.3e-32)
     (* (/ v (+ t1 u)) t_1)
     (if (<= u -2e-91)
       (/ v (- (* u (- 2.0)) t1))
       (if (<= u -1.15e-102)
         (/ (- v) (* (+ t1 u) (/ u t1)))
         (if (<= u 3.8e-14) (/ v (- t1)) (* t_1 (/ v u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 / -u;
	double tmp;
	if (u <= -6.3e-32) {
		tmp = (v / (t1 + u)) * t_1;
	} else if (u <= -2e-91) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -1.15e-102) {
		tmp = -v / ((t1 + u) * (u / t1));
	} else if (u <= 3.8e-14) {
		tmp = v / -t1;
	} else {
		tmp = t_1 * (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) :: t_1
    real(8) :: tmp
    t_1 = t1 / -u
    if (u <= (-6.3d-32)) then
        tmp = (v / (t1 + u)) * t_1
    else if (u <= (-2d-91)) then
        tmp = v / ((u * -2.0d0) - t1)
    else if (u <= (-1.15d-102)) then
        tmp = -v / ((t1 + u) * (u / t1))
    else if (u <= 3.8d-14) then
        tmp = v / -t1
    else
        tmp = t_1 * (v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 / -u;
	double tmp;
	if (u <= -6.3e-32) {
		tmp = (v / (t1 + u)) * t_1;
	} else if (u <= -2e-91) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -1.15e-102) {
		tmp = -v / ((t1 + u) * (u / t1));
	} else if (u <= 3.8e-14) {
		tmp = v / -t1;
	} else {
		tmp = t_1 * (v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 / -u
	tmp = 0
	if u <= -6.3e-32:
		tmp = (v / (t1 + u)) * t_1
	elif u <= -2e-91:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -1.15e-102:
		tmp = -v / ((t1 + u) * (u / t1))
	elif u <= 3.8e-14:
		tmp = v / -t1
	else:
		tmp = t_1 * (v / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 / Float64(-u))
	tmp = 0.0
	if (u <= -6.3e-32)
		tmp = Float64(Float64(v / Float64(t1 + u)) * t_1);
	elseif (u <= -2e-91)
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	elseif (u <= -1.15e-102)
		tmp = Float64(Float64(-v) / Float64(Float64(t1 + u) * Float64(u / t1)));
	elseif (u <= 3.8e-14)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(t_1 * Float64(v / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 / -u;
	tmp = 0.0;
	if (u <= -6.3e-32)
		tmp = (v / (t1 + u)) * t_1;
	elseif (u <= -2e-91)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -1.15e-102)
		tmp = -v / ((t1 + u) * (u / t1));
	elseif (u <= 3.8e-14)
		tmp = v / -t1;
	else
		tmp = t_1 * (v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 / (-u)), $MachinePrecision]}, If[LessEqual[u, -6.3e-32], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[u, -2e-91], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.15e-102], N[((-v) / N[(N[(t1 + u), $MachinePrecision] * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.8e-14], N[(v / (-t1)), $MachinePrecision], N[(t$95$1 * N[(v / u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -6.2999999999999997e-32

   1. Initial program 86.2%

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

      1. times-frac 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around 0 84.3%

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

      1. associate-*r/ 84.3%

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

      2. mul-1-neg 84.3%

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

   7. Simplified 84.3%

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

   if -6.2999999999999997e-32 < u < -2.00000000000000004e-91

   1. Initial program 64.2%

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

      1. associate-*l/ 65.2%

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

      2. *-commutative 65.2%

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

   3. Simplified 65.2%

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

      1. associate-*r/ 64.2%

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

      2. times-frac 99.7%

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

      3. *-commutative 99.7%

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

      4. frac-2neg 99.7%

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

      5. remove-double-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. sub-neg 99.7%

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

      9. clear-num 99.7%

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

      10. frac-2neg 99.7%

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

      11. frac-times 99.7%

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

      12. *-un-lft-identity 99.7%

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

      13. +-commutative 99.7%

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

      14. distribute-neg-in 99.7%

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

      15. sub-neg 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in u around 0 84.6%

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

      1. *-commutative 84.6%

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

   9. Simplified 84.6%

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

   if -2.00000000000000004e-91 < u < -1.14999999999999993e-102

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. times-frac 77.1%

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

      3. *-commutative 77.1%

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

      4. frac-2neg 77.1%

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

      5. remove-double-neg 77.1%

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

      6. +-commutative 77.1%

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

      7. distribute-neg-in 77.1%

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

      8. sub-neg 77.1%

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

      9. clear-num 76.3%

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

      10. frac-2neg 76.3%

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

      11. frac-times 99.6%

          \[\leadsto \color{blue}{\frac{1 \cdot \left(-v\right)}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)}} \]

      12. *-un-lft-identity 99.6%

          \[\leadsto \frac{\color{blue}{-v}}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\left(t1 + u\right)\right)} \]

      13. +-commutative 99.6%

          \[\leadsto \frac{-v}{\frac{\left(-u\right) - t1}{t1} \cdot \left(-\color{blue}{\left(u + t1\right)}\right)} \]

