Rosa's DopplerBench

Percentage Accurate: 71.5% → 98.2%

Time: 23.7s

Alternatives: 17

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: 71.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. times-frac 97.9%

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

   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. Step-by-step derivation

   1. frac-2neg 97.9%

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

   2. frac-2neg 97.9%

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

   3. frac-times 70.1%

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

   4. sub-neg 70.1%

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

   5. distribute-neg-in 70.1%

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

   6. +-commutative 70.1%

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

   7. remove-double-neg 70.1%

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

   8. frac-times 97.9%

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

   9. associate-*r/ 99.1%

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

   10. add-sqr-sqrt 46.7%

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

   11. sqrt-unprod 42.4%

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

   12. sqr-neg 42.4%

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

   13. sqrt-unprod 19.8%

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

   14. add-sqr-sqrt 39.5%

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

   15. add-sqr-sqrt 25.2%

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

   16. sqrt-unprod 58.3%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.12 \cdot 10^{-111} \lor \neg \left(u \leq 1.3 \cdot 10^{-123}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.12e-111) (not (<= u 1.3e-123)))
   (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.12e-111) || !(u <= 1.3e-123)) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.12d-111)) .or. (.not. (u <= 1.3d-123))) then
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.12e-111) || !(u <= 1.3e-123)) {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.12e-111) or not (u <= 1.3e-123):
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.12e-111) || !(u <= 1.3e-123))
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.12e-111) || ~((u <= 1.3e-123)))
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.12e-111], N[Not[LessEqual[u, 1.3e-123]], $MachinePrecision]], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.12000000000000009e-111 or 1.29999999999999998e-123 < u

   1. Initial program 76.1%

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

      1. associate-/l* 77.9%

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

      2. distribute-lft-neg-out 77.9%

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

      3. distribute-rgt-neg-in 77.9%

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

      4. associate-/r* 91.1%

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

      5. distribute-neg-frac2 91.1%

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

   3. Simplified 91.1%

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

   if -1.12000000000000009e-111 < u < 1.29999999999999998e-123

   1. Initial program 57.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. frac-2neg 100.0%

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

      3. frac-times 57.3%

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

      4. sub-neg 57.3%

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

      5. distribute-neg-in 57.3%

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

      6. +-commutative 57.3%

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

      7. remove-double-neg 57.3%

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

      8. frac-times 100.0%

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

      9. associate-*r/ 100.0%

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

      10. add-sqr-sqrt 46.0%

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

      11. sqrt-unprod 24.2%

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

      12. sqr-neg 24.2%

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

      13. sqrt-unprod 10.4%

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

      14. add-sqr-sqrt 20.2%

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

      15. add-sqr-sqrt 9.9%

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

      16. sqrt-unprod 43.9%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 94.2%

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

      1. mul-1-neg 94.2%

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

   9. Simplified 94.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.12 \cdot 10^{-111} \lor \neg \left(u \leq 1.3 \cdot 10^{-123}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+18} \lor \neg \left(u \leq 10^{-29}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5e+18) (not (<= u 1e-29)))
   (/ (/ t1 (/ u v)) (- t1 u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e+18) || !(u <= 1e-29)) {
		tmp = (t1 / (u / v)) / (t1 - 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 <= (-5d+18)) .or. (.not. (u <= 1d-29))) then
        tmp = (t1 / (u / v)) / (t1 - 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 <= -5e+18) || !(u <= 1e-29)) {
		tmp = (t1 / (u / v)) / (t1 - u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5e+18) or not (u <= 1e-29):
		tmp = (t1 / (u / v)) / (t1 - u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5e+18) || !(u <= 1e-29))
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(t1 - u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5e+18) || ~((u <= 1e-29)))
		tmp = (t1 / (u / v)) / (t1 - u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5e+18], N[Not[LessEqual[u, 1e-29]], $MachinePrecision]], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5e18 or 9.99999999999999943e-30 < u

