Rosa's DopplerBench

Percentage Accurate: 72.9% → 98.1%

Time: 17.3s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% 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

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

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(t1 * Float64(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 * N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

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

   1. associate-/l* 71.4%

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

3. Simplified 71.4%

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

   1. associate-*r/ 70.8%

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

   2. times-frac 98.8%

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

   3. frac-2neg 98.8%

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

   4. +-commutative 98.8%

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

   5. distribute-neg-in 98.8%

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

   6. sub-neg 98.8%

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

   7. associate-*r/ 98.8%

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

   8. add-sqr-sqrt 54.7%

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

   9. sqrt-unprod 48.9%

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

   10. sqr-neg 48.9%

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

   11. sqrt-unprod 17.4%

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

   12. add-sqr-sqrt 34.8%

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

   13. sub-neg 34.8%

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

   14. +-commutative 34.8%

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

   15. add-sqr-sqrt 17.5%

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

   16. sqrt-unprod 44.3%

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

   17. sqr-neg 44.3%

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

   18. sqrt-unprod 32.7%

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

   19. add-sqr-sqrt 17.3%

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

   20. sqrt-unprod 36.1%

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

   21. sqr-neg 36.1%

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

   22. sqrt-unprod 21.4%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in v around 0 80.7%

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

   1. mul-1-neg 80.7%

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

   2. associate-/l* 99.9%

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

   3. distribute-lft-neg-out 99.9%

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

   4. *-commutative 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) - u\\ t_2 := t1 \cdot \frac{v}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+97}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq -1.86 \cdot 10^{-149}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 1.72 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- t1) u)) (t_2 (* t1 (/ v (* (+ t1 u) t_1)))))
   (if (<= t1 -1.22e+97)
     (/ -1.0 (/ (+ t1 u) v))
     (if (<= t1 -1.86e-149)
       t_2
       (if (<= t1 1.72e-125)
         (/ (* t1 (/ v (- u))) u)
         (if (<= t1 1.75e+129) t_2 (/ v t_1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double t_2 = t1 * (v / ((t1 + u) * t_1));
	double tmp;
	if (t1 <= -1.22e+97) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= -1.86e-149) {
		tmp = t_2;
	} else if (t1 <= 1.72e-125) {
		tmp = (t1 * (v / -u)) / u;
	} else if (t1 <= 1.75e+129) {
		tmp = t_2;
	} else {
		tmp = v / 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 = -t1 - u
    t_2 = t1 * (v / ((t1 + u) * t_1))
    if (t1 <= (-1.22d+97)) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else if (t1 <= (-1.86d-149)) then
        tmp = t_2
    else if (t1 <= 1.72d-125) then
        tmp = (t1 * (v / -u)) / u
    else if (t1 <= 1.75d+129) then
        tmp = t_2
    else
        tmp = v / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 - u;
	double t_2 = t1 * (v / ((t1 + u) * t_1));
	double tmp;
	if (t1 <= -1.22e+97) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= -1.86e-149) {
		tmp = t_2;
	} else if (t1 <= 1.72e-125) {
		tmp = (t1 * (v / -u)) / u;
	} else if (t1 <= 1.75e+129) {
		tmp = t_2;
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 - u
	t_2 = t1 * (v / ((t1 + u) * t_1))
	tmp = 0
	if t1 <= -1.22e+97:
		tmp = -1.0 / ((t1 + u) / v)
	elif t1 <= -1.86e-149:
		tmp = t_2
	elif t1 <= 1.72e-125:
		tmp = (t1 * (v / -u)) / u
	elif t1 <= 1.75e+129:
		tmp = t_2
	else:
		tmp = v / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) - u)
	t_2 = Float64(t1 * Float64(v / Float64(Float64(t1 + u) * t_1)))
	tmp = 0.0
	if (t1 <= -1.22e+97)
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	elseif (t1 <= -1.86e-149)
		tmp = t_2;
	elseif (t1 <= 1.72e-125)
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	elseif (t1 <= 1.75e+129)
		tmp = t_2;
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 - u;
	t_2 = t1 * (v / ((t1 + u) * t_1));
	tmp = 0.0;
	if (t1 <= -1.22e+97)
		tmp = -1.0 / ((t1 + u) / v);
	elseif (t1 <= -1.86e-149)
		tmp = t_2;
	elseif (t1 <= 1.72e-125)
		tmp = (t1 * (v / -u)) / u;
	elseif (t1 <= 1.75e+129)
		tmp = t_2;
	else
		tmp = v / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) - u), $MachinePrecision]}, Block[{t$95$2 = N[(t1 * N[(v / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.22e+97], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.86e-149], t$95$2, If[LessEqual[t1, 1.72e-125], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[t1, 1.75e+129], t$95$2, N[(v / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.21999999999999997e97

   1. Initial program 48.7%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t1 around inf 92.7%

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

   if -1.21999999999999997e97 < t1 < -1.8600000000000001e-149 or 1.72000000000000013e-125 < t1 < 1.7499999999999999e129

   1. Initial program 85.5%

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

      1. associate-/l* 88.7%

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

   3. Simplified 88.7%

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

   if -1.8600000000000001e-149 < t1 < 1.72000000000000013e-125

   1. Initial program 80.7%

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

      1. associate-/l* 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 80.7%

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

      2. times-frac 94.8%

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

      3. frac-2neg 94.8%

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

      4. +-commutative 94.8%

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

      5. distribute-neg-in 94.8%

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

      6. sub-neg 94.8%

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

      7. associate-*r/ 94.8%

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

      8. add-sqr-sqrt 48.1%

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

      9. sqrt-unprod 43.5%

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

      10. sqr-neg 43.5%

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

      11. sqrt-unprod 20.2%

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

      12. add-sqr-sqrt 38.8%

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

      13. sub-neg 38.8%

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

      14. +-commutative 38.8%

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

      15. add-sqr-sqrt 18.6%

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

      16. sqrt-unprod 40.7%

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

      17. sqr-neg 40.7%

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

      18. sqrt-unprod 23.8%

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

      19. add-sqr-sqrt 11.1%

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

      20. sqrt-unprod 31.0%

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

      21. sqr-neg 31.0%

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

      22. sqrt-unprod 23.1%

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

   6. Applied egg-rr 94.8%

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

   7. Taylor expanded in t1 around 0 80.8%

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

      1. mul-1-neg 80.8%

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

      2. associate-/l* 89.5%

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

      3. distribute-lft-neg-in 89.5%

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

   9. Simplified 89.5%

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

   10. Taylor expanded in t1 around 0 93.1%

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

   if 1.7499999999999999e129 < t1

   1. Initial program 51.4%

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

      1. associate-/l* 54.9%

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

   3. Simplified 54.9%

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

      1. associate-*r/ 51.4%

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

      2. times-frac 99.9%

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

      3. frac-2neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. sub-neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 11.8%

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

      10. sqr-neg 11.8%

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

      11. sqrt-unprod 45.8%

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

      12. add-sqr-sqrt 45.8%

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

      13. sub-neg 45.8%

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

      14. +-commutative 45.8%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 56.9%

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

      17. sqr-neg 56.9%

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

      18. sqrt-unprod 87.5%

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

      19. add-sqr-sqrt 42.6%

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

      20. sqrt-unprod 85.3%

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

      21. sqr-neg 85.3%

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

      22. sqrt-unprod 52.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 86.0%

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

      1. mul-1-neg 86.0%

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

   9. Simplified 86.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+97}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq -1.86 \cdot 10^{-149}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{elif}\;t1 \leq 1.72 \cdot 10^{-125}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{+129}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+97}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.22e+97)
   (/ -1.0 (/ (+ t1 u) v))
   (if (<= t1 1.6e+188)
     (* (- t1) (/ (/ v (+ t1 u)) (+ t1 u)))
     (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.22e+97) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.6e+188) {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.22d+97)) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else if (t1 <= 1.6d+188) then
        tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.22e+97) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.6e+188) {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.22e+97:
		tmp = -1.0 / ((t1 + u) / v)
	elif t1 <= 1.6e+188:
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.22e+97)
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	elseif (t1 <= 1.6e+188)
		tmp = Float64(Float64(-t1) * Float64(Float64(v / Float64(t1 + u)) / Float64(t1 + u)));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.22e+97)
		tmp = -1.0 / ((t1 + u) / v);
	elseif (t1 <= 1.6e+188)
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.22e+97], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.6e+188], N[((-t1) * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.21999999999999997e97

   1. Initial program 48.7%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in t1 around inf 92.7%

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

   if -1.21999999999999997e97 < t1 < 1.59999999999999985e188

   1. Initial program 80.4%

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

      1. associate-/l* 81.4%

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

   3. Simplified 81.4%

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

      1. associate-/r* 92.6%

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

      2. div-inv 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. associate-*r/ 92.6%

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

      2. *-rgt-identity 92.6%

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

   8. Simplified 92.6%

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

   if 1.59999999999999985e188 < t1

   1. Initial program 45.5%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. unpow-1 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in t1 around inf 91.0%

