Rosa's DopplerBench

Percentage Accurate: 73.5% → 97.9%

Time: 12.3s

Alternatives: 19

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: 73.5% 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: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. associate-/l* 72.2%

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

   2. distribute-lft-neg-out 72.2%

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

   3. distribute-rgt-neg-in 72.2%

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

   4. associate-/r* 83.7%

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

   5. distribute-neg-frac2 83.7%

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

3. Simplified 83.7%

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

   1. associate-*r/ 99.5%

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

   2. neg-mul-1 99.5%

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

   3. associate-/r* 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -0.082 \lor \neg \left(t1 \leq -1.52 \cdot 10^{-74} \lor \neg \left(t1 \leq -1.1 \cdot 10^{-103}\right) \land t1 \leq 6.6 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(\left(-u\right) - t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -0.082)
         (not
          (or (<= t1 -1.52e-74)
              (and (not (<= t1 -1.1e-103)) (<= t1 6.6e-161)))))
   (/ v (- (- t1) (* u 2.0)))
   (* t1 (/ v (* u (- (- u) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -0.082) || !((t1 <= -1.52e-74) || (!(t1 <= -1.1e-103) && (t1 <= 6.6e-161)))) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * (v / (u * (-u - t1)));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-0.082d0)) .or. (.not. (t1 <= (-1.52d-74)) .or. (.not. (t1 <= (-1.1d-103))) .and. (t1 <= 6.6d-161))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t1 * (v / (u * (-u - t1)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -0.082) || !((t1 <= -1.52e-74) || (!(t1 <= -1.1e-103) && (t1 <= 6.6e-161)))) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * (v / (u * (-u - t1)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -0.082) or not ((t1 <= -1.52e-74) or (not (t1 <= -1.1e-103) and (t1 <= 6.6e-161))):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t1 * (v / (u * (-u - t1)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -0.082) || !((t1 <= -1.52e-74) || (!(t1 <= -1.1e-103) && (t1 <= 6.6e-161))))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(Float64(-u) - t1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -0.082) || ~(((t1 <= -1.52e-74) || (~((t1 <= -1.1e-103)) && (t1 <= 6.6e-161)))))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t1 * (v / (u * (-u - t1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -0.082], N[Not[Or[LessEqual[t1, -1.52e-74], And[N[Not[LessEqual[t1, -1.1e-103]], $MachinePrecision], LessEqual[t1, 6.6e-161]]]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(v / N[(u * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -0.0820000000000000034 or -1.51999999999999997e-74 < t1 < -1.1e-103 or 6.5999999999999997e-161 < t1

   1. Initial program 67.5%

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

      1. associate-/l* 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

      4. associate-/r* 81.2%

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

      5. distribute-neg-frac2 81.2%

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

   3. Simplified 81.2%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.7%

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

      6. clear-num 99.4%

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

      7. frac-2neg 99.4%

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

      8. frac-times 96.0%

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

      9. *-un-lft-identity 96.0%

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

      10. add-sqr-sqrt 46.6%

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

      11. sqrt-unprod 52.4%

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

      12. sqr-neg 52.4%

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

      13. sqrt-prod 16.8%

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

      14. add-sqr-sqrt 37.9%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in u around 0 83.4%

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

      1. *-commutative 83.4%

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

   9. Simplified 83.4%

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

   if -0.0820000000000000034 < t1 < -1.51999999999999997e-74 or -1.1e-103 < t1 < 6.5999999999999997e-161

   1. Initial program 74.2%

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

      1. associate-/l* 79.3%

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

      2. distribute-lft-neg-out 79.3%

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

      3. distribute-rgt-neg-in 79.3%

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

      4. associate-/r* 88.2%

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

      5. distribute-neg-frac2 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in t1 around 0 75.6%

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

   6. Taylor expanded in v around 0 70.9%

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

      1. associate-*r/ 70.9%

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

      2. neg-mul-1 70.9%

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

   8. Simplified 70.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -0.082 \lor \neg \left(t1 \leq -1.52 \cdot 10^{-74} \lor \neg \left(t1 \leq -1.1 \cdot 10^{-103}\right) \land t1 \leq 6.6 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(\left(-u\right) - t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{if}\;t1 \leq -0.16:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-73}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.1 \cdot 10^{-103} \lor \neg \left(t1 \leq 6.6 \cdot 10^{-161}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) (* u 2.0)))))
   (if (<= t1 -0.16)
     t_1
     (if (<= t1 -1.65e-73)
       (* t1 (/ (/ v (- u)) (+ t1 u)))
       (if (or (<= t1 -1.1e-103) (not (<= t1 6.6e-161)))
         t_1
         (* v (/ (/ t1 u) (- (- u) t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - (u * 2.0));
	double tmp;
	if (t1 <= -0.16) {
		tmp = t_1;
	} else if (t1 <= -1.65e-73) {
		tmp = t1 * ((v / -u) / (t1 + u));
	} else if ((t1 <= -1.1e-103) || !(t1 <= 6.6e-161)) {
		tmp = t_1;
	} else {
		tmp = v * ((t1 / u) / (-u - t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (-t1 - (u * 2.0d0))
    if (t1 <= (-0.16d0)) then
        tmp = t_1
    else if (t1 <= (-1.65d-73)) then
        tmp = t1 * ((v / -u) / (t1 + u))
    else if ((t1 <= (-1.1d-103)) .or. (.not. (t1 <= 6.6d-161))) then
        tmp = t_1
    else
        tmp = v * ((t1 / u) / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - (u * 2.0));
	double tmp;
	if (t1 <= -0.16) {
		tmp = t_1;
	} else if (t1 <= -1.65e-73) {
		tmp = t1 * ((v / -u) / (t1 + u));
	} else if ((t1 <= -1.1e-103) || !(t1 <= 6.6e-161)) {
		tmp = t_1;
	} else {
		tmp = v * ((t1 / u) / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - (u * 2.0))
	tmp = 0
	if t1 <= -0.16:
		tmp = t_1
	elif t1 <= -1.65e-73:
		tmp = t1 * ((v / -u) / (t1 + u))
	elif (t1 <= -1.1e-103) or not (t1 <= 6.6e-161):
		tmp = t_1
	else:
		tmp = v * ((t1 / u) / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -0.16)
		tmp = t_1;
	elseif (t1 <= -1.65e-73)
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / Float64(t1 + u)));
	elseif ((t1 <= -1.1e-103) || !(t1 <= 6.6e-161))
		tmp = t_1;
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - (u * 2.0));
	tmp = 0.0;
	if (t1 <= -0.16)
		tmp = t_1;
	elseif (t1 <= -1.65e-73)
		tmp = t1 * ((v / -u) / (t1 + u));
	elseif ((t1 <= -1.1e-103) || ~((t1 <= 6.6e-161)))
		tmp = t_1;
	else
		tmp = v * ((t1 / u) / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -0.16], t$95$1, If[LessEqual[t1, -1.65e-73], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -1.1e-103], N[Not[LessEqual[t1, 6.6e-161]], $MachinePrecision]], t$95$1, N[(v * N[(N[(t1 / u), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -0.160000000000000003 or -1.65000000000000002e-73 < t1 < -1.1e-103 or 6.5999999999999997e-161 < t1