   1. Initial program 78.9%

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

   3. Taylor expanded in t1 around 0 76.7%

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

   4. Taylor expanded in v around 0 76.7%

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

      1. mul-1-neg 76.7%

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

      2. *-commutative 76.7%

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

      3. times-frac 84.1%

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

      4. distribute-rgt-neg-in 84.1%

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

      5. distribute-neg-frac 84.1%

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

   6. Simplified 84.1%

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

      1. *-commutative 84.1%

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

      2. frac-2neg 84.1%

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

      3. remove-double-neg 84.1%

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

      4. associate-*l/ 85.2%

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

      5. add-sqr-sqrt 46.7%

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

      6. sqrt-unprod 65.5%

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

      7. sqr-neg 65.5%

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

      8. sqrt-unprod 29.1%

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

      9. add-sqr-sqrt 59.1%

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

      10. *-commutative 59.1%

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

      11. clear-num 59.9%

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

      12. associate-*l/ 59.9%

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

      13. *-un-lft-identity 59.9%

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

      14. add-sqr-sqrt 29.6%

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

      15. sqrt-unprod 65.4%

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

      16. sqr-neg 65.4%

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

      17. sqrt-unprod 47.1%

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

      18. add-sqr-sqrt 86.0%

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

      19. distribute-neg-in 86.0%

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

      20. add-sqr-sqrt 38.8%

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

      21. sqrt-unprod 84.9%

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

      22. sqr-neg 84.9%

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

      23. sqrt-unprod 47.0%

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

      24. add-sqr-sqrt 86.1%

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

   8. Applied egg-rr 86.1%

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

   if -5e18 < u < 9.99999999999999943e-30

   1. Initial program 61.7%

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

      1. times-frac 97.0%

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

      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 82.2%

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

      1. associate-*r/ 82.2%

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

      2. neg-mul-1 82.2%

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

   7. Simplified 82.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+18} \lor \neg \left(u \leq 10^{-29}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -70000000000000 \lor \neg \left(u \leq 1.32 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -70000000000000.0) (not (<= u 1.32e-52)))
   (/ (* v (/ t1 u)) (- t1 u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -70000000000000.0) || !(u <= 1.32e-52)) {
		tmp = (v * (t1 / u)) / (t1 - 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 <= (-70000000000000.0d0)) .or. (.not. (u <= 1.32d-52))) then
        tmp = (v * (t1 / u)) / (t1 - 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 <= -70000000000000.0) || !(u <= 1.32e-52)) {
		tmp = (v * (t1 / u)) / (t1 - u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -70000000000000.0) or not (u <= 1.32e-52):
		tmp = (v * (t1 / u)) / (t1 - u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -70000000000000.0) || !(u <= 1.32e-52))
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(t1 - u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -70000000000000.0) || ~((u <= 1.32e-52)))
		tmp = (v * (t1 / u)) / (t1 - u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -70000000000000.0], N[Not[LessEqual[u, 1.32e-52]], $MachinePrecision]], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7e13 or 1.32000000000000002e-52 < u