      14. distribute-neg-in 99.6%

          \[\leadsto \frac{-v}{\frac{\left(-u\right) - t1}{t1} \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

      15. sub-neg 99.6%

          \[\leadsto \frac{-v}{\frac{\left(-u\right) - t1}{t1} \cdot \color{blue}{\left(\left(-u\right) - t1\right)}} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{-v}{\frac{t1 + u}{t1} \cdot \left(t1 + u\right)}} \]

   7. Taylor expanded in t1 around 0 99.6%

      \[\leadsto \frac{-v}{\color{blue}{\frac{u}{t1}} \cdot \left(t1 + u\right)} \]

   if -1.14999999999999993e-102 < u < 3.8000000000000002e-14

   1. Initial program 69.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 80.1%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 80.1%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 84.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 84.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 84.2%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 84.2%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 3.8000000000000002e-14 < u

   1. Initial program 80.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 98.1%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 98.1%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 98.1%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 98.1%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 98.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 98.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around 0 81.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{t1}{u}\right)} \cdot \frac{v}{t1 + u} \]
   6. Step-by-step derivation

      1. associate-*r/ 81.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot t1}{u}} \cdot \frac{v}{t1 + u} \]

      2. mul-1-neg 81.0%

         \[\leadsto \frac{\color{blue}{-t1}}{u} \cdot \frac{v}{t1 + u} \]

   7. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{-t1}{u}} \cdot \frac{v}{t1 + u} \]

   8. Taylor expanded in t1 around 0 81.2%

      \[\leadsto \frac{-t1}{u} \cdot \color{blue}{\frac{v}{u}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq -2 \cdot 10^{-91}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;u \leq -1.15 \cdot 10^{-102}:\\ \;\;\;\;\frac{-v}{\left(t1 + u\right) \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{-155} \lor \neg \left(t1 \leq 3.2 \cdot 10^{-149}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.4e-155) (not (<= t1 3.2e-149)))
   (/ v (- (- u) t1))
   (/ v (* t1 (/ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.4e-155) || !(t1 <= 3.2e-149)) {
		tmp = v / (-u - t1);
	} else {
		tmp = v / (t1 * (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-7.4d-155)) .or. (.not. (t1 <= 3.2d-149))) then
        tmp = v / (-u - t1)
    else
        tmp = v / (t1 * (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.4e-155) || !(t1 <= 3.2e-149)) {
		tmp = v / (-u - t1);
	} else {
		tmp = v / (t1 * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7.4e-155) or not (t1 <= 3.2e-149):
		tmp = v / (-u - t1)
	else:
		tmp = v / (t1 * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.4e-155) || !(t1 <= 3.2e-149))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(v / Float64(t1 * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -7.4e-155) || ~((t1 <= 3.2e-149)))
		tmp = v / (-u - t1);
	else
		tmp = v / (t1 * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7.4e-155], N[Not[LessEqual[t1, 3.2e-149]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[(t1 * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.4000000000000001e-155 or 3.20000000000000002e-149 < t1

   1. Initial program 72.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 78.8%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 78.8%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 72.5%

         \[\leadsto \color{blue}{\frac{v \cdot \left(-t1\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

      3. *-commutative 98.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      4. frac-2neg 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

      5. +-commutative 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{-\color{blue}{\left(u + t1\right)}} \]

      6. distribute-neg-in 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      7. sub-neg 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) - t1}} \]

      8. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{-t1}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1}} \]

      9. add-sqr-sqrt 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      10. sqrt-unprod 50.3%

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      11. sqr-neg 50.3%

          \[\leadsto \frac{\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      12. sqrt-unprod 15.1%

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      13. add-sqr-sqrt 34.4%

          \[\leadsto \frac{\frac{\color{blue}{t1}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      14. sub-neg 34.4%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      15. +-commutative 34.4%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-t1\right) + \left(-u\right)}} \]

      16. add-sqr-sqrt 19.2%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)} \]

      17. sqrt-unprod 50.2%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)} \]

      18. sqr-neg 50.2%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)} \]

      19. sqrt-unprod 38.6%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)} \]

      20. add-sqr-sqrt 18.9%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{-u} \cdot \sqrt{-u}}} \]

      21. sqrt-unprod 40.1%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}} \]

      22. sqr-neg 40.1%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \sqrt{\color{blue}{u \cdot u}}} \]

      23. sqrt-unprod 23.7%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{u} \cdot \sqrt{u}}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 74.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 74.2%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 74.2%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -7.4000000000000001e-155 < t1 < 3.20000000000000002e-149

   1. Initial program 90.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 92.5%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 92.5%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 92.5%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 92.5%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 92.5%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 92.5%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 92.5%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 30.0%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. clear-num 32.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(-u\right) - t1}{t1}}} \cdot \frac{v}{t1} \]

      2. frac-times 46.5%

         \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{\left(-u\right) - t1}{t1} \cdot t1}} \]

      3. *-un-lft-identity 46.5%

         \[\leadsto \frac{\color{blue}{v}}{\frac{\left(-u\right) - t1}{t1} \cdot t1} \]

