Rosa's DopplerBench

Percentage Accurate: 72.3% → 98.0%

Time: 12.8s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = ((t1 * (v / (t1 + u))) / (-1.0d0)) / (t1 + u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 70.5%

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

   2. distribute-lft-neg-out 70.5%

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

   3. distribute-rgt-neg-in 70.5%

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

   4. associate-/r* 80.5%

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

   5. distribute-neg-frac2 80.5%

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

3. Simplified 80.5%

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

   1. associate-*r/ 97.9%

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

   2. neg-mul-1 97.9%

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

   3. associate-/r* 97.9%

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 2: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -6.6 \cdot 10^{-246}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-240}:\\ \;\;\;\;\frac{t1 \cdot v}{u} \cdot \frac{-1}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.65 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (- t1) (/ (/ v (+ t1 u)) (+ t1 u)))))
   (if (<= t1 -7.6e+59)
     (/ v (- (- u) t1))
     (if (<= t1 -6.6e-246)
       t_1
       (if (<= t1 7e-240)
         (* (/ (* t1 v) u) (/ -1.0 (- u t1)))
         (if (<= t1 1.65e+74) t_1 (/ v (- u t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -t1 * ((v / (t1 + u)) / (t1 + u));
	double tmp;
	if (t1 <= -7.6e+59) {
		tmp = v / (-u - t1);
	} else if (t1 <= -6.6e-246) {
		tmp = t_1;
	} else if (t1 <= 7e-240) {
		tmp = ((t1 * v) / u) * (-1.0 / (u - t1));
	} else if (t1 <= 1.65e+74) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = -t1 * ((v / (t1 + u)) / (t1 + u))
    if (t1 <= (-7.6d+59)) then
        tmp = v / (-u - t1)
    else if (t1 <= (-6.6d-246)) then
        tmp = t_1
    else if (t1 <= 7d-240) then
        tmp = ((t1 * v) / u) * ((-1.0d0) / (u - t1))
    else if (t1 <= 1.65d+74) then
        tmp = t_1
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -t1 * ((v / (t1 + u)) / (t1 + u));
	double tmp;
	if (t1 <= -7.6e+59) {
		tmp = v / (-u - t1);
	} else if (t1 <= -6.6e-246) {
		tmp = t_1;
	} else if (t1 <= 7e-240) {
		tmp = ((t1 * v) / u) * (-1.0 / (u - t1));
	} else if (t1 <= 1.65e+74) {
		tmp = t_1;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -t1 * ((v / (t1 + u)) / (t1 + u))
	tmp = 0
	if t1 <= -7.6e+59:
		tmp = v / (-u - t1)
	elif t1 <= -6.6e-246:
		tmp = t_1
	elif t1 <= 7e-240:
		tmp = ((t1 * v) / u) * (-1.0 / (u - t1))
	elif t1 <= 1.65e+74:
		tmp = t_1
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-t1) * Float64(Float64(v / Float64(t1 + u)) / Float64(t1 + u)))
	tmp = 0.0
	if (t1 <= -7.6e+59)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= -6.6e-246)
		tmp = t_1;
	elseif (t1 <= 7e-240)
		tmp = Float64(Float64(Float64(t1 * v) / u) * Float64(-1.0 / Float64(u - t1)));
	elseif (t1 <= 1.65e+74)
		tmp = t_1;
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -t1 * ((v / (t1 + u)) / (t1 + u));
	tmp = 0.0;
	if (t1 <= -7.6e+59)
		tmp = v / (-u - t1);
	elseif (t1 <= -6.6e-246)
		tmp = t_1;
	elseif (t1 <= 7e-240)
		tmp = ((t1 * v) / u) * (-1.0 / (u - t1));
	elseif (t1 <= 1.65e+74)
		tmp = t_1;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-t1) * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.6e+59], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -6.6e-246], t$95$1, If[LessEqual[t1, 7e-240], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] * N[(-1.0 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.65e+74], t$95$1, N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -7.6000000000000002e59

   1. Initial program 49.5%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 94.7%

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

   6. Taylor expanded in v around 0 94.7%

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

      1. associate-*r/ 94.7%

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

      2. neg-mul-1 94.7%

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

   8. Simplified 94.7%

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

   if -7.6000000000000002e59 < t1 < -6.6000000000000002e-246 or 7.00000000000000032e-240 < t1 < 1.6500000000000001e74