   1. Initial program 79.2%

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

   3. Taylor expanded in t1 around 0 76.3%

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

      1. associate-/l* 77.8%

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

      2. add-sqr-sqrt 35.1%

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

      3. sqrt-unprod 55.4%

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

      4. sqr-neg 55.4%

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

      5. sqrt-unprod 30.6%

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

      6. add-sqr-sqrt 60.5%

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

      7. associate-/r* 60.5%

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

   5. Applied egg-rr 60.5%

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

      1. associate-/l/ 60.5%

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

      2. associate-/l* 59.1%

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

      3. *-commutative 59.1%

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

      4. associate-*r/ 59.5%

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

      5. associate-/r* 58.3%

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

   7. Simplified 58.3%

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

      1. add-sqr-sqrt 57.1%

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

      2. sqrt-unprod 66.7%

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

      3. swap-sqr 54.9%

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

      4. sqr-neg 54.9%

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

      5. associate-/l/ 54.9%

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

      6. distribute-frac-neg 54.9%

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

      7. associate-/l/ 54.9%

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

      8. distribute-frac-neg 54.9%

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

      9. swap-sqr 66.0%

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

      10. sqrt-unprod 65.7%

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

      11. add-sqr-sqrt 75.6%

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

      12. distribute-frac-neg 75.6%

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

      13. distribute-frac-neg2 75.6%

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

      14. associate-*r/ 76.3%

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

   9. Applied egg-rr 76.4%

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

       1. associate-/r* 79.8%

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

       2. *-commutative 79.8%

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

       3. associate-*r/ 83.6%

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

   11. Simplified 83.6%

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

   if -7e13 < u < 1.32000000000000002e-52

   1. Initial program 61.1%

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

      1. times-frac 97.7%

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

      2. distribute-frac-neg 97.7%

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

      3. distribute-neg-frac2 97.7%

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

      4. +-commutative 97.7%

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

      5. distribute-neg-in 97.7%

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

      6. unsub-neg 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t1 around inf 82.7%

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

      1. associate-*r/ 82.7%

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

      2. neg-mul-1 82.7%

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

   7. Simplified 82.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -70000000000000 \lor \neg \left(u \leq 1.32 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{+15} \lor \neg \left(u \leq 4.1 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.9e+15) (not (<= u 4.1e-31)))
   (* (/ t1 u) (/ v (- u)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.9e+15) || !(u <= 4.1e-31)) {
		tmp = (t1 / u) * (v / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.9d+15)) .or. (.not. (u <= 4.1d-31))) then
        tmp = (t1 / u) * (v / -u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.9e+15) || !(u <= 4.1e-31)) {
		tmp = (t1 / u) * (v / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.9e+15) or not (u <= 4.1e-31):
		tmp = (t1 / u) * (v / -u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.9e+15) || !(u <= 4.1e-31))
		tmp = Float64(Float64(t1 / u) * Float64(v / Float64(-u)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.9e+15) || ~((u <= 4.1e-31)))
		tmp = (t1 / u) * (v / -u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.9e+15], N[Not[LessEqual[u, 4.1e-31]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] * N[(v / (-u)), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.9e15 or 4.0999999999999996e-31 < u

   1. Initial program 78.9%

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

   3. Taylor expanded in t1 around 0 76.7%

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

   4. Taylor expanded in v around 0 76.7%

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

      1. mul-1-neg 76.7%

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

      2. *-commutative 76.7%

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

      3. times-frac 84.1%

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

      4. distribute-rgt-neg-in 84.1%

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

      5. distribute-neg-frac 84.1%

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

   6. Simplified 84.1%

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

   7. Taylor expanded in t1 around 0 83.5%

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

      1. associate-*r/ 83.5%

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

      2. mul-1-neg 83.5%

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

   9. Simplified 83.5%

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

   if -4.9e15 < u < 4.0999999999999996e-31

   1. Initial program 61.7%

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

      1. times-frac 97.0%

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

      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 82.2%

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

      1. associate-*r/ 82.2%

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

      2. neg-mul-1 82.2%

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

   7. Simplified 82.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{+15} \lor \neg \left(u \leq 4.1 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-77}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4.8e-6)
   (/ -1.0 (/ (+ t1 u) v))
   (if (<= t1 1.1e-77) (* v (/ t1 (* u (- u)))) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4.8e-6) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.1e-77) {
		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 (t1 <= (-4.8d-6)) then
        tmp = (-1.0d0) / ((t1 + u) / v)
    else if (t1 <= 1.1d-77) 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 (t1 <= -4.8e-6) {
		tmp = -1.0 / ((t1 + u) / v);
	} else if (t1 <= 1.1e-77) {
		tmp = v * (t1 / (u * -u));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4.8e-6:
		tmp = -1.0 / ((t1 + u) / v)
	elif t1 <= 1.1e-77:
		tmp = v * (t1 / (u * -u))
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4.8e-6)
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	elseif (t1 <= 1.1e-77)
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(-u))));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4.8e-6)
		tmp = -1.0 / ((t1 + u) / v);
	elseif (t1 <= 1.1e-77)
		tmp = v * (t1 / (u * -u));
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4.8e-6], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.1e-77], N[(v * N[(t1 / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.7999999999999998e-6