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

       1. add-sqr-sqrt 65.9%

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

       2. sqrt-unprod 63.4%

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

       3. clear-num 63.5%

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

       4. mul-1-neg 63.5%

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

       5. clear-num 63.5%

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

       6. mul-1-neg 63.5%

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

       7. sqr-neg 63.5%

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

       8. sqrt-unprod 45.9%

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

       9. add-sqr-sqrt 46.7%

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

       10. frac-2neg 46.7%

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

       11. distribute-neg-in 46.7%

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

       12. add-sqr-sqrt 0.0%

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

       13. sqrt-unprod 47.6%

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

       14. sqr-neg 47.6%

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

       15. sqrt-unprod 91.9%

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

       16. add-sqr-sqrt 92.3%

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

       17. sub-neg 92.3%

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

   11. Applied egg-rr 92.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+97}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{-30} \lor \neg \left(u \leq 3.2 \cdot 10^{+109}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 - u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{\left(-t1\right) - u}}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.2e-30) (not (<= u 3.2e+109)))
   (/ (/ t1 (/ (- t1 u) v)) u)
   (/ (* t1 (/ v (- (- t1) u))) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.2e-30) || !(u <= 3.2e+109)) {
		tmp = (t1 / ((t1 - u) / v)) / u;
	} else {
		tmp = (t1 * (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 ((u <= (-7.2d-30)) .or. (.not. (u <= 3.2d+109))) then
        tmp = (t1 / ((t1 - u) / v)) / u
    else
        tmp = (t1 * (v / (-t1 - u))) / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.2e-30) || !(u <= 3.2e+109)) {
		tmp = (t1 / ((t1 - u) / v)) / u;
	} else {
		tmp = (t1 * (v / (-t1 - u))) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.2e-30) or not (u <= 3.2e+109):
		tmp = (t1 / ((t1 - u) / v)) / u
	else:
		tmp = (t1 * (v / (-t1 - u))) / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.2e-30) || !(u <= 3.2e+109))
		tmp = Float64(Float64(t1 / Float64(Float64(t1 - u) / v)) / u);
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(Float64(-t1) - u))) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.2e-30) || ~((u <= 3.2e+109)))
		tmp = (t1 / ((t1 - u) / v)) / u;
	else
		tmp = (t1 * (v / (-t1 - u))) / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.2e-30], N[Not[LessEqual[u, 3.2e+109]], $MachinePrecision]], N[(N[(t1 / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(N[(t1 * N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.2000000000000006e-30 or 3.2000000000000001e109 < u

   1. Initial program 73.7%

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

      1. associate-/l* 72.6%

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

   3. Simplified 72.6%

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

      1. associate-*r/ 73.7%

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

      2. times-frac 97.2%

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

      3. frac-2neg 97.2%

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

      4. +-commutative 97.2%

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

      5. distribute-neg-in 97.2%

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

      6. sub-neg 97.2%

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

      7. associate-*r/ 97.3%

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

      8. add-sqr-sqrt 53.9%

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

      9. sqrt-unprod 62.6%

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

      10. sqr-neg 62.6%

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

      11. sqrt-unprod 28.0%

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

      12. add-sqr-sqrt 57.1%

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

      13. sub-neg 57.1%

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

      14. +-commutative 57.1%

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

      15. add-sqr-sqrt 29.1%

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

      16. sqrt-unprod 57.4%

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

      17. sqr-neg 57.4%

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

      18. sqrt-unprod 31.7%

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

      19. add-sqr-sqrt 17.8%

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

      20. sqrt-unprod 32.6%

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

      21. sqr-neg 32.6%

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

      22. sqrt-unprod 19.1%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in v around 0 86.5%

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

      1. mul-1-neg 86.5%

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

      2. associate-/l* 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in t1 around 0 86.8%

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

       1. clear-num 87.4%

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

       2. frac-2neg 87.4%

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

       3. metadata-eval 87.4%

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

       4. associate-*l/ 87.5%

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

       5. add-sqr-sqrt 46.2%

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

       6. sqrt-unprod 60.8%

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

       7. sqr-neg 60.8%

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

       8. sqrt-unprod 28.0%

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

       9. add-sqr-sqrt 57.2%

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

       10. neg-mul-1 57.2%

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

       11. add-sqr-sqrt 29.3%

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

       12. sqrt-unprod 53.2%

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

       13. sqr-neg 53.2%

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

       14. sqrt-unprod 41.1%

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

       15. add-sqr-sqrt 87.5%

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

       16. distribute-neg-frac 87.5%

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

       17. distribute-neg-in 87.5%

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

       18. add-sqr-sqrt 46.3%

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

       19. sqrt-unprod 83.2%

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

       20. sqr-neg 83.2%

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

       21. sqrt-unprod 41.1%

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

       22. add-sqr-sqrt 87.3%

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

       23. sub-neg 87.3%

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

   12. Applied egg-rr 87.3%

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

   if -7.2000000000000006e-30 < u < 3.2000000000000001e109

   1. Initial program 68.8%

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

      1. associate-/l* 70.5%

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

   3. Simplified 70.5%

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

      1. associate-*r/ 68.8%

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

      2. times-frac 99.9%

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

      3. frac-2neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. sub-neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. add-sqr-sqrt 55.3%

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

      9. sqrt-unprod 38.7%

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

      10. sqr-neg 38.7%

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

      11. sqrt-unprod 9.5%

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

      12. add-sqr-sqrt 18.3%

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

      13. sub-neg 18.3%

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

      14. +-commutative 18.3%

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

      15. add-sqr-sqrt 8.8%

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

      16. sqrt-unprod 34.7%

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

      17. sqr-neg 34.7%

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

      18. sqrt-unprod 33.4%

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

      19. add-sqr-sqrt 17.0%

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

      20. sqrt-unprod 38.7%

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

      21. sqr-neg 38.7%

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

      22. sqrt-unprod 23.0%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 76.4%

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

      1. mul-1-neg 76.4%

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

      2. associate-/l* 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in t1 around inf 82.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{-30} \lor \neg \left(u \leq 3.2 \cdot 10^{+109}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 - u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{\left(-t1\right) - u}}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+110}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{\left(-t1\right) - u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 - u}{v}}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.5e-29)
   (/ (* t1 (/ v (+ t1 u))) (- u))
   (if (<= u 1.15e+110)
     (/ (* t1 (/ v (- (- t1) u))) t1)
     (/ (/ t1 (/ (- t1 u) v)) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.5e-29) {
		tmp = (t1 * (v / (t1 + u))) / -u;
	} else if (u <= 1.15e+110) {
		tmp = (t1 * (v / (-t1 - u))) / t1;
	} else {
		tmp = (t1 / ((t1 - u) / 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 <= (-1.5d-29)) then
        tmp = (t1 * (v / (t1 + u))) / -u
    else if (u <= 1.15d+110) then
        tmp = (t1 * (v / (-t1 - u))) / t1
    else
        tmp = (t1 / ((t1 - u) / v)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.5e-29) {
		tmp = (t1 * (v / (t1 + u))) / -u;
	} else if (u <= 1.15e+110) {
		tmp = (t1 * (v / (-t1 - u))) / t1;
	} else {
		tmp = (t1 / ((t1 - u) / v)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.5e-29:
		tmp = (t1 * (v / (t1 + u))) / -u
	elif u <= 1.15e+110:
		tmp = (t1 * (v / (-t1 - u))) / t1
	else:
		tmp = (t1 / ((t1 - u) / v)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.5e-29)
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 + u))) / Float64(-u));
	elseif (u <= 1.15e+110)
		tmp = Float64(Float64(t1 * Float64(v / Float64(Float64(-t1) - u))) / t1);
	else
		tmp = Float64(Float64(t1 / Float64(Float64(t1 - u) / v)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.5e-29)
		tmp = (t1 * (v / (t1 + u))) / -u;
	elseif (u <= 1.15e+110)
		tmp = (t1 * (v / (-t1 - u))) / t1;
	else
		tmp = (t1 / ((t1 - u) / v)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.5e-29], N[(N[(t1 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, 1.15e+110], N[(N[(t1 * N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], N[(N[(t1 / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.5000000000000001e-29

   1. Initial program 75.0%

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

      1. associate-/l* 75.5%

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

   3. Simplified 75.5%

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

      1. associate-*r/ 75.0%

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

      2. times-frac 95.3%

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

      3. frac-2neg 95.3%

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

      4. +-commutative 95.3%

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

      5. distribute-neg-in 95.3%

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

      6. sub-neg 95.3%

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

      7. associate-*r/ 95.4%

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

      8. add-sqr-sqrt 53.0%

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

      9. sqrt-unprod 64.5%

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

      10. sqr-neg 64.5%

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

      11. sqrt-unprod 29.7%

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

      12. add-sqr-sqrt 55.1%

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

      13. sub-neg 55.1%

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

      14. +-commutative 55.1%

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

      15. add-sqr-sqrt 25.4%

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

      16. sqrt-unprod 55.3%

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

      17. sqr-neg 55.3%

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

      18. sqrt-unprod 31.2%

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

      19. add-sqr-sqrt 31.2%

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

      20. sqrt-unprod 31.3%

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

      21. sqr-neg 31.3%

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

      22. sqrt-unprod 0.0%

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

   6. Applied egg-rr 95.4%

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

   7. Taylor expanded in v around 0 88.9%

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

      1. mul-1-neg 88.9%

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

      2. associate-/l* 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in t1 around 0 89.3%

