Rosa's DopplerBench

Percentage Accurate: 73.1% → 98.1%

Time: 10.0s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. associate-/l* 76.7%

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

   2. distribute-lft-neg-out 76.7%

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

   3. distribute-rgt-neg-in 76.7%

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

   4. associate-/r* 85.6%

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

   5. distribute-neg-frac2 85.6%

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

3. Simplified 85.6%

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

   1. distribute-frac-neg2 85.6%

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

   2. associate-/r* 76.7%

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

   3. distribute-rgt-neg-in 76.7%

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

   4. distribute-lft-neg-out 76.7%

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

   5. associate-*r/ 77.0%

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

   6. times-frac 97.2%

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

   7. frac-2neg 97.2%

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

   8. associate-*r/ 99.0%

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

   9. add-sqr-sqrt 50.4%

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

   10. sqrt-unprod 51.7%

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

   11. sqr-neg 51.7%

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

   12. sqrt-unprod 19.9%

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

   13. add-sqr-sqrt 39.2%

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

   14. add-sqr-sqrt 21.3%

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

   15. sqrt-unprod 63.1%

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

   16. sqr-neg 63.1%

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

   17. sqrt-prod 50.0%

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

   18. add-sqr-sqrt 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{if}\;t1 \leq -1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;t1 \leq -4.8 \cdot 10^{-154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -2.1 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 5.1 \cdot 10^{+178}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))))
   (if (<= t1 -1.35e+104)
     (/ v (- (* u (- 2.0)) t1))
     (if (<= t1 -4.8e-154)
       t_1
       (if (<= t1 -2.1e-290)
         (/ (/ (* t1 v) u) (- u))
         (if (<= t1 5.1e+178) t_1 (/ v (- t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	double tmp;
	if (t1 <= -1.35e+104) {
		tmp = v / ((u * -2.0) - t1);
	} else if (t1 <= -4.8e-154) {
		tmp = t_1;
	} else if (t1 <= -2.1e-290) {
		tmp = ((t1 * v) / u) / -u;
	} else if (t1 <= 5.1e+178) {
		tmp = t_1;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t1 * ((v / (t1 + u)) / (-u - t1))
    if (t1 <= (-1.35d+104)) then
        tmp = v / ((u * -2.0d0) - t1)
    else if (t1 <= (-4.8d-154)) then
        tmp = t_1
    else if (t1 <= (-2.1d-290)) then
        tmp = ((t1 * v) / u) / -u
    else if (t1 <= 5.1d+178) then
        tmp = t_1
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	double tmp;
	if (t1 <= -1.35e+104) {
		tmp = v / ((u * -2.0) - t1);
	} else if (t1 <= -4.8e-154) {
		tmp = t_1;
	} else if (t1 <= -2.1e-290) {
		tmp = ((t1 * v) / u) / -u;
	} else if (t1 <= 5.1e+178) {
		tmp = t_1;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * ((v / (t1 + u)) / (-u - t1))
	tmp = 0
	if t1 <= -1.35e+104:
		tmp = v / ((u * -2.0) - t1)
	elif t1 <= -4.8e-154:
		tmp = t_1
	elif t1 <= -2.1e-290:
		tmp = ((t1 * v) / u) / -u
	elif t1 <= 5.1e+178:
		tmp = t_1
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)))
	tmp = 0.0
	if (t1 <= -1.35e+104)
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	elseif (t1 <= -4.8e-154)
		tmp = t_1;
	elseif (t1 <= -2.1e-290)
		tmp = Float64(Float64(Float64(t1 * v) / u) / Float64(-u));
	elseif (t1 <= 5.1e+178)
		tmp = t_1;
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * ((v / (t1 + u)) / (-u - t1));
	tmp = 0.0;
	if (t1 <= -1.35e+104)
		tmp = v / ((u * -2.0) - t1);
	elseif (t1 <= -4.8e-154)
		tmp = t_1;
	elseif (t1 <= -2.1e-290)
		tmp = ((t1 * v) / u) / -u;
	elseif (t1 <= 5.1e+178)
		tmp = t_1;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.35e+104], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -4.8e-154], t$95$1, If[LessEqual[t1, -2.1e-290], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[t1, 5.1e+178], t$95$1, N[(v / (-t1)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.34999999999999992e104