   1. Initial program 67.5%

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

      1. associate-/l* 68.3%

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

      2. distribute-lft-neg-out 68.3%

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

      3. distribute-rgt-neg-in 68.3%

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

      4. associate-/r* 81.2%

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

      5. distribute-neg-frac2 81.2%

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

   3. Simplified 81.2%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.7%

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

      6. clear-num 99.4%

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

      7. frac-2neg 99.4%

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

      8. frac-times 96.0%

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

      9. *-un-lft-identity 96.0%

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

      10. add-sqr-sqrt 46.6%

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

      11. sqrt-unprod 52.4%

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

      12. sqr-neg 52.4%

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

      13. sqrt-prod 16.8%

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

      14. add-sqr-sqrt 37.9%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in u around 0 83.4%

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

      1. *-commutative 83.4%

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

   9. Simplified 83.4%

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

   if -0.160000000000000003 < t1 < -1.65000000000000002e-73

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. distribute-lft-neg-out 82.7%

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

      3. distribute-rgt-neg-in 82.7%

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

      4. associate-/r* 99.7%

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

      5. distribute-neg-frac2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t1 around 0 78.0%

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

   if -1.1e-103 < t1 < 6.5999999999999997e-161

   1. Initial program 73.0%

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

      1. associate-/l* 78.8%

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

      2. distribute-lft-neg-out 78.8%

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

      3. distribute-rgt-neg-in 78.8%

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

      4. associate-/r* 86.7%

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

      5. distribute-neg-frac2 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in t1 around 0 75.3%

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

   6. Taylor expanded in v around 0 70.0%

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

      1. mul-1-neg 70.0%

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

      2. associate-/r* 72.9%

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

      3. associate-*r/ 81.4%

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

      4. distribute-frac-neg 81.4%

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

      5. distribute-rgt-neg-in 81.4%

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

      6. distribute-frac-neg 81.4%

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

   8. Simplified 81.4%

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

   9. Taylor expanded in v around 0 70.0%

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

       1. mul-1-neg 70.0%

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

       2. associate-*l/ 71.2%

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

       3. *-commutative 71.2%

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

       4. distribute-lft-neg-in 71.2%

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

       5. associate-/r* 79.3%

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

   11. Simplified 79.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -0.16:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-73}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.1 \cdot 10^{-103} \lor \neg \left(t1 \leq 6.6 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-t1\right) - u \cdot 2}\\ t_2 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -8 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq -1.9 \cdot 10^{-74}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.06 \cdot 10^{-103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.6 \cdot 10^{-161}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t\_2} \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- t1) (* u 2.0)))) (t_2 (- (- u) t1)))
   (if (<= t1 -8e-6)
     t_1
     (if (<= t1 -1.9e-74)
       (* t1 (/ (/ v (- u)) (+ t1 u)))
       (if (<= t1 -1.06e-103)
         t_1
         (if (<= t1 6.6e-161)
           (* v (/ (/ t1 u) t_2))
           (* (/ t1 t_2) (/ v t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - (u * 2.0));
	double t_2 = -u - t1;
	double tmp;
	if (t1 <= -8e-6) {
		tmp = t_1;
	} else if (t1 <= -1.9e-74) {
		tmp = t1 * ((v / -u) / (t1 + u));
	} else if (t1 <= -1.06e-103) {
		tmp = t_1;
	} else if (t1 <= 6.6e-161) {
		tmp = v * ((t1 / u) / t_2);
	} else {
		tmp = (t1 / t_2) * (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) :: t_2
    real(8) :: tmp
    t_1 = v / (-t1 - (u * 2.0d0))
    t_2 = -u - t1
    if (t1 <= (-8d-6)) then
        tmp = t_1
    else if (t1 <= (-1.9d-74)) then
        tmp = t1 * ((v / -u) / (t1 + u))
    else if (t1 <= (-1.06d-103)) then
        tmp = t_1
    else if (t1 <= 6.6d-161) then
        tmp = v * ((t1 / u) / t_2)
    else
        tmp = (t1 / t_2) * (v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (-t1 - (u * 2.0));
	double t_2 = -u - t1;
	double tmp;
	if (t1 <= -8e-6) {
		tmp = t_1;
	} else if (t1 <= -1.9e-74) {
		tmp = t1 * ((v / -u) / (t1 + u));
	} else if (t1 <= -1.06e-103) {
		tmp = t_1;
	} else if (t1 <= 6.6e-161) {
		tmp = v * ((t1 / u) / t_2);
	} else {
		tmp = (t1 / t_2) * (v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-t1 - (u * 2.0))
	t_2 = -u - t1
	tmp = 0
	if t1 <= -8e-6:
		tmp = t_1
	elif t1 <= -1.9e-74:
		tmp = t1 * ((v / -u) / (t1 + u))
	elif t1 <= -1.06e-103:
		tmp = t_1
	elif t1 <= 6.6e-161:
		tmp = v * ((t1 / u) / t_2)
	else:
		tmp = (t1 / t_2) * (v / t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)))
	t_2 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (t1 <= -8e-6)
		tmp = t_1;
	elseif (t1 <= -1.9e-74)
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / Float64(t1 + u)));
	elseif (t1 <= -1.06e-103)
		tmp = t_1;
	elseif (t1 <= 6.6e-161)
		tmp = Float64(v * Float64(Float64(t1 / u) / t_2));
	else
		tmp = Float64(Float64(t1 / t_2) * Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-t1 - (u * 2.0));
	t_2 = -u - t1;
	tmp = 0.0;
	if (t1 <= -8e-6)
		tmp = t_1;
	elseif (t1 <= -1.9e-74)
		tmp = t1 * ((v / -u) / (t1 + u));
	elseif (t1 <= -1.06e-103)
		tmp = t_1;
	elseif (t1 <= 6.6e-161)
		tmp = v * ((t1 / u) / t_2);
	else
		tmp = (t1 / t_2) * (v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[t1, -8e-6], t$95$1, If[LessEqual[t1, -1.9e-74], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.06e-103], t$95$1, If[LessEqual[t1, 6.6e-161], N[(v * N[(N[(t1 / u), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / t$95$2), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -7.99999999999999964e-6 or -1.8999999999999998e-74 < t1 < -1.06000000000000004e-103