   1. Initial program 86.7%

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

      1. associate-/l* 91.2%

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

      2. distribute-lft-neg-out 91.2%

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

      3. distribute-rgt-neg-in 91.2%

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

      4. associate-/r* 95.4%

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

      5. distribute-neg-frac2 95.4%

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

   3. Simplified 95.4%

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

   if -6.6000000000000002e-246 < t1 < 7.00000000000000032e-240

   1. Initial program 69.8%

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

      1. associate-/l* 61.3%

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

      2. distribute-lft-neg-out 61.3%

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

      3. distribute-rgt-neg-in 61.3%

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

      4. associate-/r* 61.3%

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

      5. distribute-neg-frac2 61.3%

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

   3. Simplified 61.3%

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

      1. associate-*r/ 89.5%

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

      2. neg-mul-1 89.5%

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

      3. associate-/r* 89.5%

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

   6. Applied egg-rr 89.5%

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

      1. div-inv 89.2%

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

      2. associate-*r/ 92.6%

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

      3. associate-/l/ 92.6%

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

      4. neg-mul-1 92.6%

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

      5. +-commutative 92.6%

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

      6. distribute-neg-in 92.6%

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

      7. sub-neg 92.6%

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

      8. associate-*l/ 89.2%

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

      9. clear-num 89.1%

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

      10. associate-*l/ 89.2%

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

      11. *-un-lft-identity 89.2%

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

      12. add-sqr-sqrt 51.8%

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

      13. sqrt-unprod 48.2%

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

      14. sqr-neg 48.2%

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

      15. sqrt-unprod 11.9%

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

      16. add-sqr-sqrt 41.9%

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

      17. frac-2neg 41.9%

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

      18. metadata-eval 41.9%

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

      19. +-commutative 41.9%

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

      20. distribute-neg-in 41.9%

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

      21. sub-neg 41.9%

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

   8. Applied egg-rr 89.2%

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

   9. Taylor expanded in u around inf 92.6%

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

   if 1.6500000000000001e74 < t1

   1. Initial program 57.9%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 92.1%

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

      1. clear-num 92.1%

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

      2. frac-times 92.1%

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

      3. *-un-lft-identity 92.1%

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

      4. add-sqr-sqrt 41.7%

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

      5. sqrt-unprod 90.5%

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

      6. sqr-neg 90.5%

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

      7. sqrt-unprod 50.3%

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

      8. add-sqr-sqrt 92.1%

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

   7. Applied egg-rr 92.1%

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

      1. associate-/l/ 92.1%

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

      2. associate-/r/ 69.2%

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

      3. *-commutative 69.2%

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

      4. associate-/l/ 54.1%

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

      5. *-commutative 54.1%

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

      6. associate-/l* 57.8%

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

      7. times-frac 92.1%

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

      8. *-inverses 92.1%

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

      9. associate-*r/ 92.1%

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

      10. metadata-eval 92.1%

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

      11. associate-*r* 92.1%

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

      12. *-commutative 92.1%

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

      13. associate-*r* 92.1%

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

      14. *-commutative 92.1%

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

      15. associate-*l* 92.1%

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

      16. metadata-eval 92.1%

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

      17. *-rgt-identity 92.1%

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

   9. Simplified 92.1%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -6.6 \cdot 10^{-246}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{-240}:\\ \;\;\;\;\frac{t1 \cdot v}{u} \cdot \frac{-1}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.65 \cdot 10^{+74}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u - t1}\\ \mathbf{if}\;t1 \leq -7.2 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -4.4 \cdot 10^{-246}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{t1 \cdot v}{u} \cdot \frac{-1}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.65 \cdot 10^{+74}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- u t1))))
   (if (<= t1 -7.2e+116)
     t_1
     (if (<= t1 -4.4e-246)
       (* (- v) (/ t1 (* (+ t1 u) (+ t1 u))))
       (if (<= t1 4.2e-239)
         (* (/ (* t1 v) u) (/ -1.0 (- u t1)))
         (if (<= t1 1.65e+74) (* (- t1) (/ (/ v (+ t1 u)) (+ t1 u))) t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (u - t1);
	double tmp;
	if (t1 <= -7.2e+116) {
		tmp = t_1;
	} else if (t1 <= -4.4e-246) {
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	} else if (t1 <= 4.2e-239) {
		tmp = ((t1 * v) / u) * (-1.0 / (u - t1));
	} else if (t1 <= 1.65e+74) {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (u - t1)
    if (t1 <= (-7.2d+116)) then
        tmp = t_1
    else if (t1 <= (-4.4d-246)) then
        tmp = -v * (t1 / ((t1 + u) * (t1 + u)))
    else if (t1 <= 4.2d-239) then
        tmp = ((t1 * v) / u) * ((-1.0d0) / (u - t1))
    else if (t1 <= 1.65d+74) then
        tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (u - t1);
	double tmp;
	if (t1 <= -7.2e+116) {
		tmp = t_1;
	} else if (t1 <= -4.4e-246) {
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	} else if (t1 <= 4.2e-239) {
		tmp = ((t1 * v) / u) * (-1.0 / (u - t1));
	} else if (t1 <= 1.65e+74) {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (u - t1)
	tmp = 0
	if t1 <= -7.2e+116:
		tmp = t_1
	elif t1 <= -4.4e-246:
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)))
	elif t1 <= 4.2e-239:
		tmp = ((t1 * v) / u) * (-1.0 / (u - t1))
	elif t1 <= 1.65e+74:
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(u - t1))
	tmp = 0.0
	if (t1 <= -7.2e+116)
		tmp = t_1;
	elseif (t1 <= -4.4e-246)
		tmp = Float64(Float64(-v) * Float64(t1 / Float64(Float64(t1 + u) * Float64(t1 + u))));
	elseif (t1 <= 4.2e-239)
		tmp = Float64(Float64(Float64(t1 * v) / u) * Float64(-1.0 / Float64(u - t1)));
	elseif (t1 <= 1.65e+74)
		tmp = Float64(Float64(-t1) * Float64(Float64(v / Float64(t1 + u)) / Float64(t1 + u)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (u - t1);
	tmp = 0.0;
	if (t1 <= -7.2e+116)
		tmp = t_1;
	elseif (t1 <= -4.4e-246)
		tmp = -v * (t1 / ((t1 + u) * (t1 + u)));
	elseif (t1 <= 4.2e-239)
		tmp = ((t1 * v) / u) * (-1.0 / (u - t1));
	elseif (t1 <= 1.65e+74)
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.2e+116], t$95$1, If[LessEqual[t1, -4.4e-246], N[((-v) * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.2e-239], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] * N[(-1.0 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.65e+74], N[((-t1) * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -7.19999999999999941e116 or 1.6500000000000001e74 < t1