   1. Initial program 49.1%

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

      1. associate-/l* 43.4%

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

      2. distribute-lft-neg-out 43.4%

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

      3. distribute-rgt-neg-in 43.4%

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

      4. associate-/r* 66.9%

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

      5. distribute-neg-frac2 66.9%

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

   3. Simplified 66.9%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-*l/ 100.0%

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

      6. clear-num 100.0%

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

      7. frac-2neg 100.0%

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

      8. frac-times 100.0%

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

      9. *-un-lft-identity 100.0%

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

      10. add-sqr-sqrt 93.9%

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

      11. sqrt-unprod 51.0%

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

      12. sqr-neg 51.0%

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

      13. sqrt-prod 2.9%

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

      14. add-sqr-sqrt 42.4%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in u around 0 97.5%

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

      1. *-commutative 97.5%

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

   9. Simplified 97.5%

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

   if -1.34999999999999992e104 < t1 < -4.79999999999999974e-154 or -2.1000000000000001e-290 < t1 < 5.0999999999999997e178

   1. Initial program 85.9%

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

      1. associate-/l* 89.0%

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

      2. distribute-lft-neg-out 89.0%

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

      3. distribute-rgt-neg-in 89.0%

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

      4. associate-/r* 93.4%

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

      5. distribute-neg-frac2 93.4%

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

   3. Simplified 93.4%

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

   if -4.79999999999999974e-154 < t1 < -2.1000000000000001e-290

   1. Initial program 95.9%

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

      1. times-frac 84.6%

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

      2. distribute-frac-neg 84.6%

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

      3. distribute-neg-frac2 84.6%

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

      4. +-commutative 84.6%

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

      5. distribute-neg-in 84.6%

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

      6. unsub-neg 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in t1 around 0 84.5%

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

      1. associate-*r/ 84.5%

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

      2. mul-1-neg 84.5%

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

   7. Simplified 84.5%

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

      1. associate-*l/ 88.3%

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

      2. frac-2neg 88.3%

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

      3. add-sqr-sqrt 88.3%

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

      4. sqrt-unprod 50.1%

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

      5. sqr-neg 50.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 42.7%

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

      8. distribute-lft-neg-out 42.7%

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

      9. add-sqr-sqrt 42.7%

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

      10. sqrt-unprod 43.2%

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

      11. sqr-neg 43.2%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 88.3%

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

   9. Applied egg-rr 88.3%

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

   10. Taylor expanded in t1 around 0 99.7%

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

   if 5.0999999999999997e178 < t1

   1. Initial program 13.6%

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

      1. associate-/l* 16.1%

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

      2. distribute-lft-neg-out 16.1%

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

      3. distribute-rgt-neg-in 16.1%

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

      4. associate-/r* 54.6%

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

      5. distribute-neg-frac2 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in t1 around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;t1 \leq -4.8 \cdot 10^{-154}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq -2.1 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 5.1 \cdot 10^{+178}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38} \lor \neg \left(t1 \leq 4.1 \cdot 10^{-125} \lor \neg \left(t1 \leq 8.4 \cdot 10^{-91}\right) \land t1 \leq 4.5 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.9e+38)
         (not
          (or (<= t1 4.1e-125) (and (not (<= t1 8.4e-91)) (<= t1 4.5e-46)))))
   (/ v (- (* u (- 2.0)) t1))
   (* (/ v u) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.9e+38) || !((t1 <= 4.1e-125) || (!(t1 <= 8.4e-91) && (t1 <= 4.5e-46)))) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.9d+38)) .or. (.not. (t1 <= 4.1d-125) .or. (.not. (t1 <= 8.4d-91)) .and. (t1 <= 4.5d-46))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.9e+38) || !((t1 <= 4.1e-125) || (!(t1 <= 8.4e-91) && (t1 <= 4.5e-46)))) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.9e+38) or not ((t1 <= 4.1e-125) or (not (t1 <= 8.4e-91) and (t1 <= 4.5e-46))):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.9e+38) || !((t1 <= 4.1e-125) || (!(t1 <= 8.4e-91) && (t1 <= 4.5e-46))))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.9e+38) || ~(((t1 <= 4.1e-125) || (~((t1 <= 8.4e-91)) && (t1 <= 4.5e-46)))))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.9e+38], N[Not[Or[LessEqual[t1, 4.1e-125], And[N[Not[LessEqual[t1, 8.4e-91]], $MachinePrecision], LessEqual[t1, 4.5e-46]]]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.89999999999999981e38 or 4.0999999999999997e-125 < t1 < 8.3999999999999997e-91 or 4.50000000000000001e-46 < t1

