Rosa's DopplerBench

Percentage Accurate: 73.1% → 98.1%

Time: 10.8s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 98.0%

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

   2. distribute-frac-neg 98.0%

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

   3. distribute-neg-frac2 98.0%

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

   4. +-commutative 98.0%

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

   5. distribute-neg-in 98.0%

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

   6. unsub-neg 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+73} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.2e+73) (not (<= t1 7.5e+117)))
   (/ v (- (* u (- 2.0)) t1))
   (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.2e+73) || !(t1 <= 7.5e+117)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t1 * ((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) :: tmp
    if ((t1 <= (-9.2d+73)) .or. (.not. (t1 <= 7.5d+117))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9.2e+73) || !(t1 <= 7.5e+117)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9.2e+73) or not (t1 <= 7.5e+117):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9.2e+73) || !(t1 <= 7.5e+117))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9.2e+73) || ~((t1 <= 7.5e+117)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9.2e+73], N[Not[LessEqual[t1, 7.5e+117]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.199999999999999e73 or 7.5e117 < t1

   1. Initial program 46.7%

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

      1. associate-*l/ 52.5%

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

      2. *-commutative 52.5%

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

   3. Simplified 52.5%

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

      1. associate-*r/ 46.7%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. clear-num 99.9%

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

      10. frac-2neg 99.9%

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

      11. frac-times 97.5%

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

      12. *-un-lft-identity 97.5%

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

      13. +-commutative 97.5%

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

      14. distribute-neg-in 97.5%

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

      15. sub-neg 97.5%

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

      16. sub-neg 97.5%

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

      17. +-commutative 97.5%

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

      18. add-sqr-sqrt 52.5%

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

      19. sqrt-unprod 44.0%

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

      20. sqr-neg 44.0%

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

      21. sqrt-unprod 17.5%

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

      22. add-sqr-sqrt 6.8%

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

      23. sqrt-unprod 16.6%

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

      24. sqr-neg 16.6%

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

   6. Applied egg-rr 97.5%

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

   7. Taylor expanded in u around 0 89.2%

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

      1. *-commutative 89.2%

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

   9. Simplified 89.2%

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

   if -9.199999999999999e73 < t1 < 7.5e117

   1. Initial program 84.7%

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

      1. associate-/l* 87.5%

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

   3. Simplified 87.5%

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

      1. associate-/r* 93.5%

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

      2. div-inv 93.3%

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

   6. Applied egg-rr 93.3%

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

      1. associate-*r/ 93.5%

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

      2. *-rgt-identity 93.5%

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

   8. Simplified 93.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+73} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+153}:\\ \;\;\;\;v \cdot \frac{t1}{\left(\left(-u\right) - t1\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.2e+158)
   (/ v (- t1))
   (if (<= t1 4.1e+153)
     (* v (/ t1 (* (- (- u) t1) (+ t1 u))))
     (* (/ v (+ t1 u)) (+ (/ u t1) -1.0)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.2e+158) {
		tmp = v / -t1;
	} else if (t1 <= 4.1e+153) {
		tmp = v * (t1 / ((-u - t1) * (t1 + u)));
	} else {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	}
	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 (t1 <= (-2.2d+158)) then
        tmp = v / -t1
    else if (t1 <= 4.1d+153) then
        tmp = v * (t1 / ((-u - t1) * (t1 + u)))
    else
        tmp = (v / (t1 + u)) * ((u / t1) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.2e+158) {
		tmp = v / -t1;
	} else if (t1 <= 4.1e+153) {
		tmp = v * (t1 / ((-u - t1) * (t1 + u)));
	} else {
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.2e+158:
		tmp = v / -t1
	elif t1 <= 4.1e+153:
		tmp = v * (t1 / ((-u - t1) * (t1 + u)))
	else:
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.2e+158)
		tmp = Float64(v / Float64(-t1));
	elseif (t1 <= 4.1e+153)
		tmp = Float64(v * Float64(t1 / Float64(Float64(Float64(-u) - t1) * Float64(t1 + u))));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(u / t1) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.2e+158)
		tmp = v / -t1;
	elseif (t1 <= 4.1e+153)
		tmp = v * (t1 / ((-u - t1) * (t1 + u)));
	else
		tmp = (v / (t1 + u)) * ((u / t1) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.2e+158], N[(v / (-t1)), $MachinePrecision], If[LessEqual[t1, 4.1e+153], N[(v * N[(t1 / N[(N[((-u) - t1), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.2000000000000001e158

   1. Initial program 23.5%

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

      1. associate-*l/ 24.9%

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

      2. *-commutative 24.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in t1 around inf 84.1%

