Rosa's DopplerBench

Percentage Accurate: 72.0% → 97.8%

Time: 11.8s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (v / (u + t1)) / (-1.0 - (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)) / ((-1.0d0) - (u / t1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. +-commutative 67.9%

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

   2. +-commutative 67.9%

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

   3. *-commutative 67.9%

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

   4. associate-*l/ 69.9%

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

   5. distribute-rgt-neg-in 69.9%

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

   6. distribute-lft-neg-in 69.9%

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

   7. distribute-frac-neg 69.9%

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

   8. /-rgt-identity 69.9%

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

   9. metadata-eval 69.9%

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

   10. associate-/r/ 72.9%

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

   11. associate-/r* 72.9%

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

   12. times-frac 94.9%

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

   13. metadata-eval 94.9%

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

   14. /-rgt-identity 94.9%

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

   15. +-commutative 94.9%

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

   16. remove-double-neg 94.9%

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

   17. unsub-neg 94.9%

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

   18. div-sub 94.9%

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

   19. sub-neg 94.9%

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

   20. distribute-frac-neg 94.9%

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

   21. remove-double-neg 94.9%

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

   22. *-inverses 94.9%

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

3. Simplified 94.9%

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

4. Taylor expanded in v around 0 94.9%

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

   1. +-commutative 94.9%

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

   2. *-commutative 94.9%

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

   3. mul-1-neg 94.9%

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

   4. distribute-neg-frac 94.9%

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

   5. associate-/r* 98.1%

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

6. Simplified 98.1%

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

   1. frac-2neg 98.1%

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

   2. distribute-frac-neg 98.1%

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

   3. remove-double-neg 98.1%

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

   4. div-inv 98.1%

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

   5. +-commutative 98.1%

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

   6. distribute-neg-in 98.1%

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

   7. metadata-eval 98.1%

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

8. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.1%

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

   2. *-rgt-identity 98.1%

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

   3. +-commutative 98.1%

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

   4. unsub-neg 98.1%

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

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

Alternative 2: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+45} \lor \neg \left(u \leq 4.6 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u + t1}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5e+45) (not (<= u 4.6e+67)))
   (/ (* t1 (/ (- v) (+ u t1))) u)
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e+45) || !(u <= 4.6e+67)) {
		tmp = (t1 * (-v / (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 <= (-5d+45)) .or. (.not. (u <= 4.6d+67))) then
        tmp = (t1 * (-v / (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 <= -5e+45) || !(u <= 4.6e+67)) {
		tmp = (t1 * (-v / (u + t1))) / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5e+45) or not (u <= 4.6e+67):
		tmp = (t1 * (-v / (u + t1))) / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5e+45) || !(u <= 4.6e+67))
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / Float64(u + t1))) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5e+45) || ~((u <= 4.6e+67)))
		tmp = (t1 * (-v / (u + t1))) / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5e+45], N[Not[LessEqual[u, 4.6e+67]], $MachinePrecision]], N[(N[(t1 * N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5e45 or 4.5999999999999997e67 < u

   1. Initial program 74.5%

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

      1. +-commutative 74.5%

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

      2. +-commutative 74.5%

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

      3. times-frac 97.2%

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

      4. +-commutative 97.2%

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

      5. +-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around 0 85.9%

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

      1. mul-1-neg 85.9%

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

   6. Simplified 85.9%

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

      1. distribute-neg-frac 85.9%

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

      2. associate-*l/ 88.4%

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

   8. Applied egg-rr 88.4%

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

   if -5e45 < u < 4.5999999999999997e67

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

      2. +-commutative 63.3%

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

      3. *-commutative 63.3%

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

      4. associate-*l/ 65.8%

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

      5. distribute-rgt-neg-in 65.8%

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

      6. distribute-lft-neg-in 65.8%

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

      7. distribute-frac-neg 65.8%

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

      8. /-rgt-identity 65.8%

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

      9. metadata-eval 65.8%

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

      10. associate-/r/ 72.8%

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

      11. associate-/r* 72.8%

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

      12. times-frac 99.9%

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

      13. metadata-eval 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. +-commutative 99.9%

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

      16. remove-double-neg 99.9%

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

      17. unsub-neg 99.9%

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

      18. div-sub 99.9%

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

      19. sub-neg 99.9%

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

      20. distribute-frac-neg 99.9%

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

      21. remove-double-neg 99.9%

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

      22. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 80.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+45} \lor \neg \left(u \leq 4.6 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u + t1}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.25e+119)
   (/ (* t1 (/ (- v) u)) u)
   (if (<= u 4.8e+67) (/ (- v) t1) (* (/ t1 u) (/ (- v) (+ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.25e+119) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= 4.8e+67) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / u) * (-v / (u + t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.25d+119)) then
        tmp = (t1 * (-v / u)) / u
    else if (u <= 4.8d+67) then
        tmp = -v / t1
    else
        tmp = (t1 / u) * (-v / (u + t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.25e+119) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= 4.8e+67) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / u) * (-v / (u + t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.25e+119:
		tmp = (t1 * (-v / u)) / u
	elif u <= 4.8e+67:
		tmp = -v / t1
	else:
		tmp = (t1 / u) * (-v / (u + t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.25e+119)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	elseif (u <= 4.8e+67)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / Float64(u + t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.25e+119)
		tmp = (t1 * (-v / u)) / u;
	elseif (u <= 4.8e+67)
		tmp = -v / t1;
	else
		tmp = (t1 / u) * (-v / (u + t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.25e+119], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 4.8e+67], N[((-v) / t1), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.25e119