   1. Initial program 62.4%

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

      1. associate-/l* 63.4%

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

      2. distribute-lft-neg-out 63.4%

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

      3. distribute-rgt-neg-in 63.4%

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

      4. associate-/r* 77.3%

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

      5. distribute-neg-frac2 77.3%

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

   3. Simplified 77.3%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

      3. distribute-neg-in 99.3%

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

      4. sub-neg 99.3%

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

      5. associate-*l/ 99.2%

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

      6. clear-num 99.2%

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

      7. frac-2neg 99.2%

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

      8. frac-times 96.8%

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

      9. *-un-lft-identity 96.8%

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

      10. add-sqr-sqrt 44.8%

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

      11. sqrt-unprod 45.6%

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

      12. sqr-neg 45.6%

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

      13. sqrt-prod 13.7%

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

      14. add-sqr-sqrt 32.3%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in u around 0 87.5%

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

      1. *-commutative 87.5%

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

   9. Simplified 87.5%

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

   if -5.89999999999999981e38 < t1 < 4.0999999999999997e-125 or 8.3999999999999997e-91 < t1 < 4.50000000000000001e-46

   1. Initial program 91.8%

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

      1. times-frac 95.2%

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

      2. distribute-frac-neg 95.2%

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

      3. distribute-neg-frac2 95.2%

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

      4. +-commutative 95.2%

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

      5. distribute-neg-in 95.2%

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

      6. unsub-neg 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in t1 around 0 80.5%

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

      1. associate-*r/ 80.5%

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

      2. mul-1-neg 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in t1 around 0 81.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38} \lor \neg \left(t1 \leq 4.1 \cdot 10^{-125} \lor \neg \left(t1 \leq 8.4 \cdot 10^{-91}\right) \land t1 \leq 4.5 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+38} \lor \neg \left(t1 \leq 4.1 \cdot 10^{-125} \lor \neg \left(t1 \leq 1.3 \cdot 10^{-90}\right) \land t1 \leq 2.9 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7.4e+38)
         (not
          (or (<= t1 4.1e-125) (and (not (<= t1 1.3e-90)) (<= t1 2.9e-46)))))
   (/ v (- (- u) t1))
   (* (/ v u) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.4e+38) || !((t1 <= 4.1e-125) || (!(t1 <= 1.3e-90) && (t1 <= 2.9e-46)))) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-7.4d+38)) .or. (.not. (t1 <= 4.1d-125) .or. (.not. (t1 <= 1.3d-90)) .and. (t1 <= 2.9d-46))) then
        tmp = v / (-u - t1)
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7.4e+38) || !((t1 <= 4.1e-125) || (!(t1 <= 1.3e-90) && (t1 <= 2.9e-46)))) {
		tmp = v / (-u - t1);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -7.4e+38) or not ((t1 <= 4.1e-125) or (not (t1 <= 1.3e-90) and (t1 <= 2.9e-46))):
		tmp = v / (-u - t1)
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7.4e+38) || !((t1 <= 4.1e-125) || (!(t1 <= 1.3e-90) && (t1 <= 2.9e-46))))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -7.4e+38) || ~(((t1 <= 4.1e-125) || (~((t1 <= 1.3e-90)) && (t1 <= 2.9e-46)))))
		tmp = v / (-u - t1);
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7.4e+38], N[Not[Or[LessEqual[t1, 4.1e-125], And[N[Not[LessEqual[t1, 1.3e-90]], $MachinePrecision], LessEqual[t1, 2.9e-46]]]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -7.4000000000000002e38 or 4.0999999999999997e-125 < t1 < 1.3e-90 or 2.90000000000000005e-46 < t1