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

      1. associate-*r/ 84.1%

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

      2. neg-mul-1 84.1%

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

   7. Simplified 84.1%

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

   if -2.2000000000000001e158 < t1 < 4.10000000000000017e153

   1. Initial program 84.9%

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

      1. associate-*l/ 88.7%

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

      2. *-commutative 88.7%

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

   3. Simplified 88.7%

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

   if 4.10000000000000017e153 < t1

   1. Initial program 39.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 96.3%

      \[\leadsto \color{blue}{\left(\frac{u}{t1} - 1\right)} \cdot \frac{v}{t1 + u} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{elif}\;t1 \leq 4.1 \cdot 10^{+153}:\\ \;\;\;\;v \cdot \frac{t1}{\left(\left(-u\right) - t1\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{-129}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u - t1}}{t1}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.7e-129)
   (/ (* v (/ t1 (- u t1))) t1)
   (if (<= t1 1.35e-85)
     (* t1 (/ (/ v (- u)) (+ t1 u)))
     (/ v (- (* u (- 2.0)) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.7e-129) {
		tmp = (v * (t1 / (u - t1))) / t1;
	} else if (t1 <= 1.35e-85) {
		tmp = t1 * ((v / -u) / (t1 + u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.7d-129)) then
        tmp = (v * (t1 / (u - t1))) / t1
    else if (t1 <= 1.35d-85) then
        tmp = t1 * ((v / -u) / (t1 + u))
    else
        tmp = v / ((u * -2.0d0) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.7e-129) {
		tmp = (v * (t1 / (u - t1))) / t1;
	} else if (t1 <= 1.35e-85) {
		tmp = t1 * ((v / -u) / (t1 + u));
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.7e-129:
		tmp = (v * (t1 / (u - t1))) / t1
	elif t1 <= 1.35e-85:
		tmp = t1 * ((v / -u) / (t1 + u))
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.7e-129)
		tmp = Float64(Float64(v * Float64(t1 / Float64(u - t1))) / t1);
	elseif (t1 <= 1.35e-85)
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / Float64(t1 + u)));
	else
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.7e-129)
		tmp = (v * (t1 / (u - t1))) / t1;
	elseif (t1 <= 1.35e-85)
		tmp = t1 * ((v / -u) / (t1 + u));
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.7e-129], N[(N[(v * N[(t1 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, 1.35e-85], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.69999999999999999e-129

   1. Initial program 70.6%

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

      1. times-frac 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 77.3%

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

      1. associate-*r/ 78.6%

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

   7. Applied egg-rr 78.3%

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

   if -2.69999999999999999e-129 < t1 < 1.3500000000000001e-85

   1. Initial program 76.4%

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

      1. associate-/l* 81.0%

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

   3. Simplified 81.0%

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

      1. associate-/r* 89.9%

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

      2. div-inv 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. associate-*r/ 89.9%

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

      2. *-rgt-identity 89.9%

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

   8. Simplified 89.9%

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

   9. Taylor expanded in t1 around 0 84.9%

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

   if 1.3500000000000001e-85 < t1

   1. Initial program 72.4%

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

      1. associate-*l/ 77.4%

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

      2. *-commutative 77.4%

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

   3. Simplified 77.4%

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

      1. associate-*r/ 72.4%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. clear-num 100.0%

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

      10. frac-2neg 100.0%

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

      11. frac-times 96.4%

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

      12. *-un-lft-identity 96.4%

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

      13. +-commutative 96.4%

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

      14. distribute-neg-in 96.4%

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

      15. sub-neg 96.4%

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

      16. sub-neg 96.4%

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

      17. +-commutative 96.4%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 31.9%

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

      20. sqr-neg 31.9%

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

      21. sqrt-unprod 32.5%

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

      22. add-sqr-sqrt 13.1%

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

      23. sqrt-unprod 26.9%

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

      24. sqr-neg 26.9%

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

   6. Applied egg-rr 96.4%

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

   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%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{-129}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u - t1}}{t1}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{-42} \lor \neg \left(t1 \leq 1.1 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4e-42) (not (<= t1 1.1e-87)))
   (/ v (- (* u (- 2.0)) t1))
   (/ (* t1 (/ v (- u))) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4e-42) || !(t1 <= 1.1e-87)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 * (v / -u)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4d-42)) .or. (.not. (t1 <= 1.1d-87))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = (t1 * (v / -u)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4e-42) || !(t1 <= 1.1e-87)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 * (v / -u)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4e-42) or not (t1 <= 1.1e-87):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (t1 * (v / -u)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4e-42) || !(t1 <= 1.1e-87))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4e-42) || ~((t1 <= 1.1e-87)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (t1 * (v / -u)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4e-42], N[Not[LessEqual[t1, 1.1e-87]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.00000000000000015e-42 or 1.09999999999999994e-87 < t1

   1. Initial program 67.9%

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

      1. associate-*l/ 72.8%

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

      2. *-commutative 72.8%

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

   3. Simplified 72.8%

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

      1. associate-*r/ 67.9%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. clear-num 99.9%

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

      10. frac-2neg 99.9%

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

      11. frac-times 96.7%

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

      12. *-un-lft-identity 96.7%

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

      13. +-commutative 96.7%

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

      14. distribute-neg-in 96.7%

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

      15. sub-neg 96.7%

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

      16. sub-neg 96.7%

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

      17. +-commutative 96.7%

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

      18. add-sqr-sqrt 43.8%

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

      19. sqrt-unprod 48.0%

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

      20. sqr-neg 48.0%

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

      21. sqrt-unprod 17.8%

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

      22. add-sqr-sqrt 7.1%

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

      23. sqrt-unprod 14.7%

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

      24. sqr-neg 14.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in u around 0 82.6%