   1. Initial program 57.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 84.7%

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

      1. clear-num 84.7%

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

      2. un-div-inv 84.7%

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

   7. Applied egg-rr 84.7%

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

   if -4.7999999999999998e-6 < t1 < 1.10000000000000003e-77

   1. Initial program 82.8%

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

      1. associate-*l/ 87.1%

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

      2. *-commutative 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in t1 around 0 75.7%

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

   6. Taylor expanded in t1 around 0 78.0%

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

   if 1.10000000000000003e-77 < t1

   1. Initial program 66.0%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 77.7%

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

      1. add-sqr-sqrt 44.8%

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

      2. add-sqr-sqrt 44.6%

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

      3. difference-of-squares 44.6%

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

      4. add-sqr-sqrt 44.6%

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

      5. sqrt-unprod 46.9%

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

      6. sqr-neg 46.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 32.7%

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

      11. sqr-neg 32.7%

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

      12. sqrt-unprod 32.3%

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

      13. add-sqr-sqrt 32.3%

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

   7. Applied egg-rr 32.3%

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

      1. difference-of-squares 32.3%

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

      2. rem-square-sqrt 77.2%

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

      3. rem-square-sqrt 77.7%

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

   9. Simplified 77.7%

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

   10. Taylor expanded in v around 0 77.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-77}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+44} \lor \neg \left(u \leq 8 \cdot 10^{+167}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e+44) (not (<= u 8e+167)))
   (* v (/ (/ t1 u) u))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+44) || !(u <= 8e+167)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.4d+44)) .or. (.not. (u <= 8d+167))) then
        tmp = v * ((t1 / u) / u)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+44) || !(u <= 8e+167)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e+44) or not (u <= 8e+167):
		tmp = v * ((t1 / u) / u)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e+44) || !(u <= 8e+167))
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.4e+44) || ~((u <= 8e+167)))
		tmp = v * ((t1 / u) / u);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e+44], N[Not[LessEqual[u, 8e+167]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.4e44 or 8.0000000000000003e167 < u

   1. Initial program 83.4%

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

   3. Taylor expanded in t1 around 0 83.4%

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

      1. associate-/l* 86.1%

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

      2. add-sqr-sqrt 38.2%

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

      3. sqrt-unprod 69.1%

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

      4. sqr-neg 69.1%

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

      5. sqrt-unprod 39.6%

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

      6. add-sqr-sqrt 74.3%

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

      7. associate-/r* 74.2%

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

   5. Applied egg-rr 74.2%

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

      1. associate-/l/ 74.3%

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

      2. associate-/l* 74.0%

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

      3. *-commutative 74.0%

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

      4. associate-*r/ 74.4%

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

      5. associate-/r* 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in t1 around 0 74.2%

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

   if -3.4e44 < u < 8.0000000000000003e167

   1. Initial program 63.8%

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

      1. times-frac 97.7%

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

      2. distribute-frac-neg 97.7%

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

      3. distribute-neg-frac2 97.7%

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

      4. +-commutative 97.7%

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

      5. distribute-neg-in 97.7%

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

      6. unsub-neg 97.7%

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

   3. Simplified 97.7%

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

      1. frac-2neg 97.7%

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

      2. frac-2neg 97.7%

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

      3. frac-times 63.8%

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

      4. sub-neg 63.8%

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

      5. distribute-neg-in 63.8%

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

      6. +-commutative 63.8%

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

      7. remove-double-neg 63.8%

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

      8. frac-times 97.7%

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

      9. associate-*r/ 99.5%

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

      10. add-sqr-sqrt 47.5%

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

      11. sqrt-unprod 28.1%

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

      12. sqr-neg 28.1%

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

      13. sqrt-unprod 10.5%

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

      14. add-sqr-sqrt 23.2%

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

      15. add-sqr-sqrt 10.3%

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

      16. sqrt-unprod 50.8%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in t1 around inf 72.8%