   1. Initial program 48.4%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 94.2%

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

      1. clear-num 94.2%

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

      2. frac-times 94.2%

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

      3. *-un-lft-identity 94.2%

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

      4. add-sqr-sqrt 42.7%

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

      5. sqrt-unprod 90.3%

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

      6. sqr-neg 90.3%

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

      7. sqrt-unprod 51.3%

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

      8. add-sqr-sqrt 94.2%

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

   7. Applied egg-rr 94.2%

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

      1. associate-/l/ 94.2%

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

      2. associate-/r/ 66.2%

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

      3. *-commutative 66.2%

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

      4. associate-/l/ 46.7%

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

      5. *-commutative 46.7%

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

      6. associate-/l* 48.4%

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

      7. times-frac 94.2%

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

      8. *-inverses 94.2%

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

      9. associate-*r/ 94.2%

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

      10. metadata-eval 94.2%

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

      11. associate-*r* 94.2%

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

      12. *-commutative 94.2%

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

      13. associate-*r* 94.2%

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

      14. *-commutative 94.2%

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

      15. associate-*l* 94.2%

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

      16. metadata-eval 94.2%

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

      17. *-rgt-identity 94.2%

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

   9. Simplified 94.2%

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

   if -7.19999999999999941e116 < t1 < -4.39999999999999996e-246

   1. Initial program 89.3%

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

      1. associate-*l/ 95.9%

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

      2. *-commutative 95.9%

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

   3. Simplified 95.9%

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

   if -4.39999999999999996e-246 < t1 < 4.2000000000000004e-239

   1. Initial program 69.8%

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

      1. associate-/l* 61.3%

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

      2. distribute-lft-neg-out 61.3%

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

      3. distribute-rgt-neg-in 61.3%

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

      4. associate-/r* 61.3%

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

      5. distribute-neg-frac2 61.3%

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

   3. Simplified 61.3%

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

      1. associate-*r/ 89.5%

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

      2. neg-mul-1 89.5%

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

      3. associate-/r* 89.5%

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

   6. Applied egg-rr 89.5%

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

      1. div-inv 89.2%

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

      2. associate-*r/ 92.6%

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

      3. associate-/l/ 92.6%

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

      4. neg-mul-1 92.6%

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

      5. +-commutative 92.6%

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

      6. distribute-neg-in 92.6%

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

      7. sub-neg 92.6%

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

      8. associate-*l/ 89.2%

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

      9. clear-num 89.1%

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

      10. associate-*l/ 89.2%

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

      11. *-un-lft-identity 89.2%

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

      12. add-sqr-sqrt 51.8%

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

      13. sqrt-unprod 48.2%

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

      14. sqr-neg 48.2%

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

      15. sqrt-unprod 11.9%

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

      16. add-sqr-sqrt 41.9%

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

      17. frac-2neg 41.9%

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

      18. metadata-eval 41.9%

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

      19. +-commutative 41.9%

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

      20. distribute-neg-in 41.9%

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

      21. sub-neg 41.9%

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

   8. Applied egg-rr 89.2%

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

   9. Taylor expanded in u around inf 92.6%

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

   if 4.2000000000000004e-239 < t1 < 1.6500000000000001e74

   1. Initial program 83.7%

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

      1. associate-/l* 89.2%

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

      2. distribute-lft-neg-out 89.2%

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

      3. distribute-rgt-neg-in 89.2%

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

      4. associate-/r* 95.9%

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

      5. distribute-neg-frac2 95.9%

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

   3. Simplified 95.9%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -4.4 \cdot 10^{-246}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{t1 \cdot v}{u} \cdot \frac{-1}{u - t1}\\ \mathbf{elif}\;t1 \leq 1.65 \cdot 10^{+74}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{-76}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.2e-51)
   (/ -1.0 (/ (+ t1 u) v))
   (if (<= t1 6e-76)
     (* t1 (/ v (* u (- (- u) t1))))
     (/ v (* t1 (/ (- u t1) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.2e-51) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 6e-76) {
		tmp = t1 * (v / (u * (-u - t1)));
	} else {
		tmp = v / (t1 * ((u - t1) / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-5.2d-51)) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else if (t1 <= 6d-76) then
        tmp = t1 * (v / (u * (-u - t1)))
    else
        tmp = v / (t1 * ((u - t1) / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5.2e-51) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 6e-76) {
		tmp = t1 * (v / (u * (-u - t1)));
	} else {
		tmp = v / (t1 * ((u - t1) / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.2e-51:
		tmp = -1.0 / ((t1 + u) / v)
	elif t1 <= 6e-76:
		tmp = t1 * (v / (u * (-u - t1)))
	else:
		tmp = v / (t1 * ((u - t1) / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5.2e-51)
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	elseif (t1 <= 6e-76)
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(Float64(-u) - t1))));
	else
		tmp = Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5.2e-51)
		tmp = -1.0 / ((t1 + u) / v);
	elseif (t1 <= 6e-76)
		tmp = t1 * (v / (u * (-u - t1)));
	else
		tmp = v / (t1 * ((u - t1) / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5.2e-51], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6e-76], N[(t1 * N[(v / N[(u * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.2e-51