   1. Initial program 65.7%

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

      1. associate-/l* 68.1%

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

      2. distribute-lft-neg-out 68.1%

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

      3. distribute-rgt-neg-in 68.1%

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

      4. associate-/r* 81.9%

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

      5. distribute-neg-frac2 81.9%

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

   3. Simplified 81.9%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 99.5%

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

      6. clear-num 98.9%

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

      7. frac-2neg 98.9%

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

      8. frac-times 96.5%

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

      9. *-un-lft-identity 96.5%

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

      10. add-sqr-sqrt 84.6%

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

      11. sqrt-unprod 70.1%

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

      12. sqr-neg 70.1%

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

      13. sqrt-prod 7.7%

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

      14. add-sqr-sqrt 40.2%

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

   6. Applied egg-rr 96.5%

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

   7. Taylor expanded in u around 0 85.6%

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

      1. *-commutative 85.6%

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

   9. Simplified 85.6%

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

   if -7.99999999999999964e-6 < t1 < -1.8999999999999998e-74

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. distribute-lft-neg-out 82.7%

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

      3. distribute-rgt-neg-in 82.7%

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

      4. associate-/r* 99.7%

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

      5. distribute-neg-frac2 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t1 around 0 78.0%

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

   if -1.06000000000000004e-103 < t1 < 6.5999999999999997e-161

   1. Initial program 73.0%

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

      1. associate-/l* 78.8%

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

      2. distribute-lft-neg-out 78.8%

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

      3. distribute-rgt-neg-in 78.8%

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

      4. associate-/r* 86.7%

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

      5. distribute-neg-frac2 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in t1 around 0 75.3%

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

   6. Taylor expanded in v around 0 70.0%

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

      1. mul-1-neg 70.0%

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

      2. associate-/r* 72.9%

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

      3. associate-*r/ 81.4%

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

      4. distribute-frac-neg 81.4%

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

      5. distribute-rgt-neg-in 81.4%

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

      6. distribute-frac-neg 81.4%

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

   8. Simplified 81.4%

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

   9. Taylor expanded in v around 0 70.0%

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

       1. mul-1-neg 70.0%

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

       2. associate-*l/ 71.2%

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

       3. *-commutative 71.2%

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

       4. distribute-lft-neg-in 71.2%

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

       5. associate-/r* 79.3%

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

   11. Simplified 79.3%

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

   if 6.5999999999999997e-161 < t1

   1. Initial program 69.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 82.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq -1.9 \cdot 10^{-74}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.06 \cdot 10^{-103}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 6.6 \cdot 10^{-161}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+130}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4e+135)
   (/ v (- (- t1) (* u 2.0)))
   (if (<= t1 5.8e+130) (* t1 (/ (/ v (+ t1 u)) (- (- u) t1))) (/ v (- t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4e+135) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 5.8e+130) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-4d+135)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (t1 <= 5.8d+130) then
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4e+135) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 5.8e+130) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4e+135:
		tmp = v / (-t1 - (u * 2.0))
	elif t1 <= 5.8e+130:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4e+135)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (t1 <= 5.8e+130)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4e+135)
		tmp = v / (-t1 - (u * 2.0));
	elseif (t1 <= 5.8e+130)
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4e+135], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.8e+130], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.99999999999999985e135

   1. Initial program 53.2%

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

      1. associate-/l* 54.0%

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

      2. distribute-lft-neg-out 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

      4. associate-/r* 70.8%

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

      5. distribute-neg-frac2 70.8%

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

   3. Simplified 70.8%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-*l/ 100.0%

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

      6. clear-num 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 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 99.6%

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

      11. sqrt-unprod 54.0%

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

      12. sqr-neg 54.0%

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

      13. sqrt-prod 0.0%

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

      14. add-sqr-sqrt 53.3%

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

   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 95.0%

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

      1. *-commutative 95.0%

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

   9. Simplified 95.0%

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

   if -3.99999999999999985e135 < t1 < 5.7999999999999998e130

   1. Initial program 78.8%

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

      1. associate-/l* 81.1%

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

      2. distribute-lft-neg-out 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

      4. associate-/r* 89.6%

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

      5. distribute-neg-frac2 89.6%

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

   3. Simplified 89.6%

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

   if 5.7999999999999998e130 < t1

   1. Initial program 38.4%

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

      1. associate-/l* 42.6%

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

      2. distribute-lft-neg-out 42.6%

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

      3. distribute-rgt-neg-in 42.6%

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

      4. associate-/r* 65.2%

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

      5. distribute-neg-frac2 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in t1 around inf 91.5%

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

      1. associate-*r/ 91.5%

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

      2. neg-mul-1 91.5%

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

   7. Simplified 91.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+130}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -0.0085 \lor \neg \left(t1 \leq 6.6 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -0.0085) (not (<= t1 6.6e-161)))
   (/ v (- (- t1) (* u 2.0)))
   (* t1 (/ (/ v (- u)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -0.0085) || !(t1 <= 6.6e-161)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * ((v / -u) / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-0.0085d0)) .or. (.not. (t1 <= 6.6d-161))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t1 * ((v / -u) / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -0.0085) || !(t1 <= 6.6e-161)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * ((v / -u) / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -0.0085) or not (t1 <= 6.6e-161):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t1 * ((v / -u) / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -0.0085) || !(t1 <= 6.6e-161))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -0.0085) || ~((t1 <= 6.6e-161)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t1 * ((v / -u) / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -0.0085], N[Not[LessEqual[t1, 6.6e-161]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -0.0085000000000000006 or 6.5999999999999997e-161 < t1