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

      1. mul-1-neg 72.8%

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

   9. Simplified 72.8%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+44} \lor \neg \left(u \leq 8 \cdot 10^{+167}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+81} \lor \neg \left(u \leq 5.6 \cdot 10^{+137}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e+81) (not (<= u 5.6e+137)))
   (* v (/ (/ t1 u) t1))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+81) || !(u <= 5.6e+137)) {
		tmp = v * ((t1 / u) / t1);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.4d+81)) .or. (.not. (u <= 5.6d+137))) then
        tmp = v * ((t1 / u) / t1)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+81) || !(u <= 5.6e+137)) {
		tmp = v * ((t1 / u) / t1);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e+81) or not (u <= 5.6e+137):
		tmp = v * ((t1 / u) / t1)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e+81) || !(u <= 5.6e+137))
		tmp = Float64(v * Float64(Float64(t1 / u) / t1));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.4e+81) || ~((u <= 5.6e+137)))
		tmp = v * ((t1 / u) / t1);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e+81], N[Not[LessEqual[u, 5.6e+137]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.40000000000000003e81 or 5.60000000000000002e137 < u

   1. Initial program 80.7%

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

   3. Taylor expanded in t1 around 0 80.8%

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

      1. associate-/l* 83.4%

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

      2. add-sqr-sqrt 36.5%

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

      3. sqrt-unprod 65.9%

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

      4. sqr-neg 65.9%

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

      5. sqrt-unprod 37.8%

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

      6. add-sqr-sqrt 72.1%

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

      7. associate-/r* 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. associate-/l/ 72.1%

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

      2. associate-/l* 71.7%

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

      3. *-commutative 71.7%

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

      4. associate-*r/ 72.2%

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

      5. associate-/r* 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in t1 around inf 60.7%

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

   if -3.40000000000000003e81 < u < 5.60000000000000002e137

   1. Initial program 64.7%

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

      1. times-frac 97.1%

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

      2. distribute-frac-neg 97.1%

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

      3. distribute-neg-frac2 97.1%

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

      4. +-commutative 97.1%

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

      5. distribute-neg-in 97.1%

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

      6. unsub-neg 97.1%

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

   3. Simplified 97.1%

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

      1. frac-2neg 97.1%

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

      2. frac-2neg 97.1%

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

      3. frac-times 64.7%

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

      4. sub-neg 64.7%

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

      5. distribute-neg-in 64.7%

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

      6. +-commutative 64.7%

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

      7. remove-double-neg 64.7%

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

      8. frac-times 97.1%

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

      9. associate-*r/ 98.9%

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

      10. add-sqr-sqrt 46.9%

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

      11. sqrt-unprod 28.2%

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

      12. sqr-neg 28.2%

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

      13. sqrt-unprod 10.7%

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

      14. add-sqr-sqrt 23.2%

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

      15. add-sqr-sqrt 11.2%

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

      16. sqrt-unprod 50.8%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in t1 around inf 73.8%

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

      1. mul-1-neg 73.8%

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

   9. Simplified 73.8%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+81} \lor \neg \left(u \leq 5.6 \cdot 10^{+137}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+136} \lor \neg \left(u \leq 5.8 \cdot 10^{+174}\right):\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.7e+136) (not (<= u 5.8e+174)))
   (/ -1.0 (/ u v))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7e+136) || !(u <= 5.8e+174)) {
		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 <= (-1.7d+136)) .or. (.not. (u <= 5.8d+174))) 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 <= -1.7e+136) || !(u <= 5.8e+174)) {
		tmp = -1.0 / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.7e+136) or not (u <= 5.8e+174):
		tmp = -1.0 / (u / v)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.7e+136) || !(u <= 5.8e+174))
		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 <= -1.7e+136) || ~((u <= 5.8e+174)))
		tmp = -1.0 / (u / v);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.7e+136], N[Not[LessEqual[u, 5.8e+174]], $MachinePrecision]], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.69999999999999998e136 or 5.7999999999999999e174 < u

   1. Initial program 85.1%

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

      1. times-frac 99.9%

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

      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 45.6%

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

      1. clear-num 47.8%

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

      2. un-div-inv 47.8%

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

   7. Applied egg-rr 47.8%

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

   8. Taylor expanded in t1 around 0 47.4%

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

   if -1.69999999999999998e136 < u < 5.7999999999999999e174

   1. Initial program 65.0%

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

      1. times-frac 97.2%

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

      2. distribute-frac-neg 97.2%

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

      3. distribute-neg-frac2 97.2%

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

      4. +-commutative 97.2%

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

      5. distribute-neg-in 97.2%

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

      6. unsub-neg 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in t1 around inf 67.0%