      4. sub-neg 46.5%

         \[\leadsto \frac{v}{\frac{\color{blue}{\left(-u\right) + \left(-t1\right)}}{t1} \cdot t1} \]

      5. distribute-neg-in 46.5%

         \[\leadsto \frac{v}{\frac{\color{blue}{-\left(u + t1\right)}}{t1} \cdot t1} \]

      6. +-commutative 46.5%

         \[\leadsto \frac{v}{\frac{-\color{blue}{\left(t1 + u\right)}}{t1} \cdot t1} \]

      7. add-sqr-sqrt 29.6%

         \[\leadsto \frac{v}{\frac{-\left(t1 + u\right)}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}} \cdot t1} \]

      8. sqrt-unprod 53.2%

         \[\leadsto \frac{v}{\frac{-\left(t1 + u\right)}{\color{blue}{\sqrt{t1 \cdot t1}}} \cdot t1} \]

      9. sqr-neg 53.2%

         \[\leadsto \frac{v}{\frac{-\left(t1 + u\right)}{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}} \cdot t1} \]

      10. sqrt-unprod 17.1%

          \[\leadsto \frac{v}{\frac{-\left(t1 + u\right)}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}} \cdot t1} \]

      11. add-sqr-sqrt 42.6%

          \[\leadsto \frac{v}{\frac{-\left(t1 + u\right)}{\color{blue}{-t1}} \cdot t1} \]

      12. frac-2neg 42.6%

          \[\leadsto \frac{v}{\color{blue}{\frac{t1 + u}{t1}} \cdot t1} \]

   7. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{\frac{v}{\frac{t1 + u}{t1} \cdot t1}} \]

   8. Taylor expanded in t1 around 0 42.9%

      \[\leadsto \frac{v}{\color{blue}{\frac{u}{t1}} \cdot t1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{-155} \lor \neg \left(t1 \leq 3.2 \cdot 10^{-149}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-155} \lor \neg \left(t1 \leq 1.55 \cdot 10^{-144}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.6e-155) (not (<= t1 1.55e-144)))
   (/ v (- (- u) t1))
   (/ (* v (/ t1 u)) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-155) || !(t1 <= 1.55e-144)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v * (t1 / u)) / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.6d-155)) .or. (.not. (t1 <= 1.55d-144))) then
        tmp = v / (-u - t1)
    else
        tmp = (v * (t1 / u)) / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-155) || !(t1 <= 1.55e-144)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v * (t1 / u)) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.6e-155) or not (t1 <= 1.55e-144):
		tmp = v / (-u - t1)
	else:
		tmp = (v * (t1 / u)) / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.6e-155) || !(t1 <= 1.55e-144))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.6e-155) || ~((t1 <= 1.55e-144)))
		tmp = v / (-u - t1);
	else
		tmp = (v * (t1 / u)) / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.6e-155], N[Not[LessEqual[t1, 1.55e-144]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.60000000000000006e-155 or 1.55e-144 < t1

   1. Initial program 72.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 78.7%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 78.7%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 78.7%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 72.4%

         \[\leadsto \color{blue}{\frac{v \cdot \left(-t1\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      2. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

      3. *-commutative 98.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      4. frac-2neg 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

      5. +-commutative 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{-\color{blue}{\left(u + t1\right)}} \]

      6. distribute-neg-in 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      7. sub-neg 98.8%

         \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) - t1}} \]

      8. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{-t1}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1}} \]

      9. add-sqr-sqrt 54.1%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      10. sqrt-unprod 50.1%

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      11. sqr-neg 50.1%

          \[\leadsto \frac{\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      12. sqrt-unprod 14.7%

          \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      13. add-sqr-sqrt 34.0%

          \[\leadsto \frac{\frac{\color{blue}{t1}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

      14. sub-neg 34.0%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

      15. +-commutative 34.0%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-t1\right) + \left(-u\right)}} \]

      16. add-sqr-sqrt 19.3%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)} \]

      17. sqrt-unprod 49.9%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)} \]

      18. sqr-neg 49.9%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)} \]

      19. sqrt-unprod 38.2%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)} \]

      20. add-sqr-sqrt 18.4%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{-u} \cdot \sqrt{-u}}} \]

      21. sqrt-unprod 39.8%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}} \]

      22. sqr-neg 39.8%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \sqrt{\color{blue}{u \cdot u}}} \]

      23. sqrt-unprod 23.8%

          \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{u} \cdot \sqrt{u}}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{t1 + u}} \]

   7. Taylor expanded in t1 around inf 74.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
   8. Step-by-step derivation

      1. mul-1-neg 74.6%

         \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   9. Simplified 74.6%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

   if -1.60000000000000006e-155 < t1 < 1.55e-144

   1. Initial program 90.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 92.7%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 92.7%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 92.7%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 92.7%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 92.7%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 92.7%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 92.7%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 29.6%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 57.5%

         \[\leadsto \color{blue}{\frac{\frac{t1}{\left(-u\right) - t1} \cdot v}{t1}} \]

      2. add-sqr-sqrt 36.6%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{\left(-u\right) - t1} \cdot v}{t1} \]