   1. Initial program 62.4%

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

      1. associate-/l* 63.4%

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

      2. distribute-lft-neg-out 63.4%

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

      3. distribute-rgt-neg-in 63.4%

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

      4. associate-/r* 77.3%

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

      5. distribute-neg-frac2 77.3%

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

   3. Simplified 77.3%

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

      1. distribute-frac-neg2 77.3%

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

      2. associate-/r* 63.4%

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

      3. distribute-rgt-neg-in 63.4%

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

      4. distribute-lft-neg-out 63.4%

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

      5. associate-*r/ 62.4%

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

      6. times-frac 99.2%

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

      7. frac-2neg 99.2%

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

      8. associate-*r/ 100.0%

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

      9. add-sqr-sqrt 44.7%

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

      10. sqrt-unprod 31.3%

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

      11. sqr-neg 31.3%

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

      12. sqrt-unprod 14.8%

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

      13. add-sqr-sqrt 32.3%

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

      14. add-sqr-sqrt 18.6%

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

      15. sqrt-unprod 58.0%

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

      16. sqr-neg 58.0%

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

      17. sqrt-prod 53.2%

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

      18. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 87.3%

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

      1. mul-1-neg 87.3%

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

   9. Simplified 87.3%

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

   if -7.4000000000000002e38 < t1 < 4.0999999999999997e-125 or 1.3e-90 < t1 < 2.90000000000000005e-46

   1. Initial program 91.8%

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

      1. times-frac 95.2%

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

      2. distribute-frac-neg 95.2%

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

      3. distribute-neg-frac2 95.2%

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

      4. +-commutative 95.2%

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

      5. distribute-neg-in 95.2%

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

      6. unsub-neg 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in t1 around 0 80.5%

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

      1. associate-*r/ 80.5%

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

      2. mul-1-neg 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in t1 around 0 81.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+38} \lor \neg \left(t1 \leq 4.1 \cdot 10^{-125} \lor \neg \left(t1 \leq 1.3 \cdot 10^{-90}\right) \land t1 \leq 2.9 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-126}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 8.4 \cdot 10^{-91} \lor \neg \left(t1 \leq 5 \cdot 10^{-46}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u (- 2.0)) t1))))
   (if (<= t1 -5.9e+38)
     t_1
     (if (<= t1 1.35e-126)
       (/ (* t1 (/ v u)) (- u))
       (if (or (<= t1 8.4e-91) (not (<= t1 5e-46)))
         t_1
         (* (/ v u) (/ t1 (- u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = t_1;
	} else if (t1 <= 1.35e-126) {
		tmp = (t1 * (v / u)) / -u;
	} else if ((t1 <= 8.4e-91) || !(t1 <= 5e-46)) {
		tmp = t_1;
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * -2.0d0) - t1)
    if (t1 <= (-5.9d+38)) then
        tmp = t_1
    else if (t1 <= 1.35d-126) then
        tmp = (t1 * (v / u)) / -u
    else if ((t1 <= 8.4d-91) .or. (.not. (t1 <= 5d-46))) then
        tmp = t_1
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -5.9e+38) {
		tmp = t_1;
	} else if (t1 <= 1.35e-126) {
		tmp = (t1 * (v / u)) / -u;
	} else if ((t1 <= 8.4e-91) || !(t1 <= 5e-46)) {
		tmp = t_1;
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -5.9e+38:
		tmp = t_1
	elif t1 <= 1.35e-126:
		tmp = (t1 * (v / u)) / -u
	elif (t1 <= 8.4e-91) or not (t1 <= 5e-46):
		tmp = t_1
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1))
	tmp = 0.0
	if (t1 <= -5.9e+38)
		tmp = t_1;
	elseif (t1 <= 1.35e-126)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	elseif ((t1 <= 8.4e-91) || !(t1 <= 5e-46))
		tmp = t_1;
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -5.9e+38)
		tmp = t_1;
	elseif (t1 <= 1.35e-126)
		tmp = (t1 * (v / u)) / -u;
	elseif ((t1 <= 8.4e-91) || ~((t1 <= 5e-46)))
		tmp = t_1;
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.9e+38], t$95$1, If[LessEqual[t1, 1.35e-126], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[Or[LessEqual[t1, 8.4e-91], N[Not[LessEqual[t1, 5e-46]], $MachinePrecision]], t$95$1, N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.89999999999999981e38 or 1.34999999999999998e-126 < t1 < 8.3999999999999997e-91 or 4.99999999999999992e-46 < t1