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

      1. associate-*r/ 67.0%

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

      2. neg-mul-1 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+136} \lor \neg \left(u \leq 5.8 \cdot 10^{+174}\right):\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.5e+132)
   (/ -1.0 (/ u v))
   (if (<= u 2.7e+168) (/ v (- t1)) (/ 1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.5e+132) {
		tmp = -1.0 / (u / v);
	} else if (u <= 2.7e+168) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-6.5d+132)) then
        tmp = (-1.0d0) / (u / v)
    else if (u <= 2.7d+168) then
        tmp = v / -t1
    else
        tmp = 1.0d0 / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.5e+132) {
		tmp = -1.0 / (u / v);
	} else if (u <= 2.7e+168) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.5e+132:
		tmp = -1.0 / (u / v)
	elif u <= 2.7e+168:
		tmp = v / -t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.5e+132)
		tmp = Float64(-1.0 / Float64(u / v));
	elseif (u <= 2.7e+168)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.5e+132)
		tmp = -1.0 / (u / v);
	elseif (u <= 2.7e+168)
		tmp = v / -t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.5e+132], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.7e+168], N[(v / (-t1)), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -6.4999999999999994e132

   1. Initial program 87.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 45.7%

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

      1. clear-num 47.2%

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

      2. un-div-inv 47.2%

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

   7. Applied egg-rr 47.2%

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

   8. Taylor expanded in t1 around 0 47.2%

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

   if -6.4999999999999994e132 < u < 2.70000000000000016e168

   1. Initial program 65.3%

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

      1. times-frac 97.2%

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

      2. distribute-frac-neg 97.2%

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

      3. distribute-neg-frac2 97.2%

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

      4. +-commutative 97.2%

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

      5. distribute-neg-in 97.2%

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

      6. unsub-neg 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in t1 around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. neg-mul-1 67.3%

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

   7. Simplified 67.3%

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

   if 2.70000000000000016e168 < u

   1. Initial program 76.8%

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

      1. 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 43.4%

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

      1. associate-*r/ 43.4%

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

      2. neg-mul-1 43.4%

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

      3. clear-num 47.0%

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

      4. add-sqr-sqrt 27.3%

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

      5. sqrt-unprod 45.7%

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

      6. sqr-neg 45.7%

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

      7. sqrt-unprod 20.0%

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

      8. add-sqr-sqrt 46.1%

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

   7. Applied egg-rr 46.1%

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

   8. Taylor expanded in t1 around 0 46.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+134} \lor \neg \left(u \leq 9.6 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.3e+134) (not (<= u 9.6e+172))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e+134) || !(u <= 9.6e+172)) {
		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 <= (-2.3d+134)) .or. (.not. (u <= 9.6d+172))) 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 <= -2.3e+134) || !(u <= 9.6e+172)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.3e+134) or not (u <= 9.6e+172):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.3e+134) || !(u <= 9.6e+172))
		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 <= -2.3e+134) || ~((u <= 9.6e+172)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.3e+134], N[Not[LessEqual[u, 9.6e+172]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.2999999999999998e134 or 9.6000000000000002e172 < u

   1. Initial program 83.9%

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

      1. times-frac 99.9%

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

      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 59.9%

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

      1. add-sqr-sqrt 40.5%

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

      2. add-sqr-sqrt 23.1%

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

      3. difference-of-squares 23.1%

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

      4. add-sqr-sqrt 23.1%

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

      5. sqrt-unprod 26.1%

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

      6. sqr-neg 26.1%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 12.2%

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

      11. sqr-neg 12.2%

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

      12. sqrt-unprod 10.8%

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

      13. add-sqr-sqrt 10.8%

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

   7. Applied egg-rr 10.8%

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

      1. difference-of-squares 10.8%

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

      2. rem-square-sqrt 34.1%

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

      3. rem-square-sqrt 59.9%

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

   9. Simplified 59.9%

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

   10. Taylor expanded in t1 around 0 44.8%

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

   if -2.2999999999999998e134 < u < 9.6000000000000002e172

   1. Initial program 65.3%

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

      1. times-frac 97.2%

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

      2. distribute-frac-neg 97.2%

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

      3. distribute-neg-frac2 97.2%

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

      4. +-commutative 97.2%

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

      5. distribute-neg-in 97.2%

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

      6. unsub-neg 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in t1 around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. neg-mul-1 67.3%