      3. sqrt-unprod 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1 \cdot t1}}}{\left(-u\right) - t1} \cdot v}{t1} \]

      4. sqr-neg 53.9%

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}}{\left(-u\right) - t1} \cdot v}{t1} \]

      5. sqrt-unprod 21.1%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{\left(-u\right) - t1} \cdot v}{t1} \]

      6. add-sqr-sqrt 53.6%

         \[\leadsto \frac{\frac{\color{blue}{-t1}}{\left(-u\right) - t1} \cdot v}{t1} \]

      7. sub-neg 53.6%

         \[\leadsto \frac{\frac{-t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot v}{t1} \]

      8. distribute-neg-in 53.6%

         \[\leadsto \frac{\frac{-t1}{\color{blue}{-\left(u + t1\right)}} \cdot v}{t1} \]

      9. +-commutative 53.6%

         \[\leadsto \frac{\frac{-t1}{-\color{blue}{\left(t1 + u\right)}} \cdot v}{t1} \]

      10. frac-2neg 53.6%

          \[\leadsto \frac{\color{blue}{\frac{t1}{t1 + u}} \cdot v}{t1} \]

   7. Applied egg-rr 53.6%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot v}{t1}} \]

   8. Taylor expanded in t1 around 0 42.3%

      \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{u}}}{t1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 42.3%

         \[\leadsto \frac{\frac{t1 \cdot v}{\color{blue}{u \cdot 1}}}{t1} \]

      2. times-frac 53.9%

         \[\leadsto \frac{\color{blue}{\frac{t1}{u} \cdot \frac{v}{1}}}{t1} \]

      3. /-rgt-identity 53.9%

         \[\leadsto \frac{\frac{t1}{u} \cdot \color{blue}{v}}{t1} \]

   10. Simplified 53.9%

       \[\leadsto \frac{\color{blue}{\frac{t1}{u} \cdot v}}{t1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-155} \lor \neg \left(t1 \leq 1.55 \cdot 10^{-144}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 22.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{+46} \lor \neg \left(t1 \leq 2.05 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.45e+46) (not (<= t1 2.05e+95))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.45e+46) || !(t1 <= 2.05e+95)) {
		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 <= (-1.45d+46)) .or. (.not. (t1 <= 2.05d+95))) 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 <= -1.45e+46) || !(t1 <= 2.05e+95)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.45e+46) or not (t1 <= 2.05e+95):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.45e+46) || !(t1 <= 2.05e+95))
		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 <= -1.45e+46) || ~((t1 <= 2.05e+95)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.45e+46], N[Not[LessEqual[t1, 2.05e+95]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.4500000000000001e46 or 2.04999999999999993e95 < t1

   1. Initial program 48.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 88.6%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. *-commutative 88.6%

         \[\leadsto \color{blue}{\frac{v}{t1} \cdot \frac{t1}{\left(-u\right) - t1}} \]

      2. clear-num 88.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{t1}{v}}} \cdot \frac{t1}{\left(-u\right) - t1} \]

      3. frac-times 67.5%

         \[\leadsto \color{blue}{\frac{1 \cdot t1}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)}} \]

      4. *-un-lft-identity 67.5%

         \[\leadsto \frac{\color{blue}{t1}}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)} \]

      5. sub-neg 67.5%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

      6. add-sqr-sqrt 39.4%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)\right)} \]

      7. add-sqr-sqrt 26.7%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{-u} \cdot \sqrt{-u} + \color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}\right)} \]

      8. sqrt-unprod 43.9%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      9. sqr-neg 43.9%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{\color{blue}{u \cdot u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      10. sqrt-unprod 18.7%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      11. add-sqr-sqrt 45.4%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{u} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      12. sqrt-unprod 47.9%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}\right)} \]

      13. sqr-neg 47.9%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \sqrt{\color{blue}{t1 \cdot t1}}\right)} \]

      14. sqrt-unprod 12.2%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{t1} \cdot \sqrt{t1}}\right)} \]

      15. add-sqr-sqrt 37.5%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{t1}\right)} \]

      16. +-commutative 37.5%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(t1 + u\right)}} \]

   7. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{\frac{t1}{\frac{t1}{v} \cdot \left(t1 + u\right)}} \]
   8. Step-by-step derivation

      1. associate-/l/ 34.2%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{\frac{t1}{v}}} \]

      2. associate-/r/ 34.2%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{t1} \cdot v} \]

      3. /-rgt-identity 34.2%

         \[\leadsto \frac{\frac{t1}{t1 + u}}{t1} \cdot \color{blue}{\frac{v}{1}} \]

      4. times-frac 34.2%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot v}{t1 \cdot 1}} \]

      5. *-rgt-identity 34.2%

         \[\leadsto \frac{\frac{t1}{t1 + u} \cdot v}{\color{blue}{t1}} \]

      6. associate-*r/ 34.2%

         \[\leadsto \color{blue}{\frac{t1}{t1 + u} \cdot \frac{v}{t1}} \]

      7. associate-*l/ 34.2%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1}}{t1 + u}} \]