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

      1. *-commutative 82.6%

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

   9. Simplified 82.6%

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

   if -4.00000000000000015e-42 < t1 < 1.09999999999999994e-87

   1. Initial program 80.3%

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

      1. associate-*l/ 82.2%

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

      2. *-commutative 82.2%

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

   3. Simplified 82.2%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. times-frac 95.5%

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

      4. frac-2neg 95.5%

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

      5. +-commutative 95.5%

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

      6. distribute-neg-in 95.5%

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

      7. sub-neg 95.5%

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

      8. associate-*r/ 93.9%

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

      9. add-sqr-sqrt 48.1%

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

      10. sqrt-unprod 56.7%

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

      11. sqr-neg 56.7%

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

      12. sqrt-unprod 19.5%

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

      13. add-sqr-sqrt 40.5%

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

      14. sub-neg 40.5%

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

      15. +-commutative 40.5%

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

      16. add-sqr-sqrt 21.0%

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

      17. sqrt-unprod 43.2%

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

      18. sqr-neg 43.2%

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

      19. sqrt-unprod 24.8%

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

      20. add-sqr-sqrt 14.1%

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

      21. sqrt-unprod 36.4%

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

      22. sqr-neg 36.4%

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

   6. Applied egg-rr 93.9%

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

   7. Taylor expanded in t1 around 0 73.6%

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

      1. mul-1-neg 73.6%

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

      2. associate-/l* 77.2%

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

      3. distribute-rgt-neg-in 77.2%

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

      4. distribute-neg-frac2 77.2%

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

   9. Simplified 77.2%

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

   10. Taylor expanded in t1 around 0 79.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{-42} \lor \neg \left(t1 \leq 1.1 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.16 \cdot 10^{-122} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.16e-122) (not (<= t1 1.35e-85)))
   (/ v (- (* u (- 2.0)) t1))
   (/ v (* u (/ (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.16e-122) || !(t1 <= 1.35e-85)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = 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 <= (-1.16d-122)) .or. (.not. (t1 <= 1.35d-85))) then
        tmp = v / ((u * -2.0d0) - t1)
    else
        tmp = 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 <= -1.16e-122) || !(t1 <= 1.35e-85)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v / (u * (-u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.16e-122) or not (t1 <= 1.35e-85):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = v / (u * (-u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.16e-122) || !(t1 <= 1.35e-85))
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	else
		tmp = 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 <= -1.16e-122) || ~((t1 <= 1.35e-85)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = v / (u * (-u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.16e-122], N[Not[LessEqual[t1, 1.35e-85]], $MachinePrecision]], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[((-u) / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.16000000000000001e-122 or 1.3500000000000001e-85 < t1

   1. Initial program 71.3%

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

      1. associate-*l/ 75.6%

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

      2. *-commutative 75.6%

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

   3. Simplified 75.6%

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

      1. associate-*r/ 71.3%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. clear-num 99.9%

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

      10. frac-2neg 99.9%

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

      11. frac-times 96.6%

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

      12. *-un-lft-identity 96.6%

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

      13. +-commutative 96.6%

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

      14. distribute-neg-in 96.6%

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

      15. sub-neg 96.6%

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

      16. sub-neg 96.6%

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

      17. +-commutative 96.6%

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

      18. add-sqr-sqrt 50.2%

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

      19. sqrt-unprod 53.9%

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

      20. sqr-neg 53.9%

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

      21. sqrt-unprod 15.6%

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

      22. add-sqr-sqrt 6.3%

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

      23. sqrt-unprod 12.9%

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

      24. sqr-neg 12.9%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in u around 0 79.8%

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

      1. *-commutative 79.8%

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

   9. Simplified 79.8%

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

   if -1.16000000000000001e-122 < t1 < 1.3500000000000001e-85

   1. Initial program 76.7%

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

      1. associate-*l/ 79.1%

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

      2. *-commutative 79.1%

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

   3. Simplified 79.1%

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

      1. associate-*r/ 76.7%

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

      2. times-frac 94.4%

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

      3. *-commutative 94.4%

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

      4. frac-2neg 94.4%

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

      5. remove-double-neg 94.4%

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

      6. +-commutative 94.4%

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

      7. distribute-neg-in 94.4%

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

      8. sub-neg 94.4%

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

      9. clear-num 94.3%

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

      10. frac-2neg 94.3%

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

      11. frac-times 92.2%

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

      12. *-un-lft-identity 92.2%

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

      13. +-commutative 92.2%

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

      14. distribute-neg-in 92.2%

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

      15. sub-neg 92.2%

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

      16. sub-neg 92.2%

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

      17. +-commutative 92.2%

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

      18. add-sqr-sqrt 35.7%

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

      19. sqrt-unprod 82.0%

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

      20. sqr-neg 82.0%

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

      21. sqrt-unprod 48.4%

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

      22. add-sqr-sqrt 25.2%

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

      23. sqrt-unprod 33.3%

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

      24. sqr-neg 33.3%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in t1 around 0 80.9%

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

   8. Taylor expanded in t1 around 0 82.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.16 \cdot 10^{-122} \lor \neg \left(t1 \leq 1.35 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.16 \cdot 10^{-122} \lor \neg \left(t1 \leq 4.6 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.16e-122) (not (<= t1 4.6e-86)))
   (/ v (- (- u) t1))
   (/ v (* u (/ (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.16e-122) || !(t1 <= 4.6e-86)) {
		tmp = v / (-u - t1);
	} else {
		tmp = 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 <= (-1.16d-122)) .or. (.not. (t1 <= 4.6d-86))) then
        tmp = v / (-u - t1)
    else
        tmp = 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 <= -1.16e-122) || !(t1 <= 4.6e-86)) {
		tmp = v / (-u - t1);
	} else {
		tmp = v / (u * (-u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.16e-122) or not (t1 <= 4.6e-86):
		tmp = v / (-u - t1)
	else:
		tmp = v / (u * (-u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.16e-122) || !(t1 <= 4.6e-86))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = 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 <= -1.16e-122) || ~((t1 <= 4.6e-86)))
		tmp = v / (-u - t1);
	else
		tmp = v / (u * (-u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.16e-122], N[Not[LessEqual[t1, 4.6e-86]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[((-u) / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.16000000000000001e-122 or 4.59999999999999992e-86 < t1