   1. Initial program 66.6%

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

      1. associate-/l* 68.0%

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

      2. distribute-lft-neg-out 68.0%

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

      3. distribute-rgt-neg-in 68.0%

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

      4. associate-/r* 81.0%

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

      5. distribute-neg-frac2 81.0%

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

   3. Simplified 81.0%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      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 96.3%

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

      9. *-un-lft-identity 96.3%

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

      10. add-sqr-sqrt 45.4%

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

      11. sqrt-unprod 50.8%

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

      12. sqr-neg 50.8%

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

      13. sqrt-prod 16.9%

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

      14. add-sqr-sqrt 38.8%

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in u around 0 83.3%

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

      1. *-commutative 83.3%

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

   9. Simplified 83.3%

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

   if -0.0085000000000000006 < t1 < 6.5999999999999997e-161

   1. Initial program 75.1%

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

      1. associate-/l* 78.9%

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

      2. distribute-lft-neg-out 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. associate-/r* 88.1%

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

      5. distribute-neg-frac2 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in t1 around 0 72.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -0.0085 \lor \neg \left(t1 \leq 6.6 \cdot 10^{-161}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{\left(-u\right) - t1}\\ \mathbf{if}\;t1 \leq -4.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 6.6 \cdot 10^{-161}:\\ \;\;\;\;t\_1 \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ t1 (- (- u) t1))))
   (if (<= t1 -4.4e-14)
     (/ v (- (- t1) (* u 2.0)))
     (if (<= t1 6.6e-161) (* t_1 (/ v u)) (* t_1 (/ v t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 / (-u - t1);
	double tmp;
	if (t1 <= -4.4e-14) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 6.6e-161) {
		tmp = t_1 * (v / u);
	} else {
		tmp = t_1 * (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 / (-u - t1)
    if (t1 <= (-4.4d-14)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (t1 <= 6.6d-161) then
        tmp = t_1 * (v / u)
    else
        tmp = t_1 * (v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 / (-u - t1);
	double tmp;
	if (t1 <= -4.4e-14) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 6.6e-161) {
		tmp = t_1 * (v / u);
	} else {
		tmp = t_1 * (v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 / (-u - t1)
	tmp = 0
	if t1 <= -4.4e-14:
		tmp = v / (-t1 - (u * 2.0))
	elif t1 <= 6.6e-161:
		tmp = t_1 * (v / u)
	else:
		tmp = t_1 * (v / t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 / Float64(Float64(-u) - t1))
	tmp = 0.0
	if (t1 <= -4.4e-14)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (t1 <= 6.6e-161)
		tmp = Float64(t_1 * Float64(v / u));
	else
		tmp = Float64(t_1 * Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 / (-u - t1);
	tmp = 0.0;
	if (t1 <= -4.4e-14)
		tmp = v / (-t1 - (u * 2.0));
	elseif (t1 <= 6.6e-161)
		tmp = t_1 * (v / u);
	else
		tmp = t_1 * (v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -4.4e-14], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.6e-161], N[(t$95$1 * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(v / t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.4000000000000002e-14

   1. Initial program 63.6%

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

      1. associate-/l* 67.4%

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

      2. distribute-lft-neg-out 67.4%

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

      3. distribute-rgt-neg-in 67.4%

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

      4. associate-/r* 81.4%

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

      5. distribute-neg-frac2 81.4%

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

   3. Simplified 81.4%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      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 97.3%

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

      9. *-un-lft-identity 97.3%

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

      10. add-sqr-sqrt 85.7%

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

      11. sqrt-unprod 68.4%

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

      12. sqr-neg 68.4%

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

      13. sqrt-prod 7.1%

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

      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 97.3%

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

   7. Taylor expanded in u around 0 85.5%

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

      1. *-commutative 85.5%

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

   9. Simplified 85.5%

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

   if -4.4000000000000002e-14 < t1 < 6.5999999999999997e-161

   1. Initial program 75.1%

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

      1. times-frac 97.6%

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

      2. distribute-frac-neg 97.6%

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

      3. distribute-neg-frac2 97.6%

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

      4. +-commutative 97.6%

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

      5. distribute-neg-in 97.6%

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

      6. unsub-neg 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in t1 around 0 76.5%

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

   if 6.5999999999999997e-161 < t1

   1. Initial program 69.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 82.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 6.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -0.00115:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{t\_1} \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= t1 -0.00115)
     (/ v (- (- t1) (* u 2.0)))
     (if (<= t1 6.5e-161) (/ (* t1 (/ v u)) t_1) (* (/ t1 t_1) (/ v t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -0.00115) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 6.5e-161) {
		tmp = (t1 * (v / u)) / t_1;
	} else {
		tmp = (t1 / t_1) * (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 = -u - t1
    if (t1 <= (-0.00115d0)) then
        tmp = v / (-t1 - (u * 2.0d0))
    else if (t1 <= 6.5d-161) then
        tmp = (t1 * (v / u)) / t_1
    else
        tmp = (t1 / t_1) * (v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -0.00115) {
		tmp = v / (-t1 - (u * 2.0));
	} else if (t1 <= 6.5e-161) {
		tmp = (t1 * (v / u)) / t_1;
	} else {
		tmp = (t1 / t_1) * (v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if t1 <= -0.00115:
		tmp = v / (-t1 - (u * 2.0))
	elif t1 <= 6.5e-161:
		tmp = (t1 * (v / u)) / t_1
	else:
		tmp = (t1 / t_1) * (v / t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (t1 <= -0.00115)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	elseif (t1 <= 6.5e-161)
		tmp = Float64(Float64(t1 * Float64(v / u)) / t_1);
	else
		tmp = Float64(Float64(t1 / t_1) * Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (t1 <= -0.00115)
		tmp = v / (-t1 - (u * 2.0));
	elseif (t1 <= 6.5e-161)
		tmp = (t1 * (v / u)) / t_1;
	else
		tmp = (t1 / t_1) * (v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[t1, -0.00115], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.5e-161], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(t1 / t$95$1), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -0.00115