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

   if -1.5000000000000001e-29 < u < 1.15e110

   1. Initial program 68.8%

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

      1. associate-/l* 70.5%

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

   3. Simplified 70.5%

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

      1. associate-*r/ 68.8%

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

      2. times-frac 99.9%

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

      3. frac-2neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. sub-neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. add-sqr-sqrt 55.3%

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

      9. sqrt-unprod 38.7%

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

      10. sqr-neg 38.7%

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

      11. sqrt-unprod 9.5%

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

      12. add-sqr-sqrt 18.3%

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

      13. sub-neg 18.3%

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

      14. +-commutative 18.3%

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

      15. add-sqr-sqrt 8.8%

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

      16. sqrt-unprod 34.7%

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

      17. sqr-neg 34.7%

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

      18. sqrt-unprod 33.4%

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

      19. add-sqr-sqrt 17.0%

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

      20. sqrt-unprod 38.7%

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

      21. sqr-neg 38.7%

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

      22. sqrt-unprod 23.0%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 76.4%

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

      1. mul-1-neg 76.4%

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

      2. associate-/l* 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in t1 around inf 82.4%

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

   if 1.15e110 < u

   1. Initial program 71.9%

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

      1. associate-/l* 68.7%

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

   3. Simplified 68.7%

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

      1. associate-*r/ 71.9%

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

      2. times-frac 99.8%

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

      3. frac-2neg 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. sub-neg 99.8%

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

      7. associate-*r/ 99.8%

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

      8. add-sqr-sqrt 55.1%

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

      9. sqrt-unprod 60.3%

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

      10. sqr-neg 60.3%

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

      11. sqrt-unprod 25.8%

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

      12. add-sqr-sqrt 59.8%

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

      13. sub-neg 59.8%

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

      14. +-commutative 59.8%

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

      15. add-sqr-sqrt 34.0%

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

      16. sqrt-unprod 60.2%

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

      17. sqr-neg 60.2%

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

      18. sqrt-unprod 32.4%

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

      19. add-sqr-sqrt 0.0%

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

      20. sqrt-unprod 34.2%

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

      21. sqr-neg 34.2%

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

      22. sqrt-unprod 44.4%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 83.3%

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

      1. mul-1-neg 83.3%

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

      2. associate-/l* 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in t1 around 0 83.6%

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

       1. clear-num 85.1%

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

       2. frac-2neg 85.1%

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

       3. metadata-eval 85.1%

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

       4. associate-*l/ 85.1%

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

       5. add-sqr-sqrt 47.0%

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

       6. sqrt-unprod 58.0%

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

       7. sqr-neg 58.0%

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

       8. sqrt-unprod 25.8%

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

       9. add-sqr-sqrt 60.3%

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

       10. neg-mul-1 60.3%

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

       11. add-sqr-sqrt 34.5%

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

       12. sqrt-unprod 51.5%

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

       13. sqr-neg 51.5%

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

       14. sqrt-unprod 38.1%

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

       15. add-sqr-sqrt 85.1%

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

       16. distribute-neg-frac 85.1%

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

       17. distribute-neg-in 85.1%

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

       18. add-sqr-sqrt 47.0%

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

       19. sqrt-unprod 79.3%

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

       20. sqr-neg 79.3%

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

       21. sqrt-unprod 37.9%

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

       22. add-sqr-sqrt 85.3%

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

       23. sub-neg 85.3%

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

   12. Applied egg-rr 85.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+110}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{\left(-t1\right) - u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 - u}{v}}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.38 \cdot 10^{-33} \lor \neg \left(u \leq 2800000000\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 - u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.38e-33) (not (<= u 2800000000.0)))
   (/ (/ t1 (/ (- t1 u) v)) u)
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.38e-33) || !(u <= 2800000000.0)) {
		tmp = (t1 / ((t1 - u) / v)) / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.38d-33)) .or. (.not. (u <= 2800000000.0d0))) then
        tmp = (t1 / ((t1 - u) / v)) / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.38e-33) || !(u <= 2800000000.0)) {
		tmp = (t1 / ((t1 - u) / v)) / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.38e-33) or not (u <= 2800000000.0):
		tmp = (t1 / ((t1 - u) / v)) / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.38e-33) || !(u <= 2800000000.0))
		tmp = Float64(Float64(t1 / Float64(Float64(t1 - u) / v)) / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.38e-33) || ~((u <= 2800000000.0)))
		tmp = (t1 / ((t1 - u) / v)) / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.38e-33], N[Not[LessEqual[u, 2800000000.0]], $MachinePrecision]], N[(N[(t1 / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.38e-33 or 2.8e9 < u

   1. Initial program 74.3%

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

      1. associate-/l* 73.9%

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

   3. Simplified 73.9%

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

      1. associate-*r/ 74.3%

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

      2. times-frac 97.7%

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

      3. frac-2neg 97.7%

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

      4. +-commutative 97.7%

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

      5. distribute-neg-in 97.7%

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

      6. sub-neg 97.7%

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

      7. associate-*r/ 97.7%

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

      8. add-sqr-sqrt 55.5%

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

      9. sqrt-unprod 63.0%

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

      10. sqr-neg 63.0%

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

      11. sqrt-unprod 25.7%

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

      12. add-sqr-sqrt 52.4%

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

      13. sub-neg 52.4%

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

      14. +-commutative 52.4%

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

      15. add-sqr-sqrt 26.7%

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

      16. sqrt-unprod 52.7%

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

      17. sqr-neg 52.7%

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

      18. sqrt-unprod 28.8%

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

      19. add-sqr-sqrt 15.5%

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

      20. sqrt-unprod 33.2%

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

      21. sqr-neg 33.2%

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

      22. sqrt-unprod 21.2%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in v around 0 85.8%

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

      1. mul-1-neg 85.8%

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

      2. associate-/l* 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in t1 around 0 82.3%

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

       1. clear-num 82.8%

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

       2. frac-2neg 82.8%

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

       3. metadata-eval 82.8%

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

       4. associate-*l/ 82.8%

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

       5. add-sqr-sqrt 42.4%

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

       6. sqrt-unprod 56.9%

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

       7. sqr-neg 56.9%

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

       8. sqrt-unprod 25.7%

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

       9. add-sqr-sqrt 52.8%

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

       10. neg-mul-1 52.8%

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

       11. add-sqr-sqrt 27.1%

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

       12. sqrt-unprod 52.3%

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

       13. sqr-neg 52.3%

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

       14. sqrt-unprod 40.3%

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

       15. add-sqr-sqrt 82.8%

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

       16. distribute-neg-frac 82.8%

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

       17. distribute-neg-in 82.8%

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

       18. add-sqr-sqrt 42.5%

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

       19. sqrt-unprod 79.3%

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

       20. sqr-neg 79.3%

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

       21. sqrt-unprod 40.3%

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

       22. add-sqr-sqrt 83.0%

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

       23. sub-neg 83.0%

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

   12. Applied egg-rr 83.0%

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

   if -1.38e-33 < u < 2.8e9

   1. Initial program 67.2%

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

      1. associate-/l* 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in t1 around inf 85.6%

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

      1. associate-*r/ 85.6%

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

      2. neg-mul-1 85.6%

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

   7. Simplified 85.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.38 \cdot 10^{-33} \lor \neg \left(u \leq 2800000000\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 - u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{-34} \lor \neg \left(u \leq 1100000000\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.6e-34) (not (<= u 1100000000.0)))
   (/ (* t1 (/ v (- u))) u)
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.6e-34) || !(u <= 1100000000.0)) {
		tmp = (t1 * (v / -u)) / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.6d-34)) .or. (.not. (u <= 1100000000.0d0))) then
        tmp = (t1 * (v / -u)) / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.6e-34) || !(u <= 1100000000.0)) {
		tmp = (t1 * (v / -u)) / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.6e-34) or not (u <= 1100000000.0):
		tmp = (t1 * (v / -u)) / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.6e-34) || !(u <= 1100000000.0))
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.6e-34) || ~((u <= 1100000000.0)))
		tmp = (t1 * (v / -u)) / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.6e-34], N[Not[LessEqual[u, 1100000000.0]], $MachinePrecision]], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.60000000000000008e-34 or 1.1e9 < u