   1. Initial program 62.4%

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

      1. associate-/l* 63.4%

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

      2. distribute-lft-neg-out 63.4%

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

      3. distribute-rgt-neg-in 63.4%

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

      4. associate-/r* 77.3%

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

      5. distribute-neg-frac2 77.3%

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

   3. Simplified 77.3%

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

      1. associate-*r/ 99.3%

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

      2. +-commutative 99.3%

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

      3. distribute-neg-in 99.3%

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

      4. sub-neg 99.3%

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

      5. associate-*l/ 99.2%

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

      6. clear-num 99.2%

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

      7. frac-2neg 99.2%

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

      8. frac-times 96.8%

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

      9. *-un-lft-identity 96.8%

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

      10. add-sqr-sqrt 44.8%

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

      11. sqrt-unprod 45.6%

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

      12. sqr-neg 45.6%

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

      13. sqrt-prod 13.7%

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

      14. add-sqr-sqrt 32.3%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in u around 0 87.5%

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

      1. *-commutative 87.5%

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

   9. Simplified 87.5%

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

   if -5.89999999999999981e38 < t1 < 1.34999999999999998e-126

   1. Initial program 91.0%

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

      1. times-frac 94.7%

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

      2. distribute-frac-neg 94.7%

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

      3. distribute-neg-frac2 94.7%

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

      4. +-commutative 94.7%

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

      5. distribute-neg-in 94.7%

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

      6. unsub-neg 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in t1 around 0 79.3%

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

      1. associate-*r/ 79.3%

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

      2. mul-1-neg 79.3%

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

   7. Simplified 79.3%

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

      1. associate-*l/ 81.1%

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

      2. frac-2neg 81.1%

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

      3. add-sqr-sqrt 47.1%

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

      4. sqrt-unprod 60.1%

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

      5. sqr-neg 60.1%

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

      6. sqrt-unprod 21.7%

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

      7. add-sqr-sqrt 46.1%

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

      8. distribute-lft-neg-out 46.1%

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

      9. add-sqr-sqrt 24.5%

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

      10. sqrt-unprod 48.0%

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

      11. sqr-neg 48.0%

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

      12. sqrt-unprod 33.9%

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

      13. add-sqr-sqrt 81.1%

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

   9. Applied egg-rr 81.1%

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

   10. Taylor expanded in t1 around 0 82.8%

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

       1. associate-/l* 82.2%

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

   12. Simplified 82.2%

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

   if 8.3999999999999997e-91 < t1 < 4.99999999999999992e-46

   1. Initial program 99.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 91.9%

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

      1. associate-*r/ 91.9%

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

      2. mul-1-neg 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in t1 around 0 91.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-126}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 8.4 \cdot 10^{-91} \lor \neg \left(t1 \leq 5 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38} \lor \neg \left(t1 \leq 4.5 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.9e+38) (not (<= t1 4.5e-46)))
   (/ v (- (* u (- 2.0)) t1))
   (/ (/ (* t1 v) u) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.9e+38) || !(t1 <= 4.5e-46)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = ((t1 * v) / u) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.9d+38)) .or. (.not. (t1 <= 4.5d-46))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = ((t1 * v) / u) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.9e+38) || !(t1 <= 4.5e-46)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = ((t1 * v) / u) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.9e+38) or not (t1 <= 4.5e-46):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = ((t1 * v) / u) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.9e+38) || !(t1 <= 4.5e-46))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(Float64(Float64(t1 * v) / u) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.9e+38) || ~((t1 <= 4.5e-46)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = ((t1 * v) / u) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.9e+38], N[Not[LessEqual[t1, 4.5e-46]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.89999999999999981e38 or 4.50000000000000001e-46 < t1