   1. Initial program 71.3%

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

      1. associate-*l/ 75.6%

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

      2. *-commutative 75.6%

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

   3. Simplified 75.6%

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

      1. associate-*r/ 71.3%

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

      2. *-commutative 71.3%

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

      3. times-frac 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 51.7%

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

      10. sqrt-unprod 40.3%

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

      11. sqr-neg 40.3%

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

      12. sqrt-unprod 11.7%

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

      13. add-sqr-sqrt 27.5%

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

      14. sub-neg 27.5%

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

      15. +-commutative 27.5%

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

      16. add-sqr-sqrt 15.9%

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

      17. sqrt-unprod 49.6%

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

      18. sqr-neg 49.6%

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

      19. sqrt-unprod 41.5%

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

      20. add-sqr-sqrt 21.1%

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

      21. sqrt-unprod 43.8%

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

      22. sqr-neg 43.8%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 79.0%

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

      1. mul-1-neg 79.0%

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

   9. Simplified 79.0%

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

   if -1.16000000000000001e-122 < t1 < 4.59999999999999992e-86

   1. Initial program 76.7%

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

      1. associate-*l/ 79.1%

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

      2. *-commutative 79.1%

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

   3. Simplified 79.1%

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

      1. associate-*r/ 76.7%

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

      2. times-frac 94.4%

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

      3. *-commutative 94.4%

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

      4. frac-2neg 94.4%

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

      5. remove-double-neg 94.4%

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

      6. +-commutative 94.4%

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

      7. distribute-neg-in 94.4%

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

      8. sub-neg 94.4%

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

      9. clear-num 94.3%

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

      10. frac-2neg 94.3%

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

      11. frac-times 92.2%

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

      12. *-un-lft-identity 92.2%

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

      13. +-commutative 92.2%

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

      14. distribute-neg-in 92.2%

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

      15. sub-neg 92.2%

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

      16. sub-neg 92.2%

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

      17. +-commutative 92.2%

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

      18. add-sqr-sqrt 35.7%

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

      19. sqrt-unprod 82.0%

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

      20. sqr-neg 82.0%

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

      21. sqrt-unprod 48.4%

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

      22. add-sqr-sqrt 25.2%

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

      23. sqrt-unprod 33.3%

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

      24. sqr-neg 33.3%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in t1 around 0 80.9%

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

   8. Taylor expanded in t1 around 0 82.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.16 \cdot 10^{-122} \lor \neg \left(t1 \leq 4.6 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-122}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u - t1}}{t1}\\ \mathbf{elif}\;t1 \leq 1.05 \cdot 10^{-86}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.12e-122)
   (/ (* v (/ t1 (- u t1))) t1)
   (if (<= t1 1.05e-86) (/ (* t1 (/ v (- u))) u) (/ v (- (* u (- 2.0)) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.12e-122) {
		tmp = (v * (t1 / (u - t1))) / t1;
	} else if (t1 <= 1.05e-86) {
		tmp = (t1 * (v / -u)) / u;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.12d-122)) then
        tmp = (v * (t1 / (u - t1))) / t1
    else if (t1 <= 1.05d-86) then
        tmp = (t1 * (v / -u)) / u
    else
        tmp = v / ((u * -2.0d0) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.12e-122) {
		tmp = (v * (t1 / (u - t1))) / t1;
	} else if (t1 <= 1.05e-86) {
		tmp = (t1 * (v / -u)) / u;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.12e-122:
		tmp = (v * (t1 / (u - t1))) / t1
	elif t1 <= 1.05e-86:
		tmp = (t1 * (v / -u)) / u
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.12e-122)
		tmp = Float64(Float64(v * Float64(t1 / Float64(u - t1))) / t1);
	elseif (t1 <= 1.05e-86)
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	else
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.12e-122)
		tmp = (v * (t1 / (u - t1))) / t1;
	elseif (t1 <= 1.05e-86)
		tmp = (t1 * (v / -u)) / u;
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.12e-122], N[(N[(v * N[(t1 / N[(u - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], If[LessEqual[t1, 1.05e-86], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.12e-122

   1. Initial program 70.3%

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

      1. times-frac 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 77.1%

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

      1. associate-*r/ 78.4%

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

   7. Applied egg-rr 78.0%

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

   if -1.12e-122 < t1 < 1.05e-86

   1. Initial program 76.7%

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

      1. associate-*l/ 79.1%

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

      2. *-commutative 79.1%

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

   3. Simplified 79.1%

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

      1. associate-*r/ 76.7%

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

      2. *-commutative 76.7%

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

      3. times-frac 94.4%

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

      4. frac-2neg 94.4%

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

      5. +-commutative 94.4%

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

      6. distribute-neg-in 94.4%

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

      7. sub-neg 94.4%

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

      8. associate-*r/ 92.4%

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

      9. add-sqr-sqrt 35.7%

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

      10. sqrt-unprod 46.2%

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

      11. sqr-neg 46.2%

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

      12. sqrt-unprod 24.2%

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

      13. add-sqr-sqrt 40.8%

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

      14. sub-neg 40.8%

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

      15. +-commutative 40.8%

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

      16. add-sqr-sqrt 16.6%

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

      17. sqrt-unprod 44.2%

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

      18. sqr-neg 44.2%

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

      19. sqrt-unprod 30.8%

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

      20. add-sqr-sqrt 17.5%

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

      21. sqrt-unprod 45.2%

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

      22. sqr-neg 45.2%

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

   6. Applied egg-rr 92.4%

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

   7. Taylor expanded in t1 around 0 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/l* 82.1%

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

      3. distribute-rgt-neg-in 82.1%

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

      4. distribute-neg-frac2 82.1%

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

   9. Simplified 82.1%

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

   10. Taylor expanded in t1 around 0 83.4%

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

   if 1.05e-86 < t1

   1. Initial program 72.4%

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

      1. associate-*l/ 77.4%

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

      2. *-commutative 77.4%

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

   3. Simplified 77.4%

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

      1. associate-*r/ 72.4%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. clear-num 100.0%

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

      10. frac-2neg 100.0%

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

      11. frac-times 96.4%

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

      12. *-un-lft-identity 96.4%

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

      13. +-commutative 96.4%

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

      14. distribute-neg-in 96.4%

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

      15. sub-neg 96.4%

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

      16. sub-neg 96.4%

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

      17. +-commutative 96.4%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 31.9%