   1. Initial program 74.3%

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

      1. associate-/l* 73.9%

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

   3. Simplified 73.9%

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

      1. associate-*r/ 74.3%

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

      2. times-frac 97.7%

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

      3. frac-2neg 97.7%

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

      4. +-commutative 97.7%

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

      5. distribute-neg-in 97.7%

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

      6. sub-neg 97.7%

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

      7. associate-*r/ 97.7%

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

      8. add-sqr-sqrt 55.5%

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

      9. sqrt-unprod 63.0%

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

      10. sqr-neg 63.0%

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

      11. sqrt-unprod 25.7%

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

      12. add-sqr-sqrt 52.4%

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

      13. sub-neg 52.4%

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

      14. +-commutative 52.4%

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

      15. add-sqr-sqrt 26.7%

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

      16. sqrt-unprod 52.7%

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

      17. sqr-neg 52.7%

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

      18. sqrt-unprod 28.8%

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

      19. add-sqr-sqrt 15.5%

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

      20. sqrt-unprod 33.2%

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

      21. sqr-neg 33.2%

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

      22. sqrt-unprod 21.2%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in t1 around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-/l* 81.5%

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

      3. distribute-lft-neg-in 81.5%

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

   9. Simplified 81.5%

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

   10. Taylor expanded in t1 around 0 81.7%

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

   if -3.60000000000000008e-34 < u < 1.1e9

   1. Initial program 67.2%

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

      1. associate-/l* 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in t1 around inf 85.6%

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

      1. associate-*r/ 85.6%

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

      2. neg-mul-1 85.6%

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

   7. Simplified 85.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{-34} \lor \neg \left(u \leq 1100000000\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -0.0009:\\ \;\;\;\;\frac{-1}{\frac{t1 - u}{v}}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -0.0009)
   (/ -1.0 (/ (- t1 u) v))
   (if (<= t1 2.25e-38) (/ (* v (/ t1 (- u))) u) (/ v (- (- t1) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -0.0009) {
		tmp = -1.0 / ((t1 - u) / v);
	} else if (t1 <= 2.25e-38) {
		tmp = (v * (t1 / -u)) / u;
	} else {
		tmp = v / (-t1 - u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-0.0009d0)) then
        tmp = (-1.0d0) / ((t1 - u) / v)
    else if (t1 <= 2.25d-38) then
        tmp = (v * (t1 / -u)) / u
    else
        tmp = v / (-t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -0.0009) {
		tmp = -1.0 / ((t1 - u) / v);
	} else if (t1 <= 2.25e-38) {
		tmp = (v * (t1 / -u)) / u;
	} else {
		tmp = v / (-t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -0.0009:
		tmp = -1.0 / ((t1 - u) / v)
	elif t1 <= 2.25e-38:
		tmp = (v * (t1 / -u)) / u
	else:
		tmp = v / (-t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -0.0009)
		tmp = Float64(-1.0 / Float64(Float64(t1 - u) / v));
	elseif (t1 <= 2.25e-38)
		tmp = Float64(Float64(v * Float64(t1 / Float64(-u))) / u);
	else
		tmp = Float64(v / Float64(Float64(-t1) - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -0.0009)
		tmp = -1.0 / ((t1 - u) / v);
	elseif (t1 <= 2.25e-38)
		tmp = (v * (t1 / -u)) / u;
	else
		tmp = v / (-t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -0.0009], N[(-1.0 / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.25e-38], N[(N[(v * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.9999999999999998e-4

   1. Initial program 61.6%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in t1 around inf 84.7%

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

       1. add-sqr-sqrt 51.3%

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

       2. sqrt-unprod 43.1%

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

       3. clear-num 43.1%

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

       4. mul-1-neg 43.1%

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

       5. clear-num 43.1%

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

       6. mul-1-neg 43.1%

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

       7. sqr-neg 43.1%

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

       8. sqrt-unprod 20.9%

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

       9. add-sqr-sqrt 22.1%

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

       10. clear-num 22.7%

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

       11. frac-2neg 22.7%

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

       12. metadata-eval 22.7%

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

       13. distribute-neg-frac 22.7%

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

       14. distribute-neg-in 22.7%

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

       15. add-sqr-sqrt 22.7%

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

       16. sqrt-unprod 24.3%

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

       17. sqr-neg 24.3%

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

       18. sqrt-unprod 0.0%

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

       19. add-sqr-sqrt 84.6%

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

       20. sub-neg 84.6%

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

   11. Applied egg-rr 84.6%

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

   if -8.9999999999999998e-4 < t1 < 2.25000000000000004e-38

   1. Initial program 84.8%

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

      1. associate-/l* 84.7%

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

   3. Simplified 84.7%

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

      1. associate-*r/ 84.8%

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

      2. times-frac 96.9%

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

      3. frac-2neg 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. sub-neg 96.9%

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

      7. associate-*r/ 96.9%

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

      8. add-sqr-sqrt 61.6%

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

      9. sqrt-unprod 61.3%

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

      10. sqr-neg 61.3%

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

      11. sqrt-unprod 13.9%

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

      12. add-sqr-sqrt 39.1%

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

      13. sub-neg 39.1%

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

      14. +-commutative 39.1%

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

      15. add-sqr-sqrt 25.3%

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

      16. sqrt-unprod 42.3%

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

      17. sqr-neg 42.3%

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

      18. sqrt-unprod 18.0%

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

      19. add-sqr-sqrt 10.6%

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

      20. sqrt-unprod 25.1%

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

      21. sqr-neg 25.1%

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

      22. sqrt-unprod 16.4%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in v around 0 90.2%

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

      1. mul-1-neg 90.2%

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

      2. associate-/l* 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in t1 around 0 80.7%

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

   11. Taylor expanded in t1 around 0 79.7%

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

       1. mul-1-neg 79.7%

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

       2. distribute-frac-neg2 79.7%

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

       3. *-commutative 79.7%

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

       4. associate-/l* 82.0%

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

       5. distribute-frac-neg2 82.0%

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

       6. mul-1-neg 82.0%

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

       7. associate-*r/ 82.0%

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

       8. neg-mul-1 82.0%

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

   13. Simplified 82.0%

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

   if 2.25000000000000004e-38 < t1

   1. Initial program 63.1%

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

      1. associate-/l* 66.0%

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

   3. Simplified 66.0%

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

      1. associate-*r/ 63.1%

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

      2. times-frac 99.9%

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

      3. frac-2neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. sub-neg 99.9%

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

      7. associate-*r/ 99.9%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 19.3%

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

      10. sqr-neg 19.3%

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

      11. sqrt-unprod 39.8%

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

      12. add-sqr-sqrt 39.8%

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

      13. sub-neg 39.8%

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

      14. +-commutative 39.8%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 66.6%

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

      17. sqr-neg 66.6%

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

      18. sqrt-unprod 84.9%

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

      19. add-sqr-sqrt 43.7%

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

      20. sqrt-unprod 87.4%

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

      21. sqr-neg 87.4%

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

      22. sqrt-unprod 49.7%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 77.1%

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

      1. mul-1-neg 77.1%

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

   9. Simplified 77.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -0.0009:\\ \;\;\;\;\frac{-1}{\frac{t1 - u}{v}}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+182} \lor \neg \left(u \leq 2.15 \cdot 10^{+202}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.45e+182) (not (<= u 2.15e+202)))
   (* (/ v u) (/ t1 u))
   (/ -1.0 (/ (- t1 u) v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.45e+182) || !(u <= 2.15e+202)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -1.0 / ((t1 - u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.45d+182)) .or. (.not. (u <= 2.15d+202))) then
        tmp = (v / u) * (t1 / u)
    else
        tmp = (-1.0d0) / ((t1 - u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.45e+182) || !(u <= 2.15e+202)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -1.0 / ((t1 - u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.45e+182) or not (u <= 2.15e+202):
		tmp = (v / u) * (t1 / u)
	else:
		tmp = -1.0 / ((t1 - u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.45e+182) || !(u <= 2.15e+202))
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 - u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.45e+182) || ~((u <= 2.15e+202)))
		tmp = (v / u) * (t1 / u);
	else
		tmp = -1.0 / ((t1 - u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.45e+182], N[Not[LessEqual[u, 2.15e+202]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.4499999999999999e182 or 2.1500000000000001e202 < u

   1. Initial program 72.8%

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

      1. associate-/l* 73.5%

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

   3. Simplified 73.5%

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

      1. associate-*r/ 72.8%

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

      2. times-frac 98.2%

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

      3. frac-2neg 98.2%

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

      4. +-commutative 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. sub-neg 98.2%

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

      7. associate-*r/ 98.2%

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

      8. add-sqr-sqrt 56.0%

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

      9. sqrt-unprod 70.8%

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

      10. sqr-neg 70.8%

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

      11. sqrt-unprod 30.1%

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

      12. add-sqr-sqrt 73.0%

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

      13. sub-neg 73.0%

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

      14. +-commutative 73.0%

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

      15. add-sqr-sqrt 42.9%

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

      16. sqrt-unprod 73.2%

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

      17. sqr-neg 73.2%

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

      18. sqrt-unprod 30.5%

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

      19. add-sqr-sqrt 16.2%

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

      20. sqrt-unprod 30.4%

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

      21. sqr-neg 30.4%

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

      22. sqrt-unprod 19.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in t1 around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. associate-/l* 94.9%