   1. Initial program 60.4%

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

      1. associate-/l* 63.0%

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

      2. distribute-lft-neg-out 63.0%

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

      3. distribute-rgt-neg-in 63.0%

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

      4. associate-/r* 78.0%

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

      5. distribute-neg-frac2 78.0%

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

   3. Simplified 78.0%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-*l/ 99.9%

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

      6. clear-num 99.9%

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

      7. frac-2neg 99.9%

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

      8. frac-times 96.6%

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

      9. *-un-lft-identity 96.6%

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

      10. add-sqr-sqrt 47.3%

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

      11. sqrt-unprod 48.2%

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

      12. sqr-neg 48.2%

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

      13. sqrt-prod 14.7%

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

      14. add-sqr-sqrt 33.9%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in u around 0 88.1%

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

      1. *-commutative 88.1%

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

   9. Simplified 88.1%

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

   if -5.89999999999999981e38 < t1 < 4.50000000000000001e-46

   1. Initial program 91.6%

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

      1. times-frac 94.9%

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

      2. distribute-frac-neg 94.9%

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

      3. distribute-neg-frac2 94.9%

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

      4. +-commutative 94.9%

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

      5. distribute-neg-in 94.9%

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

      6. unsub-neg 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in t1 around 0 76.9%

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

      1. associate-*r/ 76.9%

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

      2. mul-1-neg 76.9%

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

   7. Simplified 76.9%

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

      1. associate-*l/ 78.4%

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

      2. frac-2neg 78.4%

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

      3. add-sqr-sqrt 39.9%

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

      4. sqrt-unprod 57.0%

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

      5. sqr-neg 57.0%

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

      6. sqrt-unprod 24.5%

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

      7. add-sqr-sqrt 45.2%

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

      8. distribute-lft-neg-out 45.2%

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

      9. add-sqr-sqrt 20.7%

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

      10. sqrt-unprod 50.4%

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

      11. sqr-neg 50.4%

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

      12. sqrt-unprod 38.5%

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

      13. add-sqr-sqrt 78.4%

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

   9. Applied egg-rr 78.4%

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

   10. Taylor expanded in t1 around 0 80.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.9 \cdot 10^{+38} \lor \neg \left(t1 \leq 4.5 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+51} \lor \neg \left(u \leq 1.4 \cdot 10^{+75}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.7e+51) (not (<= u 1.4e+75)))
   (* t1 (/ (/ v u) u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e+51) || !(u <= 1.4e+75)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.7d+51)) .or. (.not. (u <= 1.4d+75))) then
        tmp = t1 * ((v / u) / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e+51) || !(u <= 1.4e+75)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.7e+51) or not (u <= 1.4e+75):
		tmp = t1 * ((v / u) / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.7e+51) || !(u <= 1.4e+75))
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.7e+51) || ~((u <= 1.4e+75)))
		tmp = t1 * ((v / u) / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.7e+51], N[Not[LessEqual[u, 1.4e+75]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.69999999999999992e51 or 1.40000000000000006e75 < u

   1. Initial program 81.9%

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

      1. times-frac 97.8%

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

      2. distribute-frac-neg 97.8%

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

      3. distribute-neg-frac2 97.8%

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

      4. +-commutative 97.8%

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

      5. distribute-neg-in 97.8%

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

      6. unsub-neg 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in t1 around 0 86.6%

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

      1. associate-*r/ 86.6%

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

      2. mul-1-neg 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in t1 around 0 84.7%