   1. Initial program 60.6%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.1%

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

      1. clear-num 88.8%

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

      2. inv-pow 88.8%

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

   7. Applied egg-rr 88.8%

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

      1. unpow-1 88.8%

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

   9. Simplified 88.8%

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

   if -5.2e-51 < t1 < 6.00000000000000048e-76

   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* 82.0%

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

      2. distribute-lft-neg-out 82.0%

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

      3. distribute-rgt-neg-in 82.0%

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

      4. associate-/r* 84.1%

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

      5. distribute-neg-frac2 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in t1 around 0 73.7%

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

   6. Taylor expanded in v around 0 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

   8. Simplified 72.6%

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

   if 6.00000000000000048e-76 < t1

   1. Initial program 68.4%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 83.3%

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

      1. clear-num 83.3%

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

      2. frac-times 83.3%

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

      3. *-un-lft-identity 83.3%

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

      4. add-sqr-sqrt 38.2%

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

      5. sqrt-unprod 82.9%

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

      6. sqr-neg 82.9%

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

      7. sqrt-unprod 44.7%

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

      8. add-sqr-sqrt 83.3%

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

   7. Applied egg-rr 83.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{-76}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e-52)
   (/ -1.0 (/ (+ t1 u) v))
   (if (<= t1 1.2e+16)
     (* t1 (/ (/ v u) (- (- u) t1)))
     (/ v (* t1 (/ (- u t1) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e-52) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.2e+16) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else {
		tmp = v / (t1 * ((u - t1) / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.2d-52)) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else if (t1 <= 1.2d+16) then
        tmp = t1 * ((v / u) / (-u - t1))
    else
        tmp = v / (t1 * ((u - t1) / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e-52) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.2e+16) {
		tmp = t1 * ((v / u) / (-u - t1));
	} else {
		tmp = v / (t1 * ((u - t1) / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e-52:
		tmp = -1.0 / ((t1 + u) / v)
	elif t1 <= 1.2e+16:
		tmp = t1 * ((v / u) / (-u - t1))
	else:
		tmp = v / (t1 * ((u - t1) / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e-52)
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	elseif (t1 <= 1.2e+16)
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e-52)
		tmp = -1.0 / ((t1 + u) / v);
	elseif (t1 <= 1.2e+16)
		tmp = t1 * ((v / u) / (-u - t1));
	else
		tmp = v / (t1 * ((u - t1) / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e-52], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.2e+16], N[(t1 * N[(N[(v / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.2000000000000001e-52

   1. Initial program 60.6%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.1%

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

      1. clear-num 88.8%

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

      2. inv-pow 88.8%

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

   7. Applied egg-rr 88.8%

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

      1. unpow-1 88.8%

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

   9. Simplified 88.8%

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

   if -1.2000000000000001e-52 < t1 < 1.2e16

   1. Initial program 82.0%

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

      1. associate-/l* 83.3%

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

      2. distribute-lft-neg-out 83.3%

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

      3. distribute-rgt-neg-in 83.3%

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

      4. associate-/r* 86.7%

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

      5. distribute-neg-frac2 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in t1 around 0 71.6%

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

   if 1.2e16 < t1

   1. Initial program 62.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 90.3%

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

      1. clear-num 90.3%

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

      2. frac-times 90.3%

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

      3. *-un-lft-identity 90.3%

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

      4. add-sqr-sqrt 40.4%

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

      5. sqrt-unprod 90.0%

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

      6. sqr-neg 90.0%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5e-51)
   (/ -1.0 (/ (+ t1 u) v))
   (if (<= t1 1.15e+16)
     (* (/ t1 (- (- u) t1)) (/ v u))
     (/ v (* t1 (/ (- u t1) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-51) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.15e+16) {
		tmp = (t1 / (-u - t1)) * (v / u);
	} else {
		tmp = v / (t1 * ((u - t1) / 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 <= (-5d-51)) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else if (t1 <= 1.15d+16) then
        tmp = (t1 / (-u - t1)) * (v / u)
    else
        tmp = v / (t1 * ((u - t1) / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -5e-51) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.15e+16) {
		tmp = (t1 / (-u - t1)) * (v / u);
	} else {
		tmp = v / (t1 * ((u - t1) / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -5e-51:
		tmp = -1.0 / ((t1 + u) / v)
	elif t1 <= 1.15e+16:
		tmp = (t1 / (-u - t1)) * (v / u)
	else:
		tmp = v / (t1 * ((u - t1) / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -5e-51)
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	elseif (t1 <= 1.15e+16)
		tmp = Float64(Float64(t1 / Float64(Float64(-u) - t1)) * Float64(v / u));
	else
		tmp = Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -5e-51)
		tmp = -1.0 / ((t1 + u) / v);
	elseif (t1 <= 1.15e+16)
		tmp = (t1 / (-u - t1)) * (v / u);
	else
		tmp = v / (t1 * ((u - t1) / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -5e-51], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.15e+16], N[(N[(t1 / N[((-u) - t1), $MachinePrecision]), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.00000000000000004e-51