      8. associate-/l* 37.5%

         \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

   9. Simplified 37.5%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

   10. Taylor expanded in t1 around inf 33.5%

       \[\leadsto \color{blue}{\frac{v}{t1}} \]

   if -1.4500000000000001e46 < t1 < 2.04999999999999993e95

   1. Initial program 92.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 95.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 95.8%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 95.8%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 95.8%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 95.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 95.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 95.8%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 50.1%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. *-commutative 50.1%

         \[\leadsto \color{blue}{\frac{v}{t1} \cdot \frac{t1}{\left(-u\right) - t1}} \]

      2. clear-num 49.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{t1}{v}}} \cdot \frac{t1}{\left(-u\right) - t1} \]

      3. frac-times 42.4%

         \[\leadsto \color{blue}{\frac{1 \cdot t1}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)}} \]

      4. *-un-lft-identity 42.4%

         \[\leadsto \frac{\color{blue}{t1}}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)} \]

      5. sub-neg 42.4%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

      6. add-sqr-sqrt 18.9%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)\right)} \]

      7. add-sqr-sqrt 8.4%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{-u} \cdot \sqrt{-u} + \color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}\right)} \]

      8. sqrt-unprod 21.7%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      9. sqr-neg 21.7%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{\color{blue}{u \cdot u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      10. sqrt-unprod 9.4%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      11. add-sqr-sqrt 16.9%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{u} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      12. sqrt-unprod 25.0%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}\right)} \]

      13. sqr-neg 25.0%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \sqrt{\color{blue}{t1 \cdot t1}}\right)} \]

      14. sqrt-unprod 8.6%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{t1} \cdot \sqrt{t1}}\right)} \]

      15. add-sqr-sqrt 15.0%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{t1}\right)} \]

      16. +-commutative 15.0%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(t1 + u\right)}} \]

   7. Applied egg-rr 15.0%

      \[\leadsto \color{blue}{\frac{t1}{\frac{t1}{v} \cdot \left(t1 + u\right)}} \]
   8. Step-by-step derivation

      1. associate-/l/ 20.6%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{\frac{t1}{v}}} \]

      2. associate-/r/ 26.9%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{t1} \cdot v} \]

      3. /-rgt-identity 26.9%

         \[\leadsto \frac{\frac{t1}{t1 + u}}{t1} \cdot \color{blue}{\frac{v}{1}} \]

      4. times-frac 36.0%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot v}{t1 \cdot 1}} \]

      5. *-rgt-identity 36.0%

         \[\leadsto \frac{\frac{t1}{t1 + u} \cdot v}{\color{blue}{t1}} \]

      6. associate-*r/ 20.6%

         \[\leadsto \color{blue}{\frac{t1}{t1 + u} \cdot \frac{v}{t1}} \]

      7. associate-*l/ 15.0%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1}}{t1 + u}} \]

      8. associate-/l* 15.0%

         \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

   9. Simplified 15.0%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

   10. Taylor expanded in t1 around 0 15.8%

       \[\leadsto \color{blue}{\frac{v}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{+46} \lor \neg \left(t1 \leq 2.05 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{t1}{t1 + u} \cdot \frac{v}{\left(-u\right) - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (* (/ t1 (+ t1 u)) (/ v (- (- u) t1))))

C

double code(double u, double v, double t1) {
	return (t1 / (t1 + u)) * (v / (-u - t1));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (t1 / (t1 + u)) * (v / (-u - t1))
end function

Java

public static double code(double u, double v, double t1) {
	return (t1 / (t1 + u)) * (v / (-u - t1));
}

Python

def code(u, v, t1):
	return (t1 / (t1 + u)) * (v / (-u - t1))

Julia

function code(u, v, t1)
	return Float64(Float64(t1 / Float64(t1 + u)) * Float64(v / Float64(Float64(-u) - t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (t1 / (t1 + u)) * (v / (-u - t1));
end

Wolfram

code[u_, v_, t1_] := N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. Final simplification 97.2%

   \[\leadsto \frac{t1}{t1 + u} \cdot \frac{v}{\left(-u\right) - t1} \]
6. Add Preprocessing

Alternative 14: 55.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 1.6 \cdot 10^{+160}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u 1.6e+160) (/ v (- t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.6e+160) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 1.6d+160) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.6e+160) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1.6e+160:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1.6e+160)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1.6e+160)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1.6e+160], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 1.5999999999999999e160

   1. Initial program 76.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 80.4%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 80.4%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 57.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 57.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 57.5%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 1.5999999999999999e160 < u

   1. Initial program 78.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 98.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 98.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 98.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 98.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 98.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 98.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 63.4%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
   6. Step-by-step derivation

      1. *-commutative 63.4%

         \[\leadsto \color{blue}{\frac{v}{t1} \cdot \frac{t1}{\left(-u\right) - t1}} \]

      2. clear-num 63.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{t1}{v}}} \cdot \frac{t1}{\left(-u\right) - t1} \]

      3. frac-times 52.4%

         \[\leadsto \color{blue}{\frac{1 \cdot t1}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)}} \]