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

      3. distribute-lft-neg-in 94.9%

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

   9. Simplified 94.9%

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

   10. Taylor expanded in t1 around 0 95.0%

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

       1. *-commutative 95.0%

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

       2. associate-/l* 93.4%

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

       3. add-sqr-sqrt 52.8%

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

       4. sqrt-unprod 70.7%

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

       5. sqr-neg 70.7%

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

       6. sqrt-unprod 28.9%

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

       7. add-sqr-sqrt 71.8%

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

   12. Applied egg-rr 71.8%

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

   if -1.4499999999999999e182 < u < 2.1500000000000001e202

   1. Initial program 70.3%

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

      1. clear-num 98.7%

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

      2. inv-pow 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. unpow-1 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in t1 around inf 71.4%

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

       1. add-sqr-sqrt 43.6%

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

       2. sqrt-unprod 39.5%

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

       3. clear-num 39.6%

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

       4. mul-1-neg 39.6%

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

       5. clear-num 39.6%

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

       6. mul-1-neg 39.6%

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

       7. sqr-neg 39.6%

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

       8. sqrt-unprod 17.4%

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

       9. add-sqr-sqrt 19.1%

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

       10. clear-num 19.3%

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

       11. frac-2neg 19.3%

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

       12. metadata-eval 19.3%

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

       13. distribute-neg-frac 19.3%

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

       14. distribute-neg-in 19.3%

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

       15. add-sqr-sqrt 8.2%

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

       16. sqrt-unprod 31.5%

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

       17. sqr-neg 31.5%

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

       18. sqrt-unprod 30.7%

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

       19. add-sqr-sqrt 71.8%

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

       20. sub-neg 71.8%

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

   11. Applied egg-rr 71.8%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+182} \lor \neg \left(u \leq 2.15 \cdot 10^{+202}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 - u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.18 \cdot 10^{+182} \lor \neg \left(u \leq 2.35 \cdot 10^{+202}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.18e+182) (not (<= u 2.35e+202)))
   (* (/ v u) (/ t1 u))
   (/ v (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.18e+182) || !(u <= 2.35e+202)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.18d+182)) .or. (.not. (u <= 2.35d+202))) then
        tmp = (v / u) * (t1 / u)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.18e+182) || !(u <= 2.35e+202)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.18e+182) or not (u <= 2.35e+202):
		tmp = (v / u) * (t1 / u)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.18e+182) || !(u <= 2.35e+202))
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.18e+182) || ~((u <= 2.35e+202)))
		tmp = (v / u) * (t1 / u);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.18e+182], N[Not[LessEqual[u, 2.35e+202]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.1799999999999999e182 or 2.3500000000000001e202 < u

   1. Initial program 72.8%

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

      1. associate-/l* 73.5%

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

   3. Simplified 73.5%

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

      1. associate-*r/ 72.8%

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

      2. times-frac 98.2%

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

      3. frac-2neg 98.2%

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

      4. +-commutative 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. sub-neg 98.2%

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

      7. associate-*r/ 98.2%

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

      8. add-sqr-sqrt 56.0%

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

      9. sqrt-unprod 70.8%

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

      10. sqr-neg 70.8%

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

      11. sqrt-unprod 30.1%

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

      12. add-sqr-sqrt 73.0%

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

      13. sub-neg 73.0%

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

      14. +-commutative 73.0%

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

      15. add-sqr-sqrt 42.9%

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

      16. sqrt-unprod 73.2%

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

      17. sqr-neg 73.2%

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

      18. sqrt-unprod 30.5%

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

      19. add-sqr-sqrt 16.2%

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

      20. sqrt-unprod 30.4%

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

      21. sqr-neg 30.4%

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

      22. sqrt-unprod 19.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in t1 around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. associate-/l* 94.9%

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

      3. distribute-lft-neg-in 94.9%

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

   9. Simplified 94.9%

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

   10. Taylor expanded in t1 around 0 95.0%

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

       1. *-commutative 95.0%

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

       2. associate-/l* 93.4%

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

       3. add-sqr-sqrt 52.8%

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

       4. sqrt-unprod 70.7%

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

       5. sqr-neg 70.7%

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

       6. sqrt-unprod 28.9%

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

       7. add-sqr-sqrt 71.8%

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

   12. Applied egg-rr 71.8%

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

   if -1.1799999999999999e182 < u < 2.3500000000000001e202

   1. Initial program 70.3%

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

      1. clear-num 98.7%

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

      2. inv-pow 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. unpow-1 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in t1 around inf 71.4%

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

       1. add-sqr-sqrt 43.6%

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

       2. sqrt-unprod 39.5%

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

       3. clear-num 39.6%

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

       4. mul-1-neg 39.6%

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

       5. clear-num 39.6%

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

       6. mul-1-neg 39.6%

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

       7. sqr-neg 39.6%

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

       8. sqrt-unprod 17.4%

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

       9. add-sqr-sqrt 19.1%

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

       10. frac-2neg 19.1%

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

       11. distribute-neg-in 19.1%

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

       12. add-sqr-sqrt 8.2%

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

       13. sqrt-unprod 31.3%

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

       14. sqr-neg 31.3%

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

       15. sqrt-unprod 30.5%

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

       16. add-sqr-sqrt 71.8%

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

       17. sub-neg 71.8%

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

   11. Applied egg-rr 71.8%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.18 \cdot 10^{+182} \lor \neg \left(u \leq 2.35 \cdot 10^{+202}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.65 \cdot 10^{+182}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+202}:\\ \;\;\;\;\frac{-1}{\frac{t1 - u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.65e+182)
   (* (/ v u) (/ t1 u))
   (if (<= u 2.2e+202) (/ -1.0 (/ (- t1 u) v)) (/ (* t1 (/ v u)) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.65e+182) {
		tmp = (v / u) * (t1 / u);
	} else if (u <= 2.2e+202) {
		tmp = -1.0 / ((t1 - u) / v);
	} else {
		tmp = (t1 * (v / u)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.65d+182)) then
        tmp = (v / u) * (t1 / u)
    else if (u <= 2.2d+202) then
        tmp = (-1.0d0) / ((t1 - u) / v)
    else
        tmp = (t1 * (v / u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.65e+182) {
		tmp = (v / u) * (t1 / u);
	} else if (u <= 2.2e+202) {
		tmp = -1.0 / ((t1 - u) / v);
	} else {
		tmp = (t1 * (v / u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.65e+182:
		tmp = (v / u) * (t1 / u)
	elif u <= 2.2e+202:
		tmp = -1.0 / ((t1 - u) / v)
	else:
		tmp = (t1 * (v / u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.65e+182)
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	elseif (u <= 2.2e+202)
		tmp = Float64(-1.0 / Float64(Float64(t1 - u) / v));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.65e+182)
		tmp = (v / u) * (t1 / u);
	elseif (u <= 2.2e+202)
		tmp = -1.0 / ((t1 - u) / v);
	else
		tmp = (t1 * (v / u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.65e+182], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.2e+202], N[(-1.0 / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.65e182

   1. Initial program 68.1%

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

      1. associate-/l* 68.7%

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

   3. Simplified 68.7%

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

      1. associate-*r/ 68.1%

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

      2. times-frac 96.9%

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

      3. frac-2neg 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. sub-neg 96.9%

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

      7. associate-*r/ 96.8%

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

      8. add-sqr-sqrt 53.1%

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

      9. sqrt-unprod 74.1%

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

      10. sqr-neg 74.1%

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

      11. sqrt-unprod 30.8%

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

      12. add-sqr-sqrt 68.3%

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

      13. sub-neg 68.3%

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

      14. +-commutative 68.3%

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

      15. add-sqr-sqrt 37.5%

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

      16. sqrt-unprod 68.4%

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

      17. sqr-neg 68.4%

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

      18. sqrt-unprod 30.8%

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

      19. add-sqr-sqrt 30.8%

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

      20. sqrt-unprod 31.0%

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

      21. sqr-neg 31.0%

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

      22. sqrt-unprod 0.0%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in t1 around 0 86.9%

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

      1. mul-1-neg 86.9%

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

      2. associate-/l* 96.8%

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

      3. distribute-lft-neg-in 96.8%

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

   9. Simplified 96.8%

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

   10. Taylor expanded in t1 around 0 96.7%

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

       1. *-commutative 96.7%

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

       2. associate-/l* 93.7%

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

       3. add-sqr-sqrt 50.0%

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

       4. sqrt-unprod 74.0%

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

       5. sqr-neg 74.0%

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

       6. sqrt-unprod 30.8%

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

       7. add-sqr-sqrt 68.3%

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

   12. Applied egg-rr 68.3%

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

   if -1.65e182 < u < 2.19999999999999978e202

   1. Initial program 70.3%

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

      1. clear-num 98.7%

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

      2. inv-pow 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. unpow-1 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in t1 around inf 71.4%