   1. Initial program 60.6%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.1%

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

      1. clear-num 88.8%

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

      2. inv-pow 88.8%

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

   7. Applied egg-rr 88.8%

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

      1. unpow-1 88.8%

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

   9. Simplified 88.8%

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

   if -5.00000000000000004e-51 < t1 < 1.15e16

   1. Initial program 82.0%

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

      1. times-frac 94.4%

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

      2. distribute-frac-neg 94.4%

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

      3. distribute-neg-frac2 94.4%

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

      4. +-commutative 94.4%

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

      5. distribute-neg-in 94.4%

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

      6. unsub-neg 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in t1 around 0 76.9%

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

   if 1.15e16 < t1

   1. Initial program 62.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 90.3%

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

      1. clear-num 90.3%

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

      2. frac-times 90.3%

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

      3. *-un-lft-identity 90.3%

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

      4. add-sqr-sqrt 40.4%

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

      5. sqrt-unprod 90.0%

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

      6. sqr-neg 90.0%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -3.1e-53)
   (/ -1.0 (/ (+ t1 u) v))
   (if (<= t1 1.15e+16)
     (/ (* t1 (/ v (- u))) (+ t1 u))
     (/ v (* t1 (/ (- u t1) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.1e-53) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.15e+16) {
		tmp = (t1 * (v / -u)) / (t1 + u);
	} else {
		tmp = v / (t1 * ((u - t1) / 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 <= (-3.1d-53)) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else if (t1 <= 1.15d+16) then
        tmp = (t1 * (v / -u)) / (t1 + u)
    else
        tmp = v / (t1 * ((u - t1) / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -3.1e-53) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.15e+16) {
		tmp = (t1 * (v / -u)) / (t1 + u);
	} else {
		tmp = v / (t1 * ((u - t1) / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -3.1e-53:
		tmp = -1.0 / ((t1 + u) / v)
	elif t1 <= 1.15e+16:
		tmp = (t1 * (v / -u)) / (t1 + u)
	else:
		tmp = v / (t1 * ((u - t1) / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -3.1e-53)
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	elseif (t1 <= 1.15e+16)
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / Float64(t1 + u));
	else
		tmp = Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -3.1e-53)
		tmp = -1.0 / ((t1 + u) / v);
	elseif (t1 <= 1.15e+16)
		tmp = (t1 * (v / -u)) / (t1 + u);
	else
		tmp = v / (t1 * ((u - t1) / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -3.1e-53], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.15e+16], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.10000000000000015e-53

   1. Initial program 60.6%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.1%

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

      1. clear-num 88.8%

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

      2. inv-pow 88.8%

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

   7. Applied egg-rr 88.8%

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

      1. unpow-1 88.8%

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

   9. Simplified 88.8%

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

   if -3.10000000000000015e-53 < t1 < 1.15e16

   1. Initial program 82.0%

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

      1. associate-/l* 83.3%

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

      2. distribute-lft-neg-out 83.3%

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

      3. distribute-rgt-neg-in 83.3%

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

      4. associate-/r* 86.7%

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

      5. distribute-neg-frac2 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in t1 around 0 71.6%

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

      1. *-commutative 71.6%

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

      2. distribute-frac-neg2 71.6%

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

      3. distribute-frac-neg 71.6%

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

      4. associate-*l/ 77.8%

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

      5. distribute-neg-frac2 77.8%

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

      6. +-commutative 77.8%

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

   7. Applied egg-rr 77.8%

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

   if 1.15e16 < t1

   1. Initial program 62.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 90.3%

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

      1. clear-num 90.3%

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

      2. frac-times 90.3%

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

      3. *-un-lft-identity 90.3%

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

      4. add-sqr-sqrt 40.4%

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

      5. sqrt-unprod 90.0%

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

      6. sqr-neg 90.0%

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

      7. sqrt-unprod 49.8%

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

      8. add-sqr-sqrt 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+119} \lor \neg \left(u \leq 4 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2e+119) (not (<= u 4e+87)))
   (/ t1 (* (+ t1 u) (/ u v)))
   (/ v (- (- u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e+119) || !(u <= 4e+87)) {
		tmp = t1 / ((t1 + u) * (u / v));
	} 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 <= (-2d+119)) .or. (.not. (u <= 4d+87))) then
        tmp = t1 / ((t1 + u) * (u / v))
    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 <= -2e+119) || !(u <= 4e+87)) {
		tmp = t1 / ((t1 + u) * (u / v));
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2e+119) or not (u <= 4e+87):
		tmp = t1 / ((t1 + u) * (u / v))
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2e+119) || !(u <= 4e+87))
		tmp = Float64(t1 / Float64(Float64(t1 + u) * Float64(u / v)));
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2e+119) || ~((u <= 4e+87)))
		tmp = t1 / ((t1 + u) * (u / v));
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2e+119], N[Not[LessEqual[u, 4e+87]], $MachinePrecision]], N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.99999999999999989e119 or 3.9999999999999998e87 < u

   1. Initial program 78.4%

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

      1. associate-/l* 79.0%

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

      2. distribute-lft-neg-out 79.0%

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

      3. distribute-rgt-neg-in 79.0%

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

      4. associate-/r* 89.7%

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

      5. distribute-neg-frac2 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in t1 around 0 86.5%

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

      1. clear-num 86.5%

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

      2. un-div-inv 86.5%

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

      3. div-inv 86.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. sqrt-unprod 77.6%