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

      20. sqr-neg 31.9%

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

      21. sqrt-unprod 32.5%

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

      22. add-sqr-sqrt 13.1%

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

      23. sqrt-unprod 26.9%

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

      24. sqr-neg 26.9%

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

   6. Applied egg-rr 96.4%

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

   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%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-122}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u - t1}}{t1}\\ \mathbf{elif}\;t1 \leq 1.05 \cdot 10^{-86}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.85 \cdot 10^{-123}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.85e-123)
   (/ v (* t1 (/ (- u t1) t1)))
   (if (<= t1 1.35e-85) (/ (* t1 (/ v (- u))) u) (/ v (- (* u (- 2.0)) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.85e-123) {
		tmp = v / (t1 * ((u - t1) / t1));
	} else if (t1 <= 1.35e-85) {
		tmp = (t1 * (v / -u)) / u;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.85d-123)) then
        tmp = v / (t1 * ((u - t1) / t1))
    else if (t1 <= 1.35d-85) then
        tmp = (t1 * (v / -u)) / u
    else
        tmp = v / ((u * -2.0d0) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.85e-123) {
		tmp = v / (t1 * ((u - t1) / t1));
	} else if (t1 <= 1.35e-85) {
		tmp = (t1 * (v / -u)) / u;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.85e-123:
		tmp = v / (t1 * ((u - t1) / t1))
	elif t1 <= 1.35e-85:
		tmp = (t1 * (v / -u)) / u
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.85e-123)
		tmp = Float64(v / Float64(t1 * Float64(Float64(u - t1) / t1)));
	elseif (t1 <= 1.35e-85)
		tmp = Float64(Float64(t1 * Float64(v / Float64(-u))) / u);
	else
		tmp = Float64(v / Float64(Float64(u * Float64(-2.0)) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.85e-123)
		tmp = v / (t1 * ((u - t1) / t1));
	elseif (t1 <= 1.35e-85)
		tmp = (t1 * (v / -u)) / u;
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.85e-123], N[(v / N[(t1 * N[(N[(u - t1), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.35e-85], N[(N[(t1 * N[(v / (-u)), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[(N[(u * (-2.0)), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.85000000000000014e-123

   1. Initial program 70.3%

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

      1. times-frac 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 77.1%

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

      1. clear-num 77.1%

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

      2. frac-times 78.2%

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

      3. *-un-lft-identity 78.2%

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

      4. add-sqr-sqrt 41.0%

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

      5. sqrt-unprod 79.5%

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

      6. sqr-neg 79.5%

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

      7. sqrt-unprod 37.4%

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

      8. add-sqr-sqrt 77.8%

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

   7. Applied egg-rr 77.8%

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

   if -2.85000000000000014e-123 < t1 < 1.3500000000000001e-85

   1. Initial program 76.7%

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

      1. associate-*l/ 79.1%

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

      2. *-commutative 79.1%

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

   3. Simplified 79.1%

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

      1. associate-*r/ 76.7%

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

      2. *-commutative 76.7%

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

      3. times-frac 94.4%

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

      4. frac-2neg 94.4%

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

      5. +-commutative 94.4%

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

      6. distribute-neg-in 94.4%

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

      7. sub-neg 94.4%

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

      8. associate-*r/ 92.4%

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

      9. add-sqr-sqrt 35.7%

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

      10. sqrt-unprod 46.2%

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

      11. sqr-neg 46.2%

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

      12. sqrt-unprod 24.2%

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

      13. add-sqr-sqrt 40.8%

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

      14. sub-neg 40.8%

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

      15. +-commutative 40.8%

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

      16. add-sqr-sqrt 16.6%

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

      17. sqrt-unprod 44.2%

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

      18. sqr-neg 44.2%

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

      19. sqrt-unprod 30.8%

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

      20. add-sqr-sqrt 17.5%

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

      21. sqrt-unprod 45.2%

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

      22. sqr-neg 45.2%

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

   6. Applied egg-rr 92.4%

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

   7. Taylor expanded in t1 around 0 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/l* 82.1%

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

      3. distribute-rgt-neg-in 82.1%

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

      4. distribute-neg-frac2 82.1%

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

   9. Simplified 82.1%

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

   10. Taylor expanded in t1 around 0 83.4%

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

   if 1.3500000000000001e-85 < t1

   1. Initial program 72.4%

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

      1. associate-*l/ 77.4%

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

      2. *-commutative 77.4%

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

   3. Simplified 77.4%

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

      1. associate-*r/ 72.4%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. remove-double-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-neg-in 99.9%

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

      8. sub-neg 99.9%

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

      9. clear-num 100.0%

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

      10. frac-2neg 100.0%

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

      11. frac-times 96.4%

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

      12. *-un-lft-identity 96.4%

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

      13. +-commutative 96.4%

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

      14. distribute-neg-in 96.4%

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

      15. sub-neg 96.4%

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

      16. sub-neg 96.4%

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

      17. +-commutative 96.4%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 31.9%