   1. Initial program 63.6%

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

      1. associate-/l* 67.4%

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

      2. distribute-lft-neg-out 67.4%

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

      3. distribute-rgt-neg-in 67.4%

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

      4. associate-/r* 81.4%

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

      5. distribute-neg-frac2 81.4%

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

   3. Simplified 81.4%

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

      1. associate-*r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. sub-neg 99.9%

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

      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 97.3%

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

      9. *-un-lft-identity 97.3%

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

      10. add-sqr-sqrt 85.7%

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

      11. sqrt-unprod 68.4%

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

      12. sqr-neg 68.4%

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

      13. sqrt-prod 7.1%

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

      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 97.3%

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

   7. Taylor expanded in u around 0 85.5%

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

      1. *-commutative 85.5%

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

   9. Simplified 85.5%

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

   if -0.00115 < t1 < 6.50000000000000008e-161

   1. Initial program 75.1%

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

      1. associate-/l* 78.9%

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

      2. distribute-lft-neg-out 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. associate-/r* 88.1%

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

      5. distribute-neg-frac2 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in t1 around 0 72.6%

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

   6. Taylor expanded in v around 0 65.6%

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

      1. mul-1-neg 65.6%

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

      2. associate-/r* 70.7%

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

      3. associate-*r/ 77.6%

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

      4. distribute-frac-neg 77.6%

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

      5. distribute-rgt-neg-in 77.6%

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

      6. distribute-frac-neg 77.6%

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

   8. Simplified 77.6%

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

   if 6.50000000000000008e-161 < t1

   1. Initial program 69.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 82.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -0.00115:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(-u\right) - t1} \cdot \frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+112} \lor \neg \left(u \leq 7.8 \cdot 10^{+217}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4e+112) (not (<= u 7.8e+217)))
   (* t1 (/ v (* u (+ t1 u))))
   (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4e+112) || !(u <= 7.8e+217)) {
		tmp = t1 * (v / (u * (t1 + u)));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4d+112)) .or. (.not. (u <= 7.8d+217))) then
        tmp = t1 * (v / (u * (t1 + u)))
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4e+112) || !(u <= 7.8e+217)) {
		tmp = t1 * (v / (u * (t1 + u)));
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4e+112) or not (u <= 7.8e+217):
		tmp = t1 * (v / (u * (t1 + u)))
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4e+112) || !(u <= 7.8e+217))
		tmp = Float64(t1 * Float64(v / Float64(u * Float64(t1 + u))));
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4e+112) || ~((u <= 7.8e+217)))
		tmp = t1 * (v / (u * (t1 + u)));
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4e+112], N[Not[LessEqual[u, 7.8e+217]], $MachinePrecision]], N[(t1 * N[(v / N[(u * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.9999999999999997e112 or 7.79999999999999986e217 < u

   1. Initial program 74.4%

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

      1. associate-/l* 75.0%

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

      2. distribute-lft-neg-out 75.0%

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

      3. distribute-rgt-neg-in 75.0%

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

      4. associate-/r* 90.7%

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

      5. distribute-neg-frac2 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t1 around 0 84.8%

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

      1. associate-/l/ 75.0%

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

      2. associate-*r/ 74.4%

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

      3. add-sqr-sqrt 47.6%

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

      4. sqrt-unprod 74.4%

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

      5. sqr-neg 74.4%

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

      6. sqrt-unprod 26.7%

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

      7. add-sqr-sqrt 72.9%

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

   7. Applied egg-rr 72.9%

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

      1. associate-/l* 73.5%

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

      2. *-commutative 73.5%

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

   9. Simplified 73.5%

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

   if -3.9999999999999997e112 < u < 7.79999999999999986e217

   1. Initial program 68.4%

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

      1. associate-/l* 71.3%

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

      2. distribute-lft-neg-out 71.3%

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

      3. distribute-rgt-neg-in 71.3%

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

      4. associate-/r* 81.4%

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

      5. distribute-neg-frac2 81.4%

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

   3. Simplified 81.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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/ 98.9%

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

      6. clear-num 98.9%

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

      7. frac-2neg 98.9%

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

      8. frac-times 97.4%

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

      9. *-un-lft-identity 97.4%

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

      10. add-sqr-sqrt 48.1%

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

      11. sqrt-unprod 51.5%

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

      12. sqr-neg 51.5%

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

      13. sqrt-prod 13.1%

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

      14. add-sqr-sqrt 26.1%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u around 0 68.4%

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

      1. *-commutative 68.4%

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

   9. Simplified 68.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+112} \lor \neg \left(u \leq 7.8 \cdot 10^{+217}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := u \cdot \left(t1 + u\right)\\ \mathbf{if}\;u \leq -3.1 \cdot 10^{+106}:\\ \;\;\;\;v \cdot \frac{t1}{t\_1}\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* u (+ t1 u))))
   (if (<= u -3.1e+106)
     (* v (/ t1 t_1))
     (if (<= u 1.75e+217) (/ v (- (- t1) (* u 2.0))) (* t1 (/ v t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = u * (t1 + u);
	double tmp;
	if (u <= -3.1e+106) {
		tmp = v * (t1 / t_1);
	} else if (u <= 1.75e+217) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * (v / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = u * (t1 + u)
    if (u <= (-3.1d+106)) then
        tmp = v * (t1 / t_1)
    else if (u <= 1.75d+217) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t1 * (v / t_1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = u * (t1 + u);
	double tmp;
	if (u <= -3.1e+106) {
		tmp = v * (t1 / t_1);
	} else if (u <= 1.75e+217) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * (v / t_1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = u * (t1 + u)
	tmp = 0
	if u <= -3.1e+106:
		tmp = v * (t1 / t_1)
	elif u <= 1.75e+217:
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t1 * (v / t_1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(u * Float64(t1 + u))
	tmp = 0.0
	if (u <= -3.1e+106)
		tmp = Float64(v * Float64(t1 / t_1));
	elseif (u <= 1.75e+217)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(v / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = u * (t1 + u);
	tmp = 0.0;
	if (u <= -3.1e+106)
		tmp = v * (t1 / t_1);
	elseif (u <= 1.75e+217)
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t1 * (v / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(u * N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.1e+106], N[(v * N[(t1 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.75e+217], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(v / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.0999999999999999e106