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

       1. add-sqr-sqrt 43.6%

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

       2. sqrt-unprod 39.5%

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

       3. clear-num 39.6%

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

       4. mul-1-neg 39.6%

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

       5. clear-num 39.6%

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

       6. mul-1-neg 39.6%

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

       7. sqr-neg 39.6%

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

       8. sqrt-unprod 17.4%

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

       9. add-sqr-sqrt 19.1%

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

       10. clear-num 19.3%

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

       11. frac-2neg 19.3%

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

       12. metadata-eval 19.3%

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

       13. distribute-neg-frac 19.3%

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

       14. distribute-neg-in 19.3%

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

       15. add-sqr-sqrt 8.2%

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

       16. sqrt-unprod 31.5%

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

       17. sqr-neg 31.5%

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

       18. sqrt-unprod 30.7%

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

       19. add-sqr-sqrt 71.8%

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

       20. sub-neg 71.8%

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

   11. Applied egg-rr 71.8%

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

   if 2.19999999999999978e202 < u

   1. Initial program 78.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. 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 inf 56.3%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in u around inf 36.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot v + \frac{t1 \cdot v}{u}}{u}} \]
   7. Step-by-step derivation

      1. +-commutative 36.4%

         \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{u} + -1 \cdot v}}{u} \]

      2. mul-1-neg 36.4%

         \[\leadsto \frac{\frac{t1 \cdot v}{u} + \color{blue}{\left(-v\right)}}{u} \]

      3. unsub-neg 36.4%

         \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{u} - v}}{u} \]

      4. associate-/l* 36.8%

         \[\leadsto \frac{\color{blue}{t1 \cdot \frac{v}{u}} - v}{u} \]

   8. Simplified 36.8%

      \[\leadsto \color{blue}{\frac{t1 \cdot \frac{v}{u} - v}{u}} \]

   9. Taylor expanded in t1 around inf 75.5%

      \[\leadsto \frac{\color{blue}{\frac{t1 \cdot v}{u}}}{u} \]
   10. Step-by-step derivation

       1. associate-*r/ 75.7%

          \[\leadsto \frac{\color{blue}{t1 \cdot \frac{v}{u}}}{u} \]

   11. Simplified 75.7%

       \[\leadsto \frac{\color{blue}{t1 \cdot \frac{v}{u}}}{u} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.08 \cdot 10^{+204}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+202}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.08e+204)
   (/ 1.0 (/ u v))
   (if (<= u 2.2e+202) (/ v (- t1)) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.08e+204) {
		tmp = 1.0 / (u / v);
	} else if (u <= 2.2e+202) {
		tmp = v / -t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.08d+204)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 2.2d+202) then
        tmp = v / -t1
    else
        tmp = v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.08e+204) {
		tmp = 1.0 / (u / v);
	} else if (u <= 2.2e+202) {
		tmp = v / -t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.08e+204:
		tmp = 1.0 / (u / v)
	elif u <= 2.2e+202:
		tmp = v / -t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.08e+204)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 2.2e+202)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.08e+204)
		tmp = 1.0 / (u / v);
	elseif (u <= 2.2e+202)
		tmp = v / -t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.08e+204], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.2e+202], N[(v / (-t1)), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.08e204

   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. times-frac 96.1%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 96.1%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 96.1%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 96.1%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 96.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 96.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 57.7%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 42.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 42.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 42.5%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 42.5%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 29.1%

         \[\leadsto \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{u} \]

      2. sqrt-unprod 42.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{u} \]

      3. sqr-neg 42.1%

         \[\leadsto \frac{\sqrt{\color{blue}{v \cdot v}}}{u} \]

      4. sqrt-unprod 13.4%

         \[\leadsto \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{u} \]

      5. add-sqr-sqrt 42.9%

         \[\leadsto \frac{\color{blue}{v}}{u} \]

      6. clear-num 44.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

      7. inv-pow 44.8%

         \[\leadsto \color{blue}{{\left(\frac{u}{v}\right)}^{-1}} \]

   10. Applied egg-rr 44.8%

       \[\leadsto \color{blue}{{\left(\frac{u}{v}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 44.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

   12. Simplified 44.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

   if -1.08e204 < u < 2.19999999999999978e202

   1. Initial program 70.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 70.7%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 68.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 68.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 68.1%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 2.19999999999999978e202 < u

   1. Initial program 78.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. 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. Step-by-step derivation

      1. clear-num 99.0%

         \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \]

      2. inv-pow 99.0%

         \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{{\left(\frac{t1 + u}{v}\right)}^{-1}} \]

   6. Applied egg-rr 99.0%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{{\left(\frac{t1 + u}{v}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 99.0%

         \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \]

   8. Simplified 99.0%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \]

   9. Taylor expanded in t1 around inf 41.5%

      \[\leadsto \color{blue}{-1} \cdot \frac{1}{\frac{t1 + u}{v}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 40.1%

          \[\leadsto \color{blue}{\sqrt{-1 \cdot \frac{1}{\frac{t1 + u}{v}}} \cdot \sqrt{-1 \cdot \frac{1}{\frac{t1 + u}{v}}}} \]

       2. sqrt-unprod 66.0%

          \[\leadsto \color{blue}{\sqrt{\left(-1 \cdot \frac{1}{\frac{t1 + u}{v}}\right) \cdot \left(-1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)}} \]

       3. clear-num 66.0%

          \[\leadsto \sqrt{\left(-1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \cdot \left(-1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

       4. mul-1-neg 66.0%

          \[\leadsto \sqrt{\color{blue}{\left(-\frac{v}{t1 + u}\right)} \cdot \left(-1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

       5. clear-num 66.0%

          \[\leadsto \sqrt{\left(-\frac{v}{t1 + u}\right) \cdot \left(-1 \cdot \color{blue}{\frac{v}{t1 + u}}\right)} \]

       6. mul-1-neg 66.0%

          \[\leadsto \sqrt{\left(-\frac{v}{t1 + u}\right) \cdot \color{blue}{\left(-\frac{v}{t1 + u}\right)}} \]

       7. sqr-neg 66.0%

          \[\leadsto \sqrt{\color{blue}{\frac{v}{t1 + u} \cdot \frac{v}{t1 + u}}} \]

       8. sqrt-unprod 37.6%

          \[\leadsto \color{blue}{\sqrt{\frac{v}{t1 + u}} \cdot \sqrt{\frac{v}{t1 + u}}} \]

       9. add-sqr-sqrt 39.2%

          \[\leadsto \color{blue}{\frac{v}{t1 + u}} \]

   11. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\frac{v}{t1 + u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.08 \cdot 10^{+204}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+202}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+207} \lor \neg \left(u \leq 2.7 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4e+207) (not (<= u 2.7e+216))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4e+207) || !(u <= 2.7e+216)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4d+207)) .or. (.not. (u <= 2.7d+216))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4e+207) || !(u <= 2.7e+216)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4e+207) or not (u <= 2.7e+216):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4e+207) || !(u <= 2.7e+216))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4e+207) || ~((u <= 2.7e+216)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4e+207], N[Not[LessEqual[u, 2.7e+216]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.0000000000000002e207 or 2.7000000000000001e216 < u

   1. Initial program 71.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 97.7%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 97.7%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 97.7%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 97.7%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 97.7%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 97.7%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 53.3%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 38.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 38.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 38.5%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]
   9. Step-by-step derivation

      1. div-inv 38.5%

         \[\leadsto \color{blue}{\left(-v\right) \cdot \frac{1}{u}} \]

      2. add-sqr-sqrt 25.9%

         \[\leadsto \color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)} \cdot \frac{1}{u} \]

      3. sqrt-unprod 38.0%

         \[\leadsto \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}} \cdot \frac{1}{u} \]

      4. sqr-neg 38.0%

         \[\leadsto \sqrt{\color{blue}{v \cdot v}} \cdot \frac{1}{u} \]

      5. sqrt-unprod 12.6%

         \[\leadsto \color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)} \cdot \frac{1}{u} \]

      6. add-sqr-sqrt 38.6%

         \[\leadsto \color{blue}{v} \cdot \frac{1}{u} \]

   10. Applied egg-rr 38.6%

       \[\leadsto \color{blue}{v \cdot \frac{1}{u}} \]
   11. Step-by-step derivation

       1. associate-*r/ 38.6%

          \[\leadsto \color{blue}{\frac{v \cdot 1}{u}} \]

       2. *-rgt-identity 38.6%

          \[\leadsto \frac{\color{blue}{v}}{u} \]

   12. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{v}{u}} \]

   if -4.0000000000000002e207 < u < 2.7000000000000001e216

   1. Initial program 70.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 71.2%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 71.2%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 67.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 67.6%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+207} \lor \neg \left(u \leq 2.7 \cdot 10^{+216}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+201}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+221}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5e+201)
   (/ 1.0 (/ u v))
   (if (<= u 9e+221) (/ v (- t1)) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e+201) {
		tmp = 1.0 / (u / v);
	} else if (u <= 9e+221) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5d+201)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 9d+221) then
        tmp = v / -t1
    else
        tmp = v / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e+201) {
		tmp = 1.0 / (u / v);
	} else if (u <= 9e+221) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5e+201:
		tmp = 1.0 / (u / v)
	elif u <= 9e+221:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5e+201)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 9e+221)
		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 <= -5e+201)
		tmp = 1.0 / (u / v);
	elseif (u <= 9e+221)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5e+201], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 9e+221], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.9999999999999995e201