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

   7. Simplified 67.3%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+134} \lor \neg \left(u \leq 9.6 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 9.4 \cdot 10^{+167}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.45e+136) (/ v (- u)) (if (<= u 9.4e+167) (/ v (- t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.45e+136) {
		tmp = v / -u;
	} else if (u <= 9.4e+167) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.45d+136)) then
        tmp = v / -u
    else if (u <= 9.4d+167) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.45e+136) {
		tmp = v / -u;
	} else if (u <= 9.4e+167) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.45e+136:
		tmp = v / -u
	elif u <= 9.4e+167:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.45e+136)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 9.4e+167)
		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 <= -1.45e+136)
		tmp = v / -u;
	elseif (u <= 9.4e+167)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.45e+136], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 9.4e+167], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.44999999999999987e136

   1. Initial program 87.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 45.7%

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

   6. Taylor expanded in t1 around 0 45.7%

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

      1. associate-*r/ 45.7%

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

      2. mul-1-neg 45.7%

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

   8. Simplified 45.7%

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

   if -1.44999999999999987e136 < u < 9.40000000000000026e167

   1. Initial program 65.3%

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

      1. times-frac 97.2%

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

      2. distribute-frac-neg 97.2%

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

      3. distribute-neg-frac2 97.2%

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

      4. +-commutative 97.2%

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

      5. distribute-neg-in 97.2%

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

      6. unsub-neg 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in t1 around inf 67.3%

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

      1. associate-*r/ 67.3%

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

      2. neg-mul-1 67.3%

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

   7. Simplified 67.3%

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

   if 9.40000000000000026e167 < u

   1. Initial program 76.8%

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

      1. 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 61.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. difference-of-squares 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 38.4%

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

      11. sqr-neg 38.4%

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

      12. sqrt-unprod 33.9%

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

      13. add-sqr-sqrt 33.9%

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

   7. Applied egg-rr 33.9%

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

      1. difference-of-squares 33.9%

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

      2. rem-square-sqrt 33.9%

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

      3. rem-square-sqrt 61.3%

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

   9. Simplified 61.3%

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

   10. Taylor expanded in t1 around 0 43.2%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 9.4 \cdot 10^{+167}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 23.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.4 \cdot 10^{+111} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.4e+111) (not (<= t1 1.3e+169))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.4e+111) || !(t1 <= 1.3e+169)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-4.4d+111)) .or. (.not. (t1 <= 1.3d+169))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.4e+111) || !(t1 <= 1.3e+169)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.4e+111) or not (t1 <= 1.3e+169):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.4e+111) || !(t1 <= 1.3e+169))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -4.4e+111) || ~((t1 <= 1.3e+169)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.4e+111], N[Not[LessEqual[t1, 1.3e+169]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.39999999999999997e111 or 1.3e169 < t1