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

      6. sqr-neg 77.6%

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

      7. sqrt-unprod 43.3%

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

      8. add-sqr-sqrt 73.3%

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

      9. clear-num 73.3%

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

      10. +-commutative 73.3%

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

   7. Applied egg-rr 73.3%

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

   if -1.99999999999999989e119 < u < 3.9999999999999998e87

   1. Initial program 67.7%

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

      1. times-frac 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. distribute-neg-frac2 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-neg-in 97.3%

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

      6. unsub-neg 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in t1 around inf 71.5%

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

   6. Taylor expanded in v around 0 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   8. Simplified 71.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+119} \lor \neg \left(u \leq 4 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{t1}{\left(t1 + u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+171} \lor \neg \left(u \leq 1.6 \cdot 10^{+181}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.5e+171) (not (<= u 1.6e+181))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.5e+171) || !(u <= 1.6e+181)) {
		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 <= (-5.5d+171)) .or. (.not. (u <= 1.6d+181))) 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 <= -5.5e+171) || !(u <= 1.6e+181)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.5e+171) or not (u <= 1.6e+181):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.5e+171) || !(u <= 1.6e+181))
		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 <= -5.5e+171) || ~((u <= 1.6e+181)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.5e+171], N[Not[LessEqual[u, 1.6e+181]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.5000000000000003e171 or 1.6e181 < u

   1. Initial program 82.6%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around inf 70.9%

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

      1. clear-num 70.8%

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

      2. frac-times 76.3%

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

      3. *-un-lft-identity 76.3%

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

      4. add-sqr-sqrt 32.6%

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

      5. sqrt-unprod 83.1%

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

      6. sqr-neg 83.1%

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

      7. sqrt-unprod 43.6%

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

      8. add-sqr-sqrt 76.2%

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

   7. Applied egg-rr 76.2%

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

   8. Taylor expanded in u around inf 51.1%

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

   if -5.5000000000000003e171 < u < 1.6e181

   1. Initial program 68.5%

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

      1. associate-/l* 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

      4. associate-/r* 79.2%

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

      5. distribute-neg-frac2 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in t1 around inf 64.4%

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

      1. associate-*r/ 64.4%

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

      2. neg-mul-1 64.4%

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

   7. Simplified 64.4%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+171} \lor \neg \left(u \leq 1.6 \cdot 10^{+181}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.8 \cdot 10^{+174} \lor \neg \left(u \leq 1.16 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.8e+174) (not (<= u 1.16e+154))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.8e+174) || !(u <= 1.16e+154)) {
		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 <= (-7.8d+174)) .or. (.not. (u <= 1.16d+154))) 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 <= -7.8e+174) || !(u <= 1.16e+154)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.8e+174) or not (u <= 1.16e+154):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.8e+174) || !(u <= 1.16e+154))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.8e+174) || ~((u <= 1.16e+154)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.8e+174], N[Not[LessEqual[u, 1.16e+154]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.79999999999999962e174 or 1.16000000000000001e154 < u

   1. Initial program 75.8%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around inf 67.7%

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

   6. Taylor expanded in t1 around 0 48.3%

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

      1. associate-*r/ 48.3%

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

      2. neg-mul-1 48.3%

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

   8. Simplified 48.3%

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

   if -7.79999999999999962e174 < u < 1.16000000000000001e154

   1. Initial program 69.6%

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

      1. associate-/l* 69.3%

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

      2. distribute-lft-neg-out 69.3%

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

      3. distribute-rgt-neg-in 69.3%

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

      4. associate-/r* 79.1%

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

      5. distribute-neg-frac2 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in t1 around inf 65.3%

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

      1. associate-*r/ 65.3%

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

      2. neg-mul-1 65.3%

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

   7. Simplified 65.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.8 \cdot 10^{+174} \lor \neg \left(u \leq 1.16 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{+57} \lor \neg \left(t1 \leq 7.6 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.65e+57) (not (<= t1 7.6e+114))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.65e+57) || !(t1 <= 7.6e+114)) {
		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 <= (-1.65d+57)) .or. (.not. (t1 <= 7.6d+114))) 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 <= -1.65e+57) || !(t1 <= 7.6e+114)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.65e+57) or not (t1 <= 7.6e+114):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.65e+57) || !(t1 <= 7.6e+114))
		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 <= -1.65e+57) || ~((t1 <= 7.6e+114)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.65e+57], N[Not[LessEqual[t1, 7.6e+114]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6500000000000001e57 or 7.6000000000000001e114 < 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. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 94.3%

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

   6. Taylor expanded in u around inf 31.6%

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

   if -1.6500000000000001e57 < t1 < 7.6000000000000001e114

   1. Initial program 83.4%

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

      1. times-frac 95.8%

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

      2. distribute-frac-neg 95.8%

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

      3. distribute-neg-frac2 95.8%

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

      4. +-commutative 95.8%

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

      5. distribute-neg-in 95.8%

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

      6. unsub-neg 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in t1 around inf 52.4%