      4. *-un-lft-identity 52.4%

         \[\leadsto \frac{\color{blue}{t1}}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)} \]

      5. sub-neg 52.4%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)\right)} \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{-u} \cdot \sqrt{-u} + \color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}\right)} \]

      8. sqrt-unprod 31.1%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      9. sqr-neg 31.1%

         \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{\color{blue}{u \cdot u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      10. sqrt-unprod 22.6%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      11. add-sqr-sqrt 22.6%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{u} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

      12. sqrt-unprod 45.4%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}\right)} \]

      13. sqr-neg 45.4%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \sqrt{\color{blue}{t1 \cdot t1}}\right)} \]

      14. sqrt-unprod 26.2%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{t1} \cdot \sqrt{t1}}\right)} \]

      15. add-sqr-sqrt 45.3%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{t1}\right)} \]

      16. +-commutative 45.3%

          \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(t1 + u\right)}} \]

   7. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{\frac{t1}{\frac{t1}{v} \cdot \left(t1 + u\right)}} \]
   8. Step-by-step derivation

      1. associate-/l/ 56.3%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{\frac{t1}{v}}} \]

      2. associate-/r/ 67.0%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{t1} \cdot v} \]

      3. /-rgt-identity 67.0%

         \[\leadsto \frac{\frac{t1}{t1 + u}}{t1} \cdot \color{blue}{\frac{v}{1}} \]

      4. times-frac 67.0%

         \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot v}{t1 \cdot 1}} \]

      5. *-rgt-identity 67.0%

         \[\leadsto \frac{\frac{t1}{t1 + u} \cdot v}{\color{blue}{t1}} \]

      6. associate-*r/ 56.3%

         \[\leadsto \color{blue}{\frac{t1}{t1 + u} \cdot \frac{v}{t1}} \]

      7. associate-*l/ 42.8%

         \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1}}{t1 + u}} \]

      8. associate-/l* 45.3%

         \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

   9. Simplified 45.3%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

   10. Taylor expanded in t1 around 0 43.3%

       \[\leadsto \color{blue}{\frac{v}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.6 \cdot 10^{+160}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 8 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 8e+158) (/ v (- t1)) (/ v (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 8e+158) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 8d+158) then
        tmp = v / -t1
    else
        tmp = v / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 8e+158) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 8e+158:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 8e+158)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 8e+158)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 8e+158], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 7.99999999999999962e158

   1. Initial program 76.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 80.4%

         \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

      2. *-commutative 80.4%

         \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 57.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 57.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 57.5%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 7.99999999999999962e158 < u

   1. Initial program 78.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 98.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 98.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 98.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 98.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 98.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 98.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 63.4%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 43.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 43.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 43.5%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 43.5%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 8 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(-u\right) - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (- u) t1)))

C

double code(double u, double v, double t1) {
	return v / (-u - t1);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / (-u - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / (-u - t1);
}

Python

def code(u, v, t1):
	return v / (-u - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(-u) - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (-u - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-*l/ 80.2%

      \[\leadsto \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

   2. *-commutative 80.2%

      \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

3. Simplified 80.2%

   \[\leadsto \color{blue}{v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 77.0%

      \[\leadsto \color{blue}{\frac{v \cdot \left(-t1\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. times-frac 97.2%

      \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

   3. *-commutative 97.2%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   4. frac-2neg 97.2%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

   5. +-commutative 97.2%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{-\color{blue}{\left(u + t1\right)}} \]

   6. distribute-neg-in 97.2%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   7. sub-neg 97.2%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) - t1}} \]

   8. associate-*r/ 99.0%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1}} \]

   9. add-sqr-sqrt 50.4%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   10. sqrt-unprod 51.7%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   11. sqr-neg 51.7%

       \[\leadsto \frac{\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   12. sqrt-unprod 19.9%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   13. add-sqr-sqrt 39.2%

       \[\leadsto \frac{\frac{\color{blue}{t1}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   14. sub-neg 39.2%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   15. +-commutative 39.2%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-t1\right) + \left(-u\right)}} \]

   16. add-sqr-sqrt 19.3%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)} \]

   17. sqrt-unprod 51.2%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)} \]

   18. sqr-neg 51.2%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)} \]

   19. sqrt-unprod 38.5%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)} \]

   20. add-sqr-sqrt 19.1%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{-u} \cdot \sqrt{-u}}} \]

   21. sqrt-unprod 43.0%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}} \]

   22. sqr-neg 43.0%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \sqrt{\color{blue}{u \cdot u}}} \]

   23. sqrt-unprod 26.1%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{u} \cdot \sqrt{u}}} \]

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{t1 + u}} \]

7. Taylor expanded in t1 around inf 59.8%

   \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
8. Step-by-step derivation

   1. mul-1-neg 59.8%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

9. Simplified 59.8%

   \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

10. Final simplification 59.8%

    \[\leadsto \frac{v}{\left(-u\right) - t1} \]
11. Add Preprocessing

Alternative 17: 60.8% 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 77.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. Taylor expanded in t1 around inf 63.3%

   \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
6. Step-by-step derivation