   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. times-frac 96.1%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 96.1%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 96.1%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 96.1%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 96.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 96.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 57.7%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 42.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 42.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 42.5%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 42.5%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 29.1%

         \[\leadsto \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{u} \]

      2. sqrt-unprod 42.1%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{u} \]

      3. sqr-neg 42.1%

         \[\leadsto \frac{\sqrt{\color{blue}{v \cdot v}}}{u} \]

      4. sqrt-unprod 13.4%

         \[\leadsto \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{u} \]

      5. add-sqr-sqrt 42.9%

         \[\leadsto \frac{\color{blue}{v}}{u} \]

      6. clear-num 44.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

      7. inv-pow 44.8%

         \[\leadsto \color{blue}{{\left(\frac{u}{v}\right)}^{-1}} \]

   10. Applied egg-rr 44.8%

       \[\leadsto \color{blue}{{\left(\frac{u}{v}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 44.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

   12. Simplified 44.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{u}{v}}} \]

   if -4.9999999999999995e201 < u < 9.0000000000000004e221

   1. Initial program 70.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 70.8%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 67.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 67.3%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 67.3%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 9.0000000000000004e221 < u

   1. Initial program 80.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.8%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 50.4%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 35.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 35.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 35.3%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+201}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+221}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+200}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 4.1 \cdot 10^{+218}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.6e+200) (/ v u) (if (<= u 4.1e+218) (/ v (- t1)) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.6e+200) {
		tmp = v / u;
	} else if (u <= 4.1e+218) {
		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+200)) then
        tmp = v / u
    else if (u <= 4.1d+218) 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+200) {
		tmp = v / u;
	} else if (u <= 4.1e+218) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.6e+200:
		tmp = v / u
	elif u <= 4.1e+218:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.6e+200)
		tmp = Float64(v / u);
	elseif (u <= 4.1e+218)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.6e+200)
		tmp = v / u;
	elseif (u <= 4.1e+218)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.6e+200], N[(v / u), $MachinePrecision], If[LessEqual[u, 4.1e+218], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.60000000000000016e200

   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. times-frac 96.1%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 96.1%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 96.1%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 96.1%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 96.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 96.1%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 57.7%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 42.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 42.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 42.5%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 42.5%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]
   9. Step-by-step derivation

      1. div-inv 42.5%

         \[\leadsto \color{blue}{\left(-v\right) \cdot \frac{1}{u}} \]

      2. add-sqr-sqrt 29.1%

         \[\leadsto \color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)} \cdot \frac{1}{u} \]

      3. sqrt-unprod 42.1%

         \[\leadsto \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}} \cdot \frac{1}{u} \]

      4. sqr-neg 42.1%

         \[\leadsto \sqrt{\color{blue}{v \cdot v}} \cdot \frac{1}{u} \]

      5. sqrt-unprod 13.4%

         \[\leadsto \color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)} \cdot \frac{1}{u} \]

      6. add-sqr-sqrt 42.9%

         \[\leadsto \color{blue}{v} \cdot \frac{1}{u} \]

   10. Applied egg-rr 42.9%

       \[\leadsto \color{blue}{v \cdot \frac{1}{u}} \]
   11. Step-by-step derivation

       1. associate-*r/ 42.9%

          \[\leadsto \color{blue}{\frac{v \cdot 1}{u}} \]

       2. *-rgt-identity 42.9%

          \[\leadsto \frac{\color{blue}{v}}{u} \]

   12. Simplified 42.9%

       \[\leadsto \color{blue}{\frac{v}{u}} \]

   if -1.60000000000000016e200 < u < 4.09999999999999965e218

   1. Initial program 70.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 70.8%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 67.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 67.3%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   7. Simplified 67.3%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 4.09999999999999965e218 < u

   1. Initial program 80.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.8%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.8%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 50.4%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 35.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 35.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 35.3%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 35.3%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+200}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 4.1 \cdot 10^{+218}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 22.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+89} \lor \neg \left(t1 \leq 2.5 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.08e+89) (not (<= t1 2.5e+133))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.08e+89) || !(t1 <= 2.5e+133)) {
		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 <= (-1.08d+89)) .or. (.not. (t1 <= 2.5d+133))) 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 <= -1.08e+89) || !(t1 <= 2.5e+133)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.08e+89) or not (t1 <= 2.5e+133):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.08e+89) || !(t1 <= 2.5e+133))
		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 <= -1.08e+89) || ~((t1 <= 2.5e+133)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.08e+89], N[Not[LessEqual[t1, 2.5e+133]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.08000000000000006e89 or 2.4999999999999998e133 < t1

   1. Initial program 51.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 99.9%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 99.9%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 99.9%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 88.4%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{t1 + u} \]

   6. Taylor expanded in u around inf 34.1%

      \[\leadsto \color{blue}{\frac{v}{t1}} \]

   if -1.08000000000000006e89 < t1 < 2.4999999999999998e133

   1. Initial program 83.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 98.0%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

      2. distribute-frac-neg 98.0%

         \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

      3. distribute-neg-frac2 98.0%

         \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

      4. +-commutative 98.0%

         \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

      5. distribute-neg-in 98.0%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

      6. unsub-neg 98.0%

         \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
   4. Add Preprocessing

   5. Taylor expanded in t1 around inf 55.7%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{v}{t1}} \]

   6. Taylor expanded in t1 around 0 16.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   7. Step-by-step derivation

      1. associate-*r/ 16.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{u}} \]

      2. mul-1-neg 16.7%

         \[\leadsto \frac{\color{blue}{-v}}{u} \]

   8. Simplified 16.7%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]
   9. Step-by-step derivation

      1. div-inv 16.7%

         \[\leadsto \color{blue}{\left(-v\right) \cdot \frac{1}{u}} \]

      2. add-sqr-sqrt 8.8%

         \[\leadsto \color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)} \cdot \frac{1}{u} \]

      3. sqrt-unprod 21.8%

         \[\leadsto \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}} \cdot \frac{1}{u} \]

      4. sqr-neg 21.8%

         \[\leadsto \sqrt{\color{blue}{v \cdot v}} \cdot \frac{1}{u} \]

      5. sqrt-unprod 8.5%

         \[\leadsto \color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)} \cdot \frac{1}{u} \]

      6. add-sqr-sqrt 18.2%

         \[\leadsto \color{blue}{v} \cdot \frac{1}{u} \]

   10. Applied egg-rr 18.2%

       \[\leadsto \color{blue}{v \cdot \frac{1}{u}} \]
   11. Step-by-step derivation

       1. associate-*r/ 18.2%

          \[\leadsto \color{blue}{\frac{v \cdot 1}{u}} \]

       2. *-rgt-identity 18.2%

          \[\leadsto \frac{\color{blue}{v}}{u} \]

   12. Simplified 18.2%

       \[\leadsto \color{blue}{\frac{v}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+89} \lor \neg \left(t1 \leq 2.5 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{v \cdot \frac{t1}{\left(-t1\right) - u}}{t1 + u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* v (/ t1 (- (- t1) u))) (+ t1 u)))

C

double code(double u, double v, double t1) {
	return (v * (t1 / (-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 = (v * (t1 / (-t1 - u))) / (t1 + u)
end function

Java

public static double code(double u, double v, double t1) {
	return (v * (t1 / (-t1 - u))) / (t1 + u);
}

Python

def code(u, v, t1):
	return (v * (t1 / (-t1 - u))) / (t1 + u)

Julia

function code(u, v, t1)
	return Float64(Float64(v * Float64(t1 / Float64(Float64(-t1) - u))) / Float64(t1 + u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v * (t1 / (-t1 - u))) / (t1 + u);
end

Wolfram

code[u_, v_, t1_] := N[(N[(v * N[(t1 / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 71.4%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

3. Simplified 71.4%

   \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 70.8%

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. frac-2neg 98.8%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

   4. +-commutative 98.8%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{-\color{blue}{\left(u + t1\right)}} \]

   5. distribute-neg-in 98.8%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   6. sub-neg 98.8%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) - t1}} \]

   7. associate-*r/ 98.8%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1}} \]

   8. add-sqr-sqrt 54.7%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   9. sqrt-unprod 48.9%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   10. sqr-neg 48.9%

       \[\leadsto \frac{\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   11. sqrt-unprod 17.4%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   12. add-sqr-sqrt 34.8%

       \[\leadsto \frac{\frac{\color{blue}{t1}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   13. sub-neg 34.8%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   14. +-commutative 34.8%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-t1\right) + \left(-u\right)}} \]

   15. add-sqr-sqrt 17.5%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)} \]

   16. sqrt-unprod 44.3%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)} \]

   17. sqr-neg 44.3%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)} \]

   18. sqrt-unprod 32.7%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)} \]

   19. add-sqr-sqrt 17.3%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{-u} \cdot \sqrt{-u}}} \]