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

      20. sqr-neg 31.9%

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

      21. sqrt-unprod 32.5%

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

      22. add-sqr-sqrt 13.1%

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

      23. sqrt-unprod 26.9%

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

      24. sqr-neg 26.9%

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

   6. Applied egg-rr 96.4%

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

   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%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.85 \cdot 10^{-123}:\\ \;\;\;\;\frac{v}{t1 \cdot \frac{u - t1}{t1}}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-85}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+147} \lor \neg \left(u \leq 2.8 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.55e+147) (not (<= u 2.8e+123)))
   (/ (* v (/ t1 u)) u)
   (/ v (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.55e+147) || !(u <= 2.8e+123)) {
		tmp = (v * (t1 / u)) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.55d+147)) .or. (.not. (u <= 2.8d+123))) then
        tmp = (v * (t1 / u)) / u
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.55e+147) || !(u <= 2.8e+123)) {
		tmp = (v * (t1 / u)) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.55e+147) or not (u <= 2.8e+123):
		tmp = (v * (t1 / u)) / u
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.55e+147) || !(u <= 2.8e+123))
		tmp = Float64(Float64(v * Float64(t1 / u)) / u);
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.55e+147) || ~((u <= 2.8e+123)))
		tmp = (v * (t1 / u)) / u;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.55e+147], N[Not[LessEqual[u, 2.8e+123]], $MachinePrecision]], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.55e147 or 2.80000000000000011e123 < u

   1. Initial program 80.2%

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

      1. associate-*l/ 77.8%

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

      2. *-commutative 77.8%

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

   3. Simplified 77.8%

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

      1. associate-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. times-frac 97.2%

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

      4. frac-2neg 97.2%

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

      5. +-commutative 97.2%

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

      6. distribute-neg-in 97.2%

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

      7. sub-neg 97.2%

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

      8. associate-*r/ 97.2%

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

      9. add-sqr-sqrt 50.0%

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

      10. sqrt-unprod 68.5%

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

      11. sqr-neg 68.5%

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

      12. sqrt-unprod 37.2%

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

      13. add-sqr-sqrt 73.3%

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

      14. sub-neg 73.3%

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

      15. +-commutative 73.3%

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

      16. add-sqr-sqrt 36.1%

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

      17. sqrt-unprod 73.5%

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

      18. sqr-neg 73.5%

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

      19. sqrt-unprod 38.2%

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

      20. add-sqr-sqrt 20.2%

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

      21. sqrt-unprod 40.2%

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

      22. sqr-neg 40.2%

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in t1 around 0 87.2%

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

      1. mul-1-neg 87.2%

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

      2. associate-/l* 94.4%

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

      3. distribute-rgt-neg-in 94.4%

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

      4. distribute-neg-frac2 94.4%

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

   9. Simplified 94.4%

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

   10. Taylor expanded in t1 around 0 94.5%

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

       1. clear-num 94.4%

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

       2. un-div-inv 94.5%

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

       3. add-sqr-sqrt 51.2%

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

       4. sqrt-unprod 77.7%

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

       5. sqr-neg 77.7%

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

       6. sqrt-unprod 32.9%

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

       7. add-sqr-sqrt 73.2%

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

   12. Applied egg-rr 73.2%

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

       1. associate-/r/ 72.0%

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

   14. Simplified 72.0%

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

   if -1.55e147 < u < 2.80000000000000011e123

   1. Initial program 70.6%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 69.0%

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

      1. *-commutative 69.0%

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

      2. frac-times 48.3%

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

      3. add-sqr-sqrt 25.1%

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

      4. sqrt-unprod 48.7%

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

      5. sqr-neg 48.7%

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

      6. sqrt-unprod 23.0%

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

      7. add-sqr-sqrt 48.6%

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

   7. Applied egg-rr 48.6%

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

   8. Taylor expanded in v around 0 68.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+147} \lor \neg \left(u \leq 2.8 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+188} \lor \neg \left(u \leq 5.2 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.2e+188) (not (<= u 5.2e+118))) (/ 1.0 (/ u v)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.2e+188) || !(u <= 5.2e+118)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-7.2d+188)) .or. (.not. (u <= 5.2d+118))) then
        tmp = 1.0d0 / (u / v)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.2e+188) || !(u <= 5.2e+118)) {
		tmp = 1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.2e+188) or not (u <= 5.2e+118):
		tmp = 1.0 / (u / v)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.2e+188) || !(u <= 5.2e+118))
		tmp = Float64(1.0 / Float64(u / v));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.2e+188) || ~((u <= 5.2e+118)))
		tmp = 1.0 / (u / v);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.2e+188], N[Not[LessEqual[u, 5.2e+118]], $MachinePrecision]], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.20000000000000041e188 or 5.20000000000000032e118 < u

   1. Initial program 81.0%

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

      1. times-frac 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-neg-frac2 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. unsub-neg 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t1 around inf 50.7%

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

      1. *-commutative 50.7%

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

      2. frac-times 54.9%

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

      3. add-sqr-sqrt 28.1%

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

      4. sqrt-unprod 77.8%

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

      5. sqr-neg 77.8%

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

      6. sqrt-unprod 26.7%

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

      7. add-sqr-sqrt 54.8%

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

   7. Applied egg-rr 54.8%

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

   8. Taylor expanded in t1 around 0 44.0%

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

      1. clear-num 45.5%

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

      2. inv-pow 45.5%

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

   10. Applied egg-rr 45.5%

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

       1. unpow-1 45.5%

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

   12. Simplified 45.5%

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

   if -7.20000000000000041e188 < u < 5.20000000000000032e118

   1. Initial program 70.7%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+188} \lor \neg \left(u \leq 5.2 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{+186} \lor \neg \left(u \leq 10^{+117}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.7e+186) (not (<= u 1e+117))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.7e+186) || !(u <= 1e+117)) {
		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 <= (-3.7d+186)) .or. (.not. (u <= 1d+117))) 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 <= -3.7e+186) || !(u <= 1e+117)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.7e+186) or not (u <= 1e+117):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.7e+186) || !(u <= 1e+117))
		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 <= -3.7e+186) || ~((u <= 1e+117)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.7e+186], N[Not[LessEqual[u, 1e+117]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.7e186 or 1.00000000000000005e117 < u

   1. Initial program 81.0%

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

      1. times-frac 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-neg-frac2 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. unsub-neg 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t1 around inf 50.7%