   1. *-commutative 63.3%

      \[\leadsto \color{blue}{\frac{v}{t1} \cdot \frac{t1}{\left(-u\right) - t1}} \]

   2. clear-num 63.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{t1}{v}}} \cdot \frac{t1}{\left(-u\right) - t1} \]

   3. frac-times 51.1%

      \[\leadsto \color{blue}{\frac{1 \cdot t1}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)}} \]

   4. *-un-lft-identity 51.1%

      \[\leadsto \frac{\color{blue}{t1}}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)} \]

   5. sub-neg 51.1%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

   6. add-sqr-sqrt 26.0%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)\right)} \]

   7. add-sqr-sqrt 14.7%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{-u} \cdot \sqrt{-u} + \color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}\right)} \]

   8. sqrt-unprod 29.3%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

   9. sqr-neg 29.3%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{\color{blue}{u \cdot u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

   10. sqrt-unprod 12.6%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

   11. add-sqr-sqrt 26.7%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{u} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

   12. sqrt-unprod 32.9%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}\right)} \]

   13. sqr-neg 32.9%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \sqrt{\color{blue}{t1 \cdot t1}}\right)} \]

   14. sqrt-unprod 9.8%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{t1} \cdot \sqrt{t1}}\right)} \]

   15. add-sqr-sqrt 22.8%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{t1}\right)} \]

   16. +-commutative 22.8%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(t1 + u\right)}} \]

7. Applied egg-rr 22.8%

   \[\leadsto \color{blue}{\frac{t1}{\frac{t1}{v} \cdot \left(t1 + u\right)}} \]
8. Step-by-step derivation

   1. associate-/l/ 25.3%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{\frac{t1}{v}}} \]

   2. associate-/r/ 29.4%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{t1} \cdot v} \]

   3. /-rgt-identity 29.4%

      \[\leadsto \frac{\frac{t1}{t1 + u}}{t1} \cdot \color{blue}{\frac{v}{1}} \]

   4. times-frac 35.4%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot v}{t1 \cdot 1}} \]

   5. *-rgt-identity 35.4%

      \[\leadsto \frac{\frac{t1}{t1 + u} \cdot v}{\color{blue}{t1}} \]

   6. associate-*r/ 25.3%

      \[\leadsto \color{blue}{\frac{t1}{t1 + u} \cdot \frac{v}{t1}} \]

   7. associate-*l/ 21.6%

      \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1}}{t1 + u}} \]

   8. associate-/l* 22.8%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

9. Simplified 22.8%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]
10. Step-by-step derivation

    1. frac-2neg 22.8%

       \[\leadsto t1 \cdot \color{blue}{\frac{-\frac{v}{t1}}{-\left(t1 + u\right)}} \]

    2. associate-*r/ 21.6%

       \[\leadsto \color{blue}{\frac{t1 \cdot \left(-\frac{v}{t1}\right)}{-\left(t1 + u\right)}} \]

    3. distribute-neg-frac 21.6%

       \[\leadsto \frac{t1 \cdot \color{blue}{\frac{-v}{t1}}}{-\left(t1 + u\right)} \]

    4. add-sqr-sqrt 12.0%

       \[\leadsto \frac{t1 \cdot \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1}}{-\left(t1 + u\right)} \]

    5. sqrt-unprod 37.1%

       \[\leadsto \frac{t1 \cdot \frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1}}{-\left(t1 + u\right)} \]

    6. sqr-neg 37.1%

       \[\leadsto \frac{t1 \cdot \frac{\sqrt{\color{blue}{v \cdot v}}}{t1}}{-\left(t1 + u\right)} \]

    7. sqrt-unprod 30.2%

       \[\leadsto \frac{t1 \cdot \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1}}{-\left(t1 + u\right)} \]

    8. add-sqr-sqrt 59.7%

       \[\leadsto \frac{t1 \cdot \frac{\color{blue}{v}}{t1}}{-\left(t1 + u\right)} \]

    9. associate-*r/ 47.0%

       \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{t1}}}{-\left(t1 + u\right)} \]

    10. +-commutative 47.0%

        \[\leadsto \frac{\frac{t1 \cdot v}{t1}}{-\color{blue}{\left(u + t1\right)}} \]

    11. distribute-neg-in 47.0%

        \[\leadsto \frac{\frac{t1 \cdot v}{t1}}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

    12. add-sqr-sqrt 24.0%

        \[\leadsto \frac{\frac{t1 \cdot v}{t1}}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)} \]

    13. sqrt-unprod 56.1%

        \[\leadsto \frac{\frac{t1 \cdot v}{t1}}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \left(-t1\right)} \]

    14. sqr-neg 56.1%

        \[\leadsto \frac{\frac{t1 \cdot v}{t1}}{\sqrt{\color{blue}{u \cdot u}} + \left(-t1\right)} \]

    15. sqrt-unprod 23.1%

        \[\leadsto \frac{\frac{t1 \cdot v}{t1}}{\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \left(-t1\right)} \]