   1. Initial program 70.5%

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

      1. associate-/l* 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. distribute-rgt-neg-in 70.9%

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

      4. associate-/r* 90.3%

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

      5. distribute-neg-frac2 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t1 around 0 83.7%

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

      1. clear-num 83.7%

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

      2. un-div-inv 83.8%

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

      3. div-inv 83.8%

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

      4. add-sqr-sqrt 83.4%

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

      5. sqrt-unprod 70.9%

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

      6. sqr-neg 70.9%

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

      7. sqrt-unprod 0.6%

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

      8. add-sqr-sqrt 68.6%

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

      9. clear-num 68.6%

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

   7. Applied egg-rr 68.6%

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

      1. associate-/r* 68.5%

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

   9. Simplified 68.5%

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

       1. associate-/r/ 68.7%

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

       2. associate-/l/ 68.7%

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

   11. Applied egg-rr 68.7%

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

   if -3.0999999999999999e106 < u < 1.7499999999999999e217

   1. Initial program 68.4%

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

      1. associate-/l* 71.3%

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

      2. distribute-lft-neg-out 71.3%

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

      3. distribute-rgt-neg-in 71.3%

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

      4. associate-/r* 81.4%

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

      5. distribute-neg-frac2 81.4%

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

   3. Simplified 81.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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/ 98.9%

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

      6. clear-num 98.9%

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

      7. frac-2neg 98.9%

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

      8. frac-times 97.4%

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

      9. *-un-lft-identity 97.4%

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

      10. add-sqr-sqrt 48.1%

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

      11. sqrt-unprod 51.5%

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

      12. sqr-neg 51.5%

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

      13. sqrt-prod 13.1%

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

      14. add-sqr-sqrt 26.1%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u around 0 68.4%

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

      1. *-commutative 68.4%

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

   9. Simplified 68.4%

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

   if 1.7499999999999999e217 < u

   1. Initial program 81.8%

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

      1. associate-/l* 82.8%

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

      2. distribute-lft-neg-out 82.8%

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

      3. distribute-rgt-neg-in 82.8%

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

      4. associate-/r* 91.4%

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

      5. distribute-neg-frac2 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in t1 around 0 87.0%

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

      1. associate-/l/ 82.8%

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

      2. associate-*r/ 81.8%

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

      3. add-sqr-sqrt 4.5%

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

      4. sqrt-unprod 81.8%

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

      5. sqr-neg 81.8%

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

      6. sqrt-unprod 77.3%

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

      7. add-sqr-sqrt 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. associate-/l* 82.8%

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

      2. *-commutative 82.8%

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

   9. Simplified 82.8%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+106}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+112}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4e+112)
   (* v (/ t1 (* u (+ t1 u))))
   (if (<= u 1.75e+217)
     (/ v (- (- t1) (* u 2.0)))
     (/ (/ t1 (+ t1 u)) (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+112) {
		tmp = v * (t1 / (u * (t1 + u)));
	} else if (u <= 1.75e+217) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (t1 / (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 <= (-4d+112)) then
        tmp = v * (t1 / (u * (t1 + u)))
    else if (u <= 1.75d+217) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (t1 / (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 <= -4e+112) {
		tmp = v * (t1 / (u * (t1 + u)));
	} else if (u <= 1.75e+217) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (t1 / (t1 + u)) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4e+112:
		tmp = v * (t1 / (u * (t1 + u)))
	elif u <= 1.75e+217:
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (t1 / (t1 + u)) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4e+112)
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(t1 + u))));
	elseif (u <= 1.75e+217)
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4e+112)
		tmp = v * (t1 / (u * (t1 + u)));
	elseif (u <= 1.75e+217)
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (t1 / (t1 + u)) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4e+112], N[(v * N[(t1 / N[(u * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.75e+217], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.9999999999999997e112

   1. Initial program 70.5%

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

      1. associate-/l* 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. distribute-rgt-neg-in 70.9%

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

      4. associate-/r* 90.3%

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

      5. distribute-neg-frac2 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t1 around 0 83.7%

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

      1. clear-num 83.7%

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

      2. un-div-inv 83.8%

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

      3. div-inv 83.8%

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

      4. add-sqr-sqrt 83.4%

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

      5. sqrt-unprod 70.9%

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

      6. sqr-neg 70.9%

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

      7. sqrt-unprod 0.6%

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

      8. add-sqr-sqrt 68.6%

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

      9. clear-num 68.6%

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

   7. Applied egg-rr 68.6%

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

      1. associate-/r* 68.5%

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

   9. Simplified 68.5%

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

       1. associate-/r/ 68.7%

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

       2. associate-/l/ 68.7%

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

   11. Applied egg-rr 68.7%

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

   if -3.9999999999999997e112 < u < 1.7499999999999999e217

   1. Initial program 68.4%

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

      1. associate-/l* 71.3%

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

      2. distribute-lft-neg-out 71.3%

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

      3. distribute-rgt-neg-in 71.3%

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

      4. associate-/r* 81.4%

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

      5. distribute-neg-frac2 81.4%

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

   3. Simplified 81.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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/ 98.9%

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

      6. clear-num 98.9%

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

      7. frac-2neg 98.9%

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

      8. frac-times 97.4%

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

      9. *-un-lft-identity 97.4%

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

      10. add-sqr-sqrt 48.1%

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

      11. sqrt-unprod 51.5%

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

      12. sqr-neg 51.5%

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

      13. sqrt-prod 13.1%

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

      14. add-sqr-sqrt 26.1%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u around 0 68.4%

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

      1. *-commutative 68.4%

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

   9. Simplified 68.4%

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

   if 1.7499999999999999e217 < u

   1. Initial program 81.8%

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

      1. associate-/l* 82.8%

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

      2. distribute-lft-neg-out 82.8%

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

      3. distribute-rgt-neg-in 82.8%

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

      4. associate-/r* 91.4%

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

      5. distribute-neg-frac2 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in t1 around 0 87.0%

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

      1. clear-num 87.0%

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

      2. un-div-inv 87.1%

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

      3. div-inv 87.1%

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

      4. add-sqr-sqrt 4.6%

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

      5. sqrt-unprod 82.8%

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

      6. sqr-neg 82.8%

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

      7. sqrt-unprod 78.0%

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

      8. add-sqr-sqrt 83.0%

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

      9. clear-num 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. associate-/r* 82.9%