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

      1. *-commutative 50.7%

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

      2. frac-times 54.9%

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

      3. add-sqr-sqrt 28.1%

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

      4. sqrt-unprod 77.8%

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

      5. sqr-neg 77.8%

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

      6. sqrt-unprod 26.7%

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

      7. add-sqr-sqrt 54.8%

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

   7. Applied egg-rr 54.8%

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

   8. Taylor expanded in t1 around 0 44.0%

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

   if -3.7e186 < u < 1.00000000000000005e117

   1. Initial program 70.7%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{+186} \lor \neg \left(u \leq 10^{+117}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+186}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{+119}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.3e+186)
   (* (/ v u) -0.5)
   (if (<= u 2.3e+119) (/ v (- t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.3e+186) {
		tmp = (v / u) * -0.5;
	} else if (u <= 2.3e+119) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.3d+186)) then
        tmp = (v / u) * (-0.5d0)
    else if (u <= 2.3d+119) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.3e+186) {
		tmp = (v / u) * -0.5;
	} else if (u <= 2.3e+119) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.3e+186:
		tmp = (v / u) * -0.5
	elif u <= 2.3e+119:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.3e+186)
		tmp = Float64(Float64(v / u) * -0.5);
	elseif (u <= 2.3e+119)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.3e+186)
		tmp = (v / u) * -0.5;
	elseif (u <= 2.3e+119)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.3e+186], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[u, 2.3e+119], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.3e186

   1. Initial program 83.3%

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

      1. associate-*l/ 83.6%

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

      2. *-commutative 83.6%

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

   3. Simplified 83.6%

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

      1. associate-*r/ 83.3%

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

      2. times-frac 96.9%

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

      3. *-commutative 96.9%

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

      4. frac-2neg 96.9%

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

      5. remove-double-neg 96.9%

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

      6. +-commutative 96.9%

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

      7. distribute-neg-in 96.9%

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

      8. sub-neg 96.9%

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

      9. clear-num 96.9%

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

      10. frac-2neg 96.9%

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

      11. frac-times 93.5%

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

      12. *-un-lft-identity 93.5%

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

      13. +-commutative 93.5%

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

      14. distribute-neg-in 93.5%

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

      15. sub-neg 93.5%

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

      16. sub-neg 93.5%

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

      17. +-commutative 93.5%

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

      18. add-sqr-sqrt 55.4%

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

      19. sqrt-unprod 83.6%

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

      20. sqr-neg 83.6%

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

      21. sqrt-unprod 37.2%

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

      22. add-sqr-sqrt 37.2%

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

      23. sqrt-unprod 34.7%

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

      24. sqr-neg 34.7%

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in u around 0 53.3%

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

      1. *-commutative 53.3%

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

   9. Simplified 53.3%

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

   10. Taylor expanded in t1 around 0 53.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]

   if -4.3e186 < u < 2.3000000000000001e119

   1. Initial program 70.7%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

   if 2.3000000000000001e119 < u

   1. Initial program 79.0%

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

      1. times-frac 97.0%

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

      2. distribute-frac-neg 97.0%

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

      3. distribute-neg-frac2 97.0%

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

      4. +-commutative 97.0%

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

      5. distribute-neg-in 97.0%

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

      6. unsub-neg 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in t1 around inf 43.1%

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

      1. *-commutative 43.1%

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

      2. frac-times 51.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 72.8%

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

      5. sqr-neg 72.8%

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

      6. sqrt-unprod 50.8%

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

      7. add-sqr-sqrt 50.8%

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

   7. Applied egg-rr 50.8%

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

   8. Taylor expanded in t1 around 0 35.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.3 \cdot 10^{+186}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 2.3 \cdot 10^{+119}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+187}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.5e+187) (/ v (- u)) (if (<= u 4.5e+118) (/ v (- t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e+187) {
		tmp = v / -u;
	} else if (u <= 4.5e+118) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.5d+187)) then
        tmp = v / -u
    else if (u <= 4.5d+118) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e+187) {
		tmp = v / -u;
	} else if (u <= 4.5e+118) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.5e+187:
		tmp = v / -u
	elif u <= 4.5e+118:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.5e+187)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 4.5e+118)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.5e+187)
		tmp = v / -u;
	elseif (u <= 4.5e+118)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.5e+187], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 4.5e+118], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.49999999999999997e187

   1. Initial program 83.3%

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

      1. times-frac 96.9%

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

      2. distribute-frac-neg 96.9%

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

      3. distribute-neg-frac2 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. unsub-neg 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in t1 around inf 59.1%

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

   6. Taylor expanded in t1 around 0 53.3%

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

      1. associate-*r/ 53.3%

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

      2. mul-1-neg 53.3%

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

   8. Simplified 53.3%

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

   if -5.49999999999999997e187 < u < 4.50000000000000002e118

   1. Initial program 70.7%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in t1 around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. neg-mul-1 64.3%

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

   7. Simplified 64.3%

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

   if 4.50000000000000002e118 < u

   1. Initial program 79.0%

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

      1. times-frac 97.0%

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

      2. distribute-frac-neg 97.0%

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

      3. distribute-neg-frac2 97.0%

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

      4. +-commutative 97.0%

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

      5. distribute-neg-in 97.0%

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

      6. unsub-neg 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in t1 around inf 43.1%