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

      1. clear-num 83.5%

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

      2. frac-times 74.7%

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

      3. *-un-lft-identity 74.7%

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

      4. add-sqr-sqrt 32.9%

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

      5. sqrt-unprod 67.1%

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

      6. sqr-neg 67.1%

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

      7. sqrt-unprod 39.5%

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

      8. add-sqr-sqrt 69.0%

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

   10. Applied egg-rr 69.0%

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

       1. clear-num 69.0%

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

       2. *-commutative 69.0%

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

       3. associate-/l* 68.9%

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

       4. associate-/r* 68.7%

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

       5. associate-/l/ 68.7%

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

       6. clear-num 68.7%

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

       7. associate-/l* 69.7%

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

       8. associate-/l* 70.8%

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

   12. Applied egg-rr 70.8%

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

   if -2.69999999999999992e51 < u < 1.40000000000000006e75

   1. Initial program 73.9%

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

      1. associate-/l* 73.1%

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

      2. distribute-lft-neg-out 73.1%

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

      3. distribute-rgt-neg-in 73.1%

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

      4. associate-/r* 82.4%

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

      5. distribute-neg-frac2 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   7. Simplified 71.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+51} \lor \neg \left(u \leq 1.4 \cdot 10^{+75}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+50}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 2.55 \cdot 10^{+81}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.5e+50)
   (* t1 (/ (/ v u) u))
   (if (<= u 2.55e+81) (/ v (- t1)) (/ t1 (* u (/ u v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e+50) {
		tmp = t1 * ((v / u) / u);
	} else if (u <= 2.55e+81) {
		tmp = v / -t1;
	} else {
		tmp = t1 / (u * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.5d+50)) then
        tmp = t1 * ((v / u) / u)
    else if (u <= 2.55d+81) then
        tmp = v / -t1
    else
        tmp = t1 / (u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e+50) {
		tmp = t1 * ((v / u) / u);
	} else if (u <= 2.55e+81) {
		tmp = v / -t1;
	} else {
		tmp = t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.5e+50:
		tmp = t1 * ((v / u) / u)
	elif u <= 2.55e+81:
		tmp = v / -t1
	else:
		tmp = t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.5e+50)
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	elseif (u <= 2.55e+81)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.5e+50)
		tmp = t1 * ((v / u) / u);
	elseif (u <= 2.55e+81)
		tmp = v / -t1;
	else
		tmp = t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.5e+50], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.55e+81], N[(v / (-t1)), $MachinePrecision], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.4999999999999998e50

   1. Initial program 84.2%

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

      1. times-frac 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. distribute-neg-frac2 98.1%

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

      4. +-commutative 98.1%

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

      5. distribute-neg-in 98.1%

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

      6. unsub-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in t1 around 0 86.5%

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

      1. associate-*r/ 86.5%

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

      2. mul-1-neg 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in t1 around 0 82.8%

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

      1. clear-num 82.8%

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

      2. frac-times 79.2%

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

      3. *-un-lft-identity 79.2%

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

      4. add-sqr-sqrt 40.6%

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

      5. sqrt-unprod 72.8%

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

      6. sqr-neg 72.8%

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

      7. sqrt-unprod 36.4%

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

      8. add-sqr-sqrt 73.0%

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

   10. Applied egg-rr 73.0%

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

       1. clear-num 73.0%

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

       2. *-commutative 73.0%

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

       3. associate-/l* 73.0%

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

       4. associate-/r* 72.8%

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

       5. associate-/l/ 72.8%

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

       6. clear-num 72.8%

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

       7. associate-/l* 72.9%

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

       8. associate-/l* 74.9%

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

   12. Applied egg-rr 74.9%

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

   if -5.4999999999999998e50 < u < 2.5500000000000001e81

   1. Initial program 73.9%

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

      1. associate-/l* 73.1%

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

      2. distribute-lft-neg-out 73.1%

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

      3. distribute-rgt-neg-in 73.1%

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

      4. associate-/r* 82.4%

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

      5. distribute-neg-frac2 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   7. Simplified 71.6%

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

   if 2.5500000000000001e81 < u

   1. Initial program 79.5%

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

      1. times-frac 97.4%

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

      2. distribute-frac-neg 97.4%

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

      3. distribute-neg-frac2 97.4%

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

      4. +-commutative 97.4%

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

      5. distribute-neg-in 97.4%

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

      6. unsub-neg 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in t1 around 0 86.7%

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

      1. associate-*r/ 86.7%

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

      2. mul-1-neg 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in t1 around 0 86.7%

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

      1. *-commutative 86.7%

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

      2. clear-num 86.7%

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

      3. frac-times 84.7%

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

      4. *-un-lft-identity 84.7%

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

      5. add-sqr-sqrt 29.0%

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

      6. sqrt-unprod 66.8%

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

      7. sqr-neg 66.8%

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

      8. sqrt-unprod 44.5%

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

      9. add-sqr-sqrt 66.6%

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

   10. Applied egg-rr 66.6%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+50}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 2.55 \cdot 10^{+81}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. times-frac 97.2%