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

   9. Simplified 82.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+112}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+110} \lor \neg \left(u \leq 3.6 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.7e+110) (not (<= u 3.6e+217)))
   (/ (/ t1 u) (/ u v))
   (/ v (- (- t1) (* u 2.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7e+110) || !(u <= 3.6e+217)) {
		tmp = (t1 / u) / (u / v);
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.7d+110)) .or. (.not. (u <= 3.6d+217))) then
        tmp = (t1 / u) / (u / v)
    else
        tmp = v / (-t1 - (u * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7e+110) || !(u <= 3.6e+217)) {
		tmp = (t1 / u) / (u / v);
	} else {
		tmp = v / (-t1 - (u * 2.0));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.7e+110) or not (u <= 3.6e+217):
		tmp = (t1 / u) / (u / v)
	else:
		tmp = v / (-t1 - (u * 2.0))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.7e+110) || !(u <= 3.6e+217))
		tmp = Float64(Float64(t1 / u) / Float64(u / v));
	else
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.7e+110) || ~((u <= 3.6e+217)))
		tmp = (t1 / u) / (u / v);
	else
		tmp = v / (-t1 - (u * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.7e+110], N[Not[LessEqual[u, 3.6e+217]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.7000000000000001e110 or 3.6000000000000002e217 < u

   1. Initial program 74.4%

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

      1. associate-/l* 75.0%

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

      2. distribute-lft-neg-out 75.0%

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

      3. distribute-rgt-neg-in 75.0%

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

      4. associate-/r* 90.7%

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

      5. distribute-neg-frac2 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t1 around 0 84.8%

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

      1. clear-num 84.8%

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

      2. un-div-inv 84.9%

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

      3. div-inv 84.9%

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

      4. add-sqr-sqrt 56.3%

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

      5. sqrt-unprod 75.0%

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

      6. sqr-neg 75.0%

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

      7. sqrt-unprod 27.2%

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

      8. add-sqr-sqrt 73.6%

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

      9. clear-num 73.6%

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

   7. Applied egg-rr 73.6%

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

      1. associate-/r* 73.5%

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

   9. Simplified 73.5%

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

   10. Taylor expanded in t1 around 0 73.0%

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

   if -1.7000000000000001e110 < u < 3.6000000000000002e217

   1. Initial program 68.4%

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

      1. associate-/l* 71.3%

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

      2. distribute-lft-neg-out 71.3%

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

      3. distribute-rgt-neg-in 71.3%

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

      4. associate-/r* 81.4%

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

      5. distribute-neg-frac2 81.4%

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

   3. Simplified 81.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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/ 98.9%

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

      6. clear-num 98.9%

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

      7. frac-2neg 98.9%

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

      8. frac-times 97.4%

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

      9. *-un-lft-identity 97.4%

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

      10. add-sqr-sqrt 48.1%

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

      11. sqrt-unprod 51.5%

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

      12. sqr-neg 51.5%

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

      13. sqrt-prod 13.1%

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

      14. add-sqr-sqrt 26.1%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u around 0 68.4%

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

      1. *-commutative 68.4%

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

   9. Simplified 68.4%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+110} \lor \neg \left(u \leq 3.6 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+110} \lor \neg \left(u \leq 1.75 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2e+110) (not (<= u 1.75e+217)))
   (/ (/ t1 u) (/ u v))
   (/ v (- (- u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e+110) || !(u <= 1.75e+217)) {
		tmp = (t1 / u) / (u / v);
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2d+110)) .or. (.not. (u <= 1.75d+217))) then
        tmp = (t1 / u) / (u / v)
    else
        tmp = v / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e+110) || !(u <= 1.75e+217)) {
		tmp = (t1 / u) / (u / v);
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2e+110) or not (u <= 1.75e+217):
		tmp = (t1 / u) / (u / v)
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2e+110) || !(u <= 1.75e+217))
		tmp = Float64(Float64(t1 / u) / Float64(u / v));
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2e+110) || ~((u <= 1.75e+217)))
		tmp = (t1 / u) / (u / v);
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2e+110], N[Not[LessEqual[u, 1.75e+217]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2e110 or 1.7499999999999999e217 < u

   1. Initial program 74.4%

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

      1. associate-/l* 75.0%

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

      2. distribute-lft-neg-out 75.0%

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

      3. distribute-rgt-neg-in 75.0%

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

      4. associate-/r* 90.7%

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

      5. distribute-neg-frac2 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in t1 around 0 84.8%

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

      1. clear-num 84.8%

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

      2. un-div-inv 84.9%

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

      3. div-inv 84.9%

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

      4. add-sqr-sqrt 56.3%

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

      5. sqrt-unprod 75.0%

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

      6. sqr-neg 75.0%

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

      7. sqrt-unprod 27.2%

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

      8. add-sqr-sqrt 73.6%

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

      9. clear-num 73.6%

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

   7. Applied egg-rr 73.6%

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

      1. associate-/r* 73.5%

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

   9. Simplified 73.5%

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

   10. Taylor expanded in t1 around 0 73.0%

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

   if -2e110 < u < 1.7499999999999999e217

   1. Initial program 68.4%

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

      1. associate-/l* 71.3%

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

      2. distribute-lft-neg-out 71.3%

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

      3. distribute-rgt-neg-in 71.3%

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

      4. associate-/r* 81.4%

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

      5. distribute-neg-frac2 81.4%

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

   3. Simplified 81.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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. neg-mul-1 99.3%

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

      3. associate-/r* 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in t1 around inf 67.6%

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

   8. Taylor expanded in v around 0 67.6%

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

      1. neg-mul-1 67.6%

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

      2. distribute-neg-frac 67.6%

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

   10. Simplified 67.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+110} \lor \neg \left(u \leq 1.75 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+176} \lor \neg \left(u \leq 2.7 \cdot 10^{+199}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6e+176) (not (<= u 2.7e+199))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6e+176) || !(u <= 2.7e+199)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-6d+176)) .or. (.not. (u <= 2.7d+199))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6e+176) || !(u <= 2.7e+199)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6e+176) or not (u <= 2.7e+199):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6e+176) || !(u <= 2.7e+199))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6e+176) || ~((u <= 2.7e+199)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6e+176], N[Not[LessEqual[u, 2.7e+199]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6e176 or 2.6999999999999999e199 < u