   1. Initial program 44.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 91.8%

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

      1. associate-*r/ 91.8%

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

      2. neg-mul-1 91.8%

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

   7. Simplified 91.8%

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

      1. distribute-frac-neg 91.8%

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

      2. div-inv 91.6%

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

      3. distribute-rgt-neg-in 91.6%

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

      4. frac-2neg 91.6%

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

      5. metadata-eval 91.6%

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

      6. add-sqr-sqrt 52.5%

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

      7. sqrt-unprod 45.1%

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

      8. sqr-neg 45.1%

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

      9. sqrt-unprod 16.5%

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

      10. add-sqr-sqrt 36.4%

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

   9. Applied egg-rr 36.4%

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

       1. distribute-rgt-neg-out 36.4%

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

       2. *-commutative 36.4%

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

       3. associate-*l/ 36.4%

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

       4. mul-1-neg 36.4%

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

       5. distribute-neg-frac 36.4%

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

       6. remove-double-neg 36.4%

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

   11. Simplified 36.4%

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

   if -4.39999999999999997e111 < t1 < 1.3e169

   1. Initial program 80.9%

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

      1. 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 55.4%

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

      1. add-sqr-sqrt 32.1%

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

      2. add-sqr-sqrt 19.8%

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

      3. difference-of-squares 19.8%

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

      4. add-sqr-sqrt 19.8%

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

      5. sqrt-unprod 20.9%

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

      6. sqr-neg 20.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 13.6%

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

      11. sqr-neg 13.6%

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

      12. sqrt-unprod 13.0%

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

      13. add-sqr-sqrt 13.0%

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

   7. Applied egg-rr 13.0%

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

      1. difference-of-squares 13.0%

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

      2. rem-square-sqrt 33.6%

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

      3. rem-square-sqrt 55.3%

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

   9. Simplified 55.3%

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

   10. Taylor expanded in t1 around 0 19.3%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.4 \cdot 10^{+111} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. times-frac 97.9%

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

   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. Final simplification 97.9%

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

Alternative 15: 61.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. times-frac 97.9%

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

   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. Step-by-step derivation

   1. frac-2neg 97.9%

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

   2. frac-2neg 97.9%

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

   3. frac-times 70.1%

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

   4. sub-neg 70.1%

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

   5. distribute-neg-in 70.1%

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

   6. +-commutative 70.1%

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

   7. remove-double-neg 70.1%

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

   8. frac-times 97.9%

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

   9. associate-*r/ 99.1%

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

   10. add-sqr-sqrt 46.7%

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

   11. sqrt-unprod 42.4%

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

   12. sqr-neg 42.4%

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

   13. sqrt-unprod 19.8%

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

   14. add-sqr-sqrt 39.5%

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

   15. add-sqr-sqrt 25.2%

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

   16. sqrt-unprod 58.3%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in t1 around inf 62.2%

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

   1. mul-1-neg 62.2%

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

9. Simplified 62.2%

   \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]
10. Add Preprocessing

Alternative 16: 61.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. times-frac 97.9%

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

   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 66.8%

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

   1. add-sqr-sqrt 34.4%

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

   2. add-sqr-sqrt 20.1%

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

   3. difference-of-squares 20.1%

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

   4. add-sqr-sqrt 20.1%

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

   5. sqrt-unprod 20.9%

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

   6. sqr-neg 20.9%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 0.0%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 14.6%

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

   11. sqr-neg 14.6%

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

   12. sqrt-unprod 14.5%

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

   13. add-sqr-sqrt 14.5%

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

7. Applied egg-rr 14.5%

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

   1. difference-of-squares 14.5%

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

   2. rem-square-sqrt 35.2%

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

   3. rem-square-sqrt 66.7%

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

9. Simplified 66.7%

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

10. Taylor expanded in v around 0 62.1%

    \[\leadsto \color{blue}{\frac{v}{u - t1}} \]
11. Add Preprocessing

Alternative 17: 14.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

   1. times-frac 97.9%

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

   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 54.1%

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

   1. associate-*r/ 54.1%

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

   2. neg-mul-1 54.1%

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

7. Simplified 54.1%

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

   1. distribute-frac-neg 54.1%

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

   2. div-inv 54.0%

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

   3. distribute-rgt-neg-in 54.0%

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

   4. frac-2neg 54.0%

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

   5. metadata-eval 54.0%

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

   6. add-sqr-sqrt 26.0%

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

   7. sqrt-unprod 25.2%

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

   8. sqr-neg 25.2%

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

   9. sqrt-unprod 7.1%

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

   10. add-sqr-sqrt 15.1%

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

9. Applied egg-rr 15.1%

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

    1. distribute-rgt-neg-out 15.1%

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

    2. *-commutative 15.1%

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

    3. associate-*l/ 15.1%

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

    4. mul-1-neg 15.1%

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

    5. distribute-neg-frac 15.1%

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

    6. remove-double-neg 15.1%

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

11. Simplified 15.1%

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

Reproduce

herbie shell --seed 2024143 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))