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

      1. clear-num 52.4%

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

      2. frac-times 54.5%

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

      3. *-un-lft-identity 54.5%

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

      4. add-sqr-sqrt 26.5%

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

      5. sqrt-unprod 57.8%

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

      6. sqr-neg 57.8%

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

      7. sqrt-unprod 27.7%

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

      8. add-sqr-sqrt 54.0%

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

   7. Applied egg-rr 54.0%

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

   8. Taylor expanded in u around inf 17.1%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.65 \cdot 10^{+57} \lor \neg \left(t1 \leq 7.6 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 1.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 1.7e+90) (/ v (- (- u) t1)) (/ (* v (/ t1 (- u))) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.7e+90) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v * (t1 / -u)) / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 1.7d+90) then
        tmp = v / (-u - t1)
    else
        tmp = (v * (t1 / -u)) / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.7e+90) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v * (t1 / -u)) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1.7e+90:
		tmp = v / (-u - t1)
	else:
		tmp = (v * (t1 / -u)) / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1.7e+90)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(v * Float64(t1 / Float64(-u))) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1.7e+90)
		tmp = v / (-u - t1);
	else
		tmp = (v * (t1 / -u)) / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1.7e+90], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 1.70000000000000009e90

   1. Initial program 69.9%

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

      1. times-frac 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. distribute-neg-frac2 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-neg-in 97.3%

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

      6. unsub-neg 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in t1 around inf 70.7%

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

   6. Taylor expanded in v around 0 68.7%

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

      1. associate-*r/ 68.7%

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

      2. neg-mul-1 68.7%

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

   8. Simplified 68.7%

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

   if 1.70000000000000009e90 < u

   1. Initial program 74.7%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around inf 61.0%

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

   6. Taylor expanded in t1 around 0 54.2%

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

      1. associate-*r/ 54.2%

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

      2. mul-1-neg 54.2%

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

   8. Simplified 54.2%

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

      1. frac-2neg 54.2%

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

      2. associate-*r/ 64.7%

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

      3. add-sqr-sqrt 24.8%

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

      4. sqrt-unprod 47.8%

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

      5. sqr-neg 47.8%

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

      6. sqrt-unprod 39.2%

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

      7. add-sqr-sqrt 64.0%

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

      8. add-sqr-sqrt 24.8%

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

      9. sqrt-unprod 51.1%

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

      10. sqr-neg 51.1%

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

      11. sqrt-unprod 39.8%

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

      12. add-sqr-sqrt 64.7%

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

   10. Applied egg-rr 64.7%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{-u}}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 8 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 8e+217) (/ v (- (- u) t1)) (* v (/ (/ t1 u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 8e+217) {
		tmp = v / (-u - t1);
	} else {
		tmp = v * ((t1 / u) / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 8d+217) then
        tmp = v / (-u - t1)
    else
        tmp = v * ((t1 / u) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 8e+217) {
		tmp = v / (-u - t1);
	} else {
		tmp = v * ((t1 / u) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 8e+217:
		tmp = v / (-u - t1)
	else:
		tmp = v * ((t1 / u) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 8e+217)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 8e+217)
		tmp = v / (-u - t1);
	else
		tmp = v * ((t1 / u) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 8e+217], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(t1 / u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 7.99999999999999968e217

   1. Initial program 69.4%

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

      1. times-frac 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. distribute-neg-frac2 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-neg-in 97.3%

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

      6. unsub-neg 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in t1 around inf 69.4%

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

   6. Taylor expanded in v around 0 66.3%

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

      1. associate-*r/ 66.3%

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

      2. neg-mul-1 66.3%

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

   8. Simplified 66.3%

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

   if 7.99999999999999968e217 < u

   1. Initial program 88.2%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 66.7%

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

   6. Taylor expanded in t1 around 0 66.7%

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

      1. associate-*r/ 66.7%

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

      2. mul-1-neg 66.7%

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

   8. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. clear-num 66.7%

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

      3. frac-times 60.6%

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

      4. *-un-lft-identity 60.6%

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

      5. add-sqr-sqrt 18.5%

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

      6. sqrt-unprod 36.4%

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

      7. sqr-neg 36.4%

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

      8. sqrt-unprod 41.9%

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

      9. add-sqr-sqrt 60.5%

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

   10. Applied egg-rr 60.5%

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

       1. associate-/l/ 66.6%

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

       2. associate-/r/ 78.3%

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

   12. Simplified 78.3%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 8 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 1.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 1.7e+90) (/ v (- (- u) t1)) (/ (* v (/ t1 u)) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.7e+90) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v * (t1 / u)) / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 1.7d+90) then
        tmp = v / (-u - t1)
    else
        tmp = (v * (t1 / u)) / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 1.7e+90) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v * (t1 / u)) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 1.7e+90:
		tmp = v / (-u - t1)
	else:
		tmp = (v * (t1 / u)) / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 1.7e+90)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 1.7e+90)
		tmp = v / (-u - t1);
	else
		tmp = (v * (t1 / u)) / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 1.7e+90], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 1.70000000000000009e90

   1. Initial program 69.9%

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

      1. times-frac 97.3%

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

      2. distribute-frac-neg 97.3%

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

      3. distribute-neg-frac2 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-neg-in 97.3%

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

      6. unsub-neg 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in t1 around inf 70.7%