    16. add-sqr-sqrt 46.6%

        \[\leadsto \frac{\frac{t1 \cdot v}{t1}}{\color{blue}{u} + \left(-t1\right)} \]

11. Applied egg-rr 46.6%

    \[\leadsto \color{blue}{\frac{\frac{t1 \cdot v}{t1}}{u + \left(-t1\right)}} \]
12. Step-by-step derivation

    1. *-lft-identity 46.6%

       \[\leadsto \frac{\frac{t1 \cdot v}{\color{blue}{1 \cdot t1}}}{u + \left(-t1\right)} \]

    2. *-commutative 46.6%

       \[\leadsto \frac{\frac{\color{blue}{v \cdot t1}}{1 \cdot t1}}{u + \left(-t1\right)} \]

    3. times-frac 59.3%

       \[\leadsto \frac{\color{blue}{\frac{v}{1} \cdot \frac{t1}{t1}}}{u + \left(-t1\right)} \]

    4. /-rgt-identity 59.3%

       \[\leadsto \frac{\color{blue}{v} \cdot \frac{t1}{t1}}{u + \left(-t1\right)} \]

    5. *-inverses 59.3%

       \[\leadsto \frac{v \cdot \color{blue}{1}}{u + \left(-t1\right)} \]

    6. *-rgt-identity 59.3%

       \[\leadsto \frac{\color{blue}{v}}{u + \left(-t1\right)} \]

    7. sub-neg 59.3%

       \[\leadsto \frac{v}{\color{blue}{u - t1}} \]

13. Simplified 59.3%

    \[\leadsto \color{blue}{\frac{v}{u - t1}} \]

14. Final simplification 59.3%

    \[\leadsto \frac{v}{u - t1} \]
15. Add Preprocessing

Alternative 18: 13.4% 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 77.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. Taylor expanded in t1 around inf 63.3%

   \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]
6. Step-by-step derivation

   1. *-commutative 63.3%

      \[\leadsto \color{blue}{\frac{v}{t1} \cdot \frac{t1}{\left(-u\right) - t1}} \]

   2. clear-num 63.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{t1}{v}}} \cdot \frac{t1}{\left(-u\right) - t1} \]

   3. frac-times 51.1%

      \[\leadsto \color{blue}{\frac{1 \cdot t1}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)}} \]

   4. *-un-lft-identity 51.1%

      \[\leadsto \frac{\color{blue}{t1}}{\frac{t1}{v} \cdot \left(\left(-u\right) - t1\right)} \]

   5. sub-neg 51.1%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(\left(-u\right) + \left(-t1\right)\right)}} \]

   6. add-sqr-sqrt 26.0%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{-u} \cdot \sqrt{-u}} + \left(-t1\right)\right)} \]

   7. add-sqr-sqrt 14.7%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{-u} \cdot \sqrt{-u} + \color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}\right)} \]

   8. sqrt-unprod 29.3%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

   9. sqr-neg 29.3%

      \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\sqrt{\color{blue}{u \cdot u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

   10. sqrt-unprod 12.6%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{\sqrt{u} \cdot \sqrt{u}} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

   11. add-sqr-sqrt 26.7%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(\color{blue}{u} + \sqrt{-t1} \cdot \sqrt{-t1}\right)} \]

   12. sqrt-unprod 32.9%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}\right)} \]

   13. sqr-neg 32.9%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \sqrt{\color{blue}{t1 \cdot t1}}\right)} \]

   14. sqrt-unprod 9.8%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{\sqrt{t1} \cdot \sqrt{t1}}\right)} \]

   15. add-sqr-sqrt 22.8%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \left(u + \color{blue}{t1}\right)} \]

   16. +-commutative 22.8%

       \[\leadsto \frac{t1}{\frac{t1}{v} \cdot \color{blue}{\left(t1 + u\right)}} \]

7. Applied egg-rr 22.8%

   \[\leadsto \color{blue}{\frac{t1}{\frac{t1}{v} \cdot \left(t1 + u\right)}} \]
8. Step-by-step derivation

   1. associate-/l/ 25.3%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{\frac{t1}{v}}} \]

   2. associate-/r/ 29.4%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u}}{t1} \cdot v} \]

   3. /-rgt-identity 29.4%

      \[\leadsto \frac{\frac{t1}{t1 + u}}{t1} \cdot \color{blue}{\frac{v}{1}} \]

   4. times-frac 35.4%

      \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot v}{t1 \cdot 1}} \]

   5. *-rgt-identity 35.4%

      \[\leadsto \frac{\frac{t1}{t1 + u} \cdot v}{\color{blue}{t1}} \]

   6. associate-*r/ 25.3%

      \[\leadsto \color{blue}{\frac{t1}{t1 + u} \cdot \frac{v}{t1}} \]

   7. associate-*l/ 21.6%

      \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{t1}}{t1 + u}} \]

   8. associate-/l* 22.8%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

9. Simplified 22.8%

   \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1}}{t1 + u}} \]

10. Taylor expanded in t1 around inf 13.8%

    \[\leadsto \color{blue}{\frac{v}{t1}} \]

11. Final simplification 13.8%

    \[\leadsto \frac{v}{t1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))