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

   2. distribute-frac-neg 97.2%

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

   3. distribute-neg-frac2 97.2%

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

   4. +-commutative 97.2%

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

   5. distribute-neg-in 97.2%

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

   6. unsub-neg 97.2%

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

3. Simplified 97.2%

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

5. Final simplification 97.2%

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

Alternative 10: 55.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 6.1 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 6.1e+159) (/ v (- t1)) (/ v (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 6.1e+159) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 6.1d+159) then
        tmp = v / -t1
    else
        tmp = v / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 6.1e+159) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 6.1e+159:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 6.1e+159)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 6.1e+159)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 6.1e+159], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 6.1e159

   1. Initial program 76.8%

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

      1. associate-/l* 76.5%

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

      2. distribute-lft-neg-out 76.5%

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

      3. distribute-rgt-neg-in 76.5%

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

      4. associate-/r* 84.7%

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

      5. distribute-neg-frac2 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in t1 around inf 57.5%

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

      1. associate-*r/ 57.5%

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

      2. neg-mul-1 57.5%

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

   7. Simplified 57.5%

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

   if 6.1e159 < u

   1. Initial program 78.2%

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

      1. associate-/l* 78.5%

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

      2. distribute-lft-neg-out 78.5%

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

      3. distribute-rgt-neg-in 78.5%

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

      4. associate-/r* 92.1%

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

      5. distribute-neg-frac2 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in t1 around 0 85.4%

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

   6. Taylor expanded in t1 around inf 43.5%

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

      1. associate-*r/ 43.5%

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

      2. mul-1-neg 43.5%

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

   8. Simplified 43.5%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 6.1 \cdot 10^{+159}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. associate-/l* 76.7%

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

   2. distribute-lft-neg-out 76.7%

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

   3. distribute-rgt-neg-in 76.7%

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

   4. associate-/r* 85.6%

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

   5. distribute-neg-frac2 85.6%

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

3. Simplified 85.6%

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

   1. distribute-frac-neg2 85.6%

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

   2. associate-/r* 76.7%

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

   3. distribute-rgt-neg-in 76.7%

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

   4. distribute-lft-neg-out 76.7%

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

   5. associate-*r/ 77.0%

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

   6. times-frac 97.2%

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

   7. frac-2neg 97.2%

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

   8. associate-*r/ 99.0%

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

   9. add-sqr-sqrt 50.4%

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

   10. sqrt-unprod 51.7%

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

   11. sqr-neg 51.7%

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

   12. sqrt-unprod 19.9%

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

   13. add-sqr-sqrt 39.2%

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

   14. add-sqr-sqrt 21.3%

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

   15. sqrt-unprod 63.1%

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

   16. sqr-neg 63.1%

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

   17. sqrt-prod 50.0%

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

   18. add-sqr-sqrt 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in t1 around inf 59.8%

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

   1. mul-1-neg 59.8%

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

9. Simplified 59.8%

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

10. Final simplification 59.8%

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

Alternative 12: 53.6% accurate, 3.0× 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 / Float64(-t1))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. associate-/l* 76.7%

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

   2. distribute-lft-neg-out 76.7%

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

   3. distribute-rgt-neg-in 76.7%

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

   4. associate-/r* 85.6%

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

   5. distribute-neg-frac2 85.6%

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

3. Simplified 85.6%

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

5. Taylor expanded in t1 around inf 53.3%

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

   1. associate-*r/ 53.3%

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

   2. neg-mul-1 53.3%

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

7. Simplified 53.3%

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

8. Final simplification 53.3%

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

Alternative 13: 13.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. times-frac 97.2%

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

   2. distribute-frac-neg 97.2%

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

   3. distribute-neg-frac2 97.2%

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

   4. +-commutative 97.2%

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

   5. distribute-neg-in 97.2%

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

   6. unsub-neg 97.2%

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

3. Simplified 97.2%

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

5. Taylor expanded in t1 around inf 55.4%

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

6. Taylor expanded in u around inf 13.8%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))