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

   6. Taylor expanded in v around 0 68.7%

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

      1. associate-*r/ 68.7%

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

      2. neg-mul-1 68.7%

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

   8. Simplified 68.7%

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

   if 1.70000000000000009e90 < u

   1. Initial program 74.7%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around inf 61.0%

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

   6. Taylor expanded in t1 around 0 54.2%

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

      1. associate-*r/ 54.2%

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

      2. mul-1-neg 54.2%

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

   8. Simplified 54.2%

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

      1. *-commutative 54.2%

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

      2. associate-*l/ 64.7%

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

      3. add-sqr-sqrt 24.8%

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

      4. sqrt-unprod 47.8%

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

      5. sqr-neg 47.8%

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

      6. sqrt-unprod 39.2%

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

      7. add-sqr-sqrt 64.0%

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

   10. Applied egg-rr 64.0%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. times-frac 97.5%

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

   2. distribute-frac-neg 97.5%

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

   3. distribute-neg-frac2 97.5%

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

   4. +-commutative 97.5%

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

   5. distribute-neg-in 97.5%

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

   6. unsub-neg 97.5%

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

3. Simplified 97.5%

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

5. Final simplification 97.5%

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

Alternative 16: 69.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. times-frac 97.5%

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

   2. distribute-frac-neg 97.5%

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

   3. distribute-neg-frac2 97.5%

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

   4. +-commutative 97.5%

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

   5. distribute-neg-in 97.5%

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

   6. unsub-neg 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in t1 around inf 69.2%

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

   1. clear-num 69.2%

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

   2. frac-times 70.5%

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

   3. *-un-lft-identity 70.5%

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

   4. add-sqr-sqrt 32.8%

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

   5. sqrt-unprod 71.0%

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

   6. sqr-neg 71.0%

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

   7. sqrt-unprod 37.5%

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

   8. add-sqr-sqrt 70.2%

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

7. Applied egg-rr 70.2%

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

8. Final simplification 70.2%

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

Alternative 17: 71.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (v / ((u - t1) / t1)) / 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) / t1)) / t1
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. times-frac 97.5%

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

   2. distribute-frac-neg 97.5%

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

   3. distribute-neg-frac2 97.5%

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

   4. +-commutative 97.5%

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

   5. distribute-neg-in 97.5%

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

   6. unsub-neg 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in t1 around inf 69.2%

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

   1. associate-*r/ 73.7%

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

   2. clear-num 73.7%

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

   3. associate-*l/ 73.7%

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

   4. *-un-lft-identity 73.7%

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

   5. add-sqr-sqrt 34.7%

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

   6. sqrt-unprod 73.9%

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

   7. sqr-neg 73.9%

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

   8. sqrt-unprod 38.9%

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

   9. add-sqr-sqrt 73.4%

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

7. Applied egg-rr 73.4%

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

8. Final simplification 73.4%

   \[\leadsto \frac{\frac{v}{\frac{u - t1}{t1}}}{t1} \]
9. Add Preprocessing

Alternative 18: 61.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. times-frac 97.5%

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

   2. distribute-frac-neg 97.5%

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

   3. distribute-neg-frac2 97.5%

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

   4. +-commutative 97.5%

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

   5. distribute-neg-in 97.5%

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

   6. unsub-neg 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in t1 around inf 69.2%

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

6. Taylor expanded in v around 0 65.3%

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

   1. associate-*r/ 65.3%

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

   2. neg-mul-1 65.3%

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

8. Simplified 65.3%

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

9. Final simplification 65.3%

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

Alternative 19: 61.6% 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 70.6%

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

   1. times-frac 97.5%

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

   2. distribute-frac-neg 97.5%

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

   3. distribute-neg-frac2 97.5%

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

   4. +-commutative 97.5%

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

   5. distribute-neg-in 97.5%

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

   6. unsub-neg 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in t1 around inf 69.2%

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

   1. clear-num 69.2%

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

   2. frac-times 70.5%

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

   3. *-un-lft-identity 70.5%

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

   4. add-sqr-sqrt 32.8%

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

   5. sqrt-unprod 71.0%

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

   6. sqr-neg 71.0%

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

   7. sqrt-unprod 37.5%

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

   8. add-sqr-sqrt 70.2%

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

7. Applied egg-rr 70.2%

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

   1. associate-/l/ 68.9%

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

   2. associate-/r/ 52.7%

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

   3. *-commutative 52.7%

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

   4. associate-/l/ 44.9%

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

   5. *-commutative 44.9%

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

   6. associate-/l* 47.3%

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

   7. times-frac 64.9%

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

   8. *-inverses 64.9%

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

   9. associate-*r/ 64.9%

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

   10. metadata-eval 64.9%

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

   11. associate-*r* 64.9%

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

   12. *-commutative 64.9%

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

   13. associate-*r* 64.9%

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

   14. *-commutative 64.9%

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

   15. associate-*l* 64.9%

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

   16. metadata-eval 64.9%

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

   17. *-rgt-identity 64.9%

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

9. Simplified 64.9%

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

10. Final simplification 64.9%

    \[\leadsto \frac{v}{u - t1} \]
11. Add Preprocessing

Alternative 20: 14.3% 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 70.6%

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

   1. times-frac 97.5%

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

   2. distribute-frac-neg 97.5%

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

   3. distribute-neg-frac2 97.5%

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

   4. +-commutative 97.5%

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

   5. distribute-neg-in 97.5%

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

   6. unsub-neg 97.5%

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

3. Simplified 97.5%

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

5. Taylor expanded in t1 around inf 60.9%

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

6. Taylor expanded in u around inf 14.7%

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

7. Final simplification 14.7%

   \[\leadsto \frac{v}{t1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))