   1. Initial program 72.7%

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

      1. associate-/l* 73.5%

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

      2. distribute-lft-neg-out 73.5%

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

      3. distribute-rgt-neg-in 73.5%

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

      4. associate-/r* 89.7%

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

      5. distribute-neg-frac2 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in t1 around 0 87.4%

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

      1. clear-num 87.4%

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

      2. un-div-inv 87.4%

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

      3. div-inv 87.4%

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

      4. add-sqr-sqrt 43.6%

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

      5. sqrt-unprod 73.5%

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

      6. sqr-neg 73.5%

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

      7. sqrt-unprod 39.5%

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

      8. add-sqr-sqrt 73.4%

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

      9. clear-num 73.4%

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

   7. Applied egg-rr 73.4%

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

      1. associate-/r* 73.2%

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

   9. Simplified 73.2%

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

   10. Taylor expanded in t1 around inf 46.6%

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

   if -6e176 < u < 2.6999999999999999e199

   1. Initial program 69.3%

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

      1. associate-/l* 71.9%

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

      2. distribute-lft-neg-out 71.9%

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

      3. distribute-rgt-neg-in 71.9%

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

      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 62.0%

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

      1. associate-*r/ 62.0%

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

      2. neg-mul-1 62.0%

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

   7. Simplified 62.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+176} \lor \neg \left(u \leq 2.7 \cdot 10^{+199}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+176} \lor \neg \left(u \leq 3.3 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.4e+176) (not (<= u 3.3e+198))) (/ v (- u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+176) || !(u <= 3.3e+198)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.4d+176)) .or. (.not. (u <= 3.3d+198))) then
        tmp = v / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+176) || !(u <= 3.3e+198)) {
		tmp = v / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.4e+176) or not (u <= 3.3e+198):
		tmp = v / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.4e+176) || !(u <= 3.3e+198))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.4e+176) || ~((u <= 3.3e+198)))
		tmp = v / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.4e+176], N[Not[LessEqual[u, 3.3e+198]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.4000000000000001e176 or 3.29999999999999994e198 < u

   1. Initial program 72.7%

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

      1. associate-/l* 73.5%

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

      2. distribute-lft-neg-out 73.5%

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

      3. distribute-rgt-neg-in 73.5%

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

      4. associate-/r* 89.7%

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

      5. distribute-neg-frac2 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in t1 around 0 87.4%

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

   6. Taylor expanded in t1 around inf 46.7%

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

      1. associate-*r/ 46.7%

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

      2. neg-mul-1 46.7%

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

   8. Simplified 46.7%

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

   if -1.4000000000000001e176 < u < 3.29999999999999994e198

   1. Initial program 69.3%

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

      1. associate-/l* 71.9%

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

      2. distribute-lft-neg-out 71.9%

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

      3. distribute-rgt-neg-in 71.9%

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

      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 62.0%

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

      1. associate-*r/ 62.0%

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

      2. neg-mul-1 62.0%

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

   7. Simplified 62.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+176} \lor \neg \left(u \leq 3.3 \cdot 10^{+198}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 23.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+78} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.7e+78) (not (<= t1 3.6e+130))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.7e+78) || !(t1 <= 3.6e+130)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.7d+78)) .or. (.not. (t1 <= 3.6d+130))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.7e+78) || !(t1 <= 3.6e+130)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.7e+78) or not (t1 <= 3.6e+130):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.7e+78) || !(t1 <= 3.6e+130))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.7e+78) || ~((t1 <= 3.6e+130)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.7e+78], N[Not[LessEqual[t1, 3.6e+130]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.70000000000000006e78 or 3.6000000000000001e130 < t1

   1. Initial program 48.2%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 89.8%

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

   6. Taylor expanded in u around inf 38.7%

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

   if -4.70000000000000006e78 < t1 < 3.6000000000000001e130

   1. Initial program 80.9%

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

      1. associate-/l* 81.8%

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

      2. distribute-lft-neg-out 81.8%

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

      3. distribute-rgt-neg-in 81.8%

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

      4. associate-/r* 90.0%

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

      5. distribute-neg-frac2 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in t1 around 0 63.0%

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

      1. clear-num 63.0%

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

      2. un-div-inv 63.9%

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

      3. div-inv 63.8%

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

      4. add-sqr-sqrt 33.5%

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

      5. sqrt-unprod 48.7%

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

      6. sqr-neg 48.7%

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

      7. sqrt-unprod 17.7%

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

      8. add-sqr-sqrt 38.5%

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

      9. clear-num 38.5%

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

   7. Applied egg-rr 38.5%

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

      1. associate-/r* 38.2%

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

   9. Simplified 38.2%

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

   10. Taylor expanded in t1 around inf 19.6%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.7 \cdot 10^{+78} \lor \neg \left(t1 \leq 3.6 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. times-frac 99.0%

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

   2. distribute-frac-neg 99.0%

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

   3. distribute-neg-frac2 99.0%

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

   4. +-commutative 99.0%

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

   5. distribute-neg-in 99.0%

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

   6. unsub-neg 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 18: 62.1% 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 69.9%

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

   1. associate-/l* 72.2%

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

   2. distribute-lft-neg-out 72.2%

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

   3. distribute-rgt-neg-in 72.2%

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

   4. associate-/r* 83.7%

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

   5. distribute-neg-frac2 83.7%

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

3. Simplified 83.7%

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

   1. associate-*r/ 99.5%

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

   2. neg-mul-1 99.5%

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

   3. associate-/r* 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in t1 around inf 62.7%

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

8. Taylor expanded in v around 0 62.7%

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

   1. neg-mul-1 62.7%

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

   2. distribute-neg-frac 62.7%

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

10. Simplified 62.7%

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

11. Final simplification 62.7%

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

Alternative 19: 13.9% 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 69.9%

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

   1. times-frac 99.0%

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

   2. distribute-frac-neg 99.0%

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

   3. distribute-neg-frac2 99.0%

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

   4. +-commutative 99.0%

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

   5. distribute-neg-in 99.0%

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

   6. unsub-neg 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in t1 around inf 58.4%

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

6. Taylor expanded in u around inf 15.6%

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

7. Final simplification 15.6%

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

Reproduce

herbie shell --seed 2024058 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))