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

      1. *-commutative 43.1%

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

      2. frac-times 51.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 72.8%

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

      5. sqr-neg 72.8%

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

      6. sqrt-unprod 50.8%

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

      7. add-sqr-sqrt 50.8%

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

   7. Applied egg-rr 50.8%

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

   8. Taylor expanded in t1 around 0 35.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+187}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3.1 \cdot 10^{+211}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot v}{t1 \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 3.1e+211) (/ v (- u t1)) (/ (* t1 v) (* t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.1e+211) {
		tmp = v / (u - t1);
	} else {
		tmp = (t1 * v) / (t1 * u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 3.1d+211) then
        tmp = v / (u - t1)
    else
        tmp = (t1 * v) / (t1 * u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.1e+211) {
		tmp = v / (u - t1);
	} else {
		tmp = (t1 * v) / (t1 * u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3.1e+211:
		tmp = v / (u - t1)
	else:
		tmp = (t1 * v) / (t1 * u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3.1e+211)
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(Float64(t1 * v) / Float64(t1 * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 3.1e+211)
		tmp = v / (u - t1);
	else
		tmp = (t1 * v) / (t1 * u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3.1e+211], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * v), $MachinePrecision] / N[(t1 * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.1000000000000002e211

   1. Initial program 72.2%

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

      1. times-frac 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-neg-frac2 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. unsub-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around inf 65.4%

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

      1. *-commutative 65.4%

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

      2. frac-times 47.5%

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

      3. add-sqr-sqrt 27.8%

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

      4. sqrt-unprod 52.5%

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

      5. sqr-neg 52.5%

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

      6. sqrt-unprod 19.6%

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

      7. add-sqr-sqrt 47.7%

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

   7. Applied egg-rr 47.7%

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

   8. Taylor expanded in v around 0 63.8%

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

   if 3.1000000000000002e211 < u

   1. Initial program 84.5%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 44.9%

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

      1. *-commutative 44.9%

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

      2. frac-times 63.1%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 84.5%

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

      5. sqr-neg 84.5%

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

      6. sqrt-unprod 63.0%

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

      7. add-sqr-sqrt 63.0%

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

   7. Applied egg-rr 63.0%

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

   8. Taylor expanded in t1 around 0 63.0%

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

      1. *-commutative 63.0%

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

   10. Simplified 63.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 3.1 \cdot 10^{+211}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot v}{t1 \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 19.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq 1.95 \cdot 10^{+192}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= t1 1.95e+192) (/ v u) (/ v t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= 1.95e+192) {
		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 (t1 <= 1.95d+192) 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 (t1 <= 1.95e+192) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= 1.95e+192:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= 1.95e+192)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= 1.95e+192)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, 1.95e+192], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < 1.9499999999999999e192

   1. Initial program 75.0%

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

      1. times-frac 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-neg-frac2 97.9%

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

      4. +-commutative 97.9%

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

      5. distribute-neg-in 97.9%

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

      6. unsub-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around inf 60.9%

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

      1. *-commutative 60.9%

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

      2. frac-times 48.5%

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

      3. add-sqr-sqrt 26.5%

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

      4. sqrt-unprod 55.2%

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

      5. sqr-neg 55.2%

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

      6. sqrt-unprod 21.8%

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

      7. add-sqr-sqrt 48.7%

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

   7. Applied egg-rr 48.7%

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

   8. Taylor expanded in t1 around 0 17.0%

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

   if 1.9499999999999999e192 < t1

   1. Initial program 51.1%

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

      1. associate-*l/ 52.4%

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

      2. *-commutative 52.4%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in t1 around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. neg-mul-1 98.5%

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

   7. Simplified 98.5%

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

      1. add-sqr-sqrt 44.7%

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

      2. sqrt-unprod 61.4%

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

      3. sqr-neg 61.4%

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

      4. sqrt-unprod 26.2%

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

      5. add-sqr-sqrt 51.7%

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

      6. div-inv 51.7%

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

   9. Applied egg-rr 51.7%

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

       1. associate-*r/ 51.7%

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

       2. *-rgt-identity 51.7%

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

   11. Simplified 51.7%

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

Alternative 17: 61.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 98.0%

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

   2. distribute-frac-neg 98.0%

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

   3. distribute-neg-frac2 98.0%

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

   4. +-commutative 98.0%

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

   5. distribute-neg-in 98.0%

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

   6. unsub-neg 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in t1 around inf 63.9%

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

   1. *-commutative 63.9%

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

   2. frac-times 48.7%

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

   3. add-sqr-sqrt 25.7%

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

   4. sqrt-unprod 54.9%

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

   5. sqr-neg 54.9%

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

   6. sqrt-unprod 22.8%

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

   7. add-sqr-sqrt 48.9%

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

7. Applied egg-rr 48.9%

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

8. Taylor expanded in v around 0 61.8%

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

Alternative 18: 13.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. associate-*l/ 76.7%

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

   2. *-commutative 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in t1 around inf 53.6%

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

   1. associate-*r/ 53.6%

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

   2. neg-mul-1 53.6%

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

7. Simplified 53.6%

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

   1. add-sqr-sqrt 26.3%

      \[\leadsto \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

   2. sqrt-unprod 33.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{t1} \]

   3. sqr-neg 33.2%

      \[\leadsto \frac{\sqrt{\color{blue}{v \cdot v}}}{t1} \]

   4. sqrt-unprod 5.7%

      \[\leadsto \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

   5. add-sqr-sqrt 10.3%

      \[\leadsto \frac{\color{blue}{v}}{t1} \]

   6. div-inv 10.3%

      \[\leadsto \color{blue}{v \cdot \frac{1}{t1}} \]

9. Applied egg-rr 10.3%

   \[\leadsto \color{blue}{v \cdot \frac{1}{t1}} \]
10. Step-by-step derivation

    1. associate-*r/ 10.3%

       \[\leadsto \color{blue}{\frac{v \cdot 1}{t1}} \]

    2. *-rgt-identity 10.3%

       \[\leadsto \frac{\color{blue}{v}}{t1} \]

11. Simplified 10.3%

    \[\leadsto \color{blue}{\frac{v}{t1}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024188 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))