   20. sqrt-unprod 36.1%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}} \]

   21. sqr-neg 36.1%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \sqrt{\color{blue}{u \cdot u}}} \]

   22. sqrt-unprod 21.4%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{u} \cdot \sqrt{u}}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{t1 + u}} \]

7. Final simplification 98.8%

   \[\leadsto \frac{v \cdot \frac{t1}{\left(-t1\right) - u}}{t1 + u} \]
8. Add Preprocessing

Alternative 18: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{t1 + u} \cdot \frac{t1}{\left(-t1\right) - u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (* (/ v (+ t1 u)) (/ t1 (- (- t1) u))))

C

double code(double u, double v, double t1) {
	return (v / (t1 + u)) * (t1 / (-t1 - u));
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / (t1 + u)) * (t1 / (-t1 - u))
end function

Java

public static double code(double u, double v, double t1) {
	return (v / (t1 + u)) * (t1 / (-t1 - u));
}

Python

def code(u, v, t1):
	return (v / (t1 + u)) * (t1 / (-t1 - u))

Julia

function code(u, v, t1)
	return Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(Float64(-t1) - u)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (v / (t1 + u)) * (t1 / (-t1 - u));
end

Wolfram

code[u_, v_, t1_] := N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 98.8%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 98.8%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 98.8%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 98.8%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 98.8%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Final simplification 98.8%

   \[\leadsto \frac{v}{t1 + u} \cdot \frac{t1}{\left(-t1\right) - u} \]
6. Add Preprocessing

Alternative 19: 62.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(-t1\right) - u} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (- t1) u)))

C

double code(double u, double v, double t1) {
	return v / (-t1 - u);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / (-t1 - u)
end function

Java

public static double code(double u, double v, double t1) {
	return v / (-t1 - u);
}

Python

def code(u, v, t1):
	return v / (-t1 - u)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(-t1) - u))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / (-t1 - u);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. associate-/l* 71.4%

      \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

3. Simplified 71.4%

   \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 70.8%

      \[\leadsto \color{blue}{\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

   2. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. frac-2neg 98.8%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

   4. +-commutative 98.8%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{-\color{blue}{\left(u + t1\right)}} \]

   5. distribute-neg-in 98.8%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   6. sub-neg 98.8%

      \[\leadsto \frac{-t1}{t1 + u} \cdot \frac{-v}{\color{blue}{\left(-u\right) - t1}} \]

   7. associate-*r/ 98.8%

      \[\leadsto \color{blue}{\frac{\frac{-t1}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1}} \]

   8. add-sqr-sqrt 54.7%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   9. sqrt-unprod 48.9%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   10. sqr-neg 48.9%

       \[\leadsto \frac{\frac{\sqrt{\color{blue}{t1 \cdot t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   11. sqrt-unprod 17.4%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   12. add-sqr-sqrt 34.8%

       \[\leadsto \frac{\frac{\color{blue}{t1}}{t1 + u} \cdot \left(-v\right)}{\left(-u\right) - t1} \]

   13. sub-neg 34.8%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \]

   14. +-commutative 34.8%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\left(-t1\right) + \left(-u\right)}} \]

   15. add-sqr-sqrt 17.5%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)} \]

   16. sqrt-unprod 44.3%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)} \]

   17. sqr-neg 44.3%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)} \]

   18. sqrt-unprod 32.7%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)} \]

   19. add-sqr-sqrt 17.3%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{-u} \cdot \sqrt{-u}}} \]

   20. sqrt-unprod 36.1%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}} \]

   21. sqr-neg 36.1%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \sqrt{\color{blue}{u \cdot u}}} \]

   22. sqrt-unprod 21.4%

       \[\leadsto \frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{\sqrt{t1} \cdot \sqrt{t1} + \color{blue}{\sqrt{u} \cdot \sqrt{u}}} \]

6. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{t1 + u}} \]

7. Taylor expanded in t1 around inf 64.4%

   \[\leadsto \frac{\color{blue}{-1 \cdot v}}{t1 + u} \]
8. Step-by-step derivation

   1. mul-1-neg 64.4%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

9. Simplified 64.4%

   \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]

10. Final simplification 64.4%

    \[\leadsto \frac{v}{\left(-t1\right) - u} \]
11. Add Preprocessing

Alternative 20: 62.0% 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 70.8%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 98.8%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 98.8%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 98.8%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 98.8%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 98.8%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 98.5%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \]

   2. inv-pow 98.5%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{{\left(\frac{t1 + u}{v}\right)}^{-1}} \]

6. Applied egg-rr 98.5%

   \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{{\left(\frac{t1 + u}{v}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 98.5%

      \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \]

8. Simplified 98.5%

   \[\leadsto \frac{t1}{\left(-u\right) - t1} \cdot \color{blue}{\frac{1}{\frac{t1 + u}{v}}} \]

9. Taylor expanded in t1 around inf 64.6%

   \[\leadsto \color{blue}{-1} \cdot \frac{1}{\frac{t1 + u}{v}} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 42.2%

       \[\leadsto \color{blue}{\sqrt{-1 \cdot \frac{1}{\frac{t1 + u}{v}}} \cdot \sqrt{-1 \cdot \frac{1}{\frac{t1 + u}{v}}}} \]

    2. sqrt-unprod 45.3%

       \[\leadsto \color{blue}{\sqrt{\left(-1 \cdot \frac{1}{\frac{t1 + u}{v}}\right) \cdot \left(-1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)}} \]

    3. clear-num 45.3%

       \[\leadsto \sqrt{\left(-1 \cdot \color{blue}{\frac{v}{t1 + u}}\right) \cdot \left(-1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

    4. mul-1-neg 45.3%

       \[\leadsto \sqrt{\color{blue}{\left(-\frac{v}{t1 + u}\right)} \cdot \left(-1 \cdot \frac{1}{\frac{t1 + u}{v}}\right)} \]

    5. clear-num 45.4%

       \[\leadsto \sqrt{\left(-\frac{v}{t1 + u}\right) \cdot \left(-1 \cdot \color{blue}{\frac{v}{t1 + u}}\right)} \]

    6. mul-1-neg 45.4%

       \[\leadsto \sqrt{\left(-\frac{v}{t1 + u}\right) \cdot \color{blue}{\left(-\frac{v}{t1 + u}\right)}} \]

    7. sqr-neg 45.4%

       \[\leadsto \sqrt{\color{blue}{\frac{v}{t1 + u} \cdot \frac{v}{t1 + u}}} \]

    8. sqrt-unprod 21.5%

       \[\leadsto \color{blue}{\sqrt{\frac{v}{t1 + u}} \cdot \sqrt{\frac{v}{t1 + u}}} \]

    9. add-sqr-sqrt 23.1%

       \[\leadsto \color{blue}{\frac{v}{t1 + u}} \]

    10. frac-2neg 23.1%

        \[\leadsto \color{blue}{\frac{-v}{-\left(t1 + u\right)}} \]

    11. distribute-neg-in 23.1%

        \[\leadsto \frac{-v}{\color{blue}{\left(-t1\right) + \left(-u\right)}} \]

    12. add-sqr-sqrt 11.6%

        \[\leadsto \frac{-v}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)} \]

    13. sqrt-unprod 33.4%

        \[\leadsto \frac{-v}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)} \]

    14. sqr-neg 33.4%

        \[\leadsto \frac{-v}{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)} \]

    15. sqrt-unprod 26.8%

        \[\leadsto \frac{-v}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)} \]

    16. add-sqr-sqrt 64.8%

        \[\leadsto \frac{-v}{\color{blue}{t1} + \left(-u\right)} \]

    17. sub-neg 64.8%

        \[\leadsto \frac{-v}{\color{blue}{t1 - u}} \]

11. Applied egg-rr 64.8%

    \[\leadsto \color{blue}{\frac{-v}{t1 - u}} \]

12. Final simplification 64.8%

    \[\leadsto \frac{v}{u - t1} \]
13. Add Preprocessing

Alternative 21: 13.8% 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 70.8%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 98.8%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   2. distribute-frac-neg 98.8%

      \[\leadsto \color{blue}{\left(-\frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u} \]

   3. distribute-neg-frac2 98.8%

      \[\leadsto \color{blue}{\frac{t1}{-\left(t1 + u\right)}} \cdot \frac{v}{t1 + u} \]

   4. +-commutative 98.8%

      \[\leadsto \frac{t1}{-\color{blue}{\left(u + t1\right)}} \cdot \frac{v}{t1 + u} \]

   5. distribute-neg-in 98.8%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) + \left(-t1\right)}} \cdot \frac{v}{t1 + u} \]

   6. unsub-neg 98.8%

      \[\leadsto \frac{t1}{\color{blue}{\left(-u\right) - t1}} \cdot \frac{v}{t1 + u} \]

3. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1 + u}} \]
4. Add Preprocessing

5. Taylor expanded in t1 around inf 60.5%

   \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{t1 + u} \]

6. Taylor expanded in u around inf 15.1%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))