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. +-commutative 72.0%

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

      3. times-frac 95.0%

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

      4. +-commutative 95.0%

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

      5. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in t1 around 0 92.0%

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

      1. mul-1-neg 92.0%

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

   6. Simplified 92.0%

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

      1. distribute-neg-frac 92.0%

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

      2. associate-*l/ 96.8%

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

   8. Applied egg-rr 96.8%

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

   9. Taylor expanded in t1 around 0 97.0%

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

   if -1.25e119 < u < 4.80000000000000004e67

   1. Initial program 63.6%

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

      1. +-commutative 63.6%

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

      2. +-commutative 63.6%

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

      3. *-commutative 63.6%

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

      4. associate-*l/ 65.4%

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

      5. distribute-rgt-neg-in 65.4%

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

      6. distribute-lft-neg-in 65.4%

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

      7. distribute-frac-neg 65.4%

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

      8. /-rgt-identity 65.4%

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

      9. metadata-eval 65.4%

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

      10. associate-/r/ 72.4%

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

      11. associate-/r* 72.4%

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

      12. times-frac 99.9%

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

      13. metadata-eval 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. +-commutative 99.9%

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

      16. remove-double-neg 99.9%

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

      17. unsub-neg 99.9%

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

      18. div-sub 99.9%

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

      19. sub-neg 99.9%

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

      20. distribute-frac-neg 99.9%

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

      21. remove-double-neg 99.9%

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

      22. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 78.7%

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

   if 4.80000000000000004e67 < u

   1. Initial program 77.7%

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

      1. +-commutative 77.7%

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

      2. +-commutative 77.7%

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

      3. times-frac 98.1%

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

      4. +-commutative 98.1%

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

      5. +-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

   6. Simplified 86.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u + t1}\\ \end{array} \]

Alternative 4: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{elif}\;u \leq 4.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.1e+47)
   (/ t1 (* (/ u v) (- (- t1) u)))
   (if (<= u 4.6e+67) (/ (- v) t1) (* (/ t1 u) (/ (- v) (+ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+47) {
		tmp = t1 / ((u / v) * (-t1 - u));
	} else if (u <= 4.6e+67) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / u) * (-v / (u + t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.1d+47)) then
        tmp = t1 / ((u / v) * (-t1 - u))
    else if (u <= 4.6d+67) then
        tmp = -v / t1
    else
        tmp = (t1 / u) * (-v / (u + t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+47) {
		tmp = t1 / ((u / v) * (-t1 - u));
	} else if (u <= 4.6e+67) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / u) * (-v / (u + t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.1e+47:
		tmp = t1 / ((u / v) * (-t1 - u))
	elif u <= 4.6e+67:
		tmp = -v / t1
	else:
		tmp = (t1 / u) * (-v / (u + t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.1e+47)
		tmp = Float64(t1 / Float64(Float64(u / v) * Float64(Float64(-t1) - u)));
	elseif (u <= 4.6e+67)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / Float64(u + t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.1e+47)
		tmp = t1 / ((u / v) * (-t1 - u));
	elseif (u <= 4.6e+67)
		tmp = -v / t1;
	else
		tmp = (t1 / u) * (-v / (u + t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.1e+47], N[(t1 / N[(N[(u / v), $MachinePrecision] * N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.6e+67], N[((-v) / t1), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.1e47

   1. Initial program 70.9%

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

      1. +-commutative 70.9%

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

      2. +-commutative 70.9%

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

      3. times-frac 96.2%

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

      4. distribute-frac-neg 96.2%

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

      5. distribute-lft-neg-in 96.2%

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

      6. distribute-rgt-neg-in 96.2%

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

      7. distribute-frac-neg 96.2%

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

      8. associate-/r/ 92.1%

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

      9. associate-/r/ 92.0%

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

      10. *-commutative 92.0%

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

      11. +-commutative 92.0%

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

      12. +-commutative 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in t1 around 0 87.2%

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

      1. mul-1-neg 87.2%

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

   6. Simplified 87.2%

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

   if -2.1e47 < u < 4.5999999999999997e67

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

      2. +-commutative 63.3%

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

      3. *-commutative 63.3%

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

      4. associate-*l/ 65.8%

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

      5. distribute-rgt-neg-in 65.8%

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

      6. distribute-lft-neg-in 65.8%

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

      7. distribute-frac-neg 65.8%

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

      8. /-rgt-identity 65.8%

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

      9. metadata-eval 65.8%

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

      10. associate-/r/ 72.8%

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

      11. associate-/r* 72.8%

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

      12. times-frac 99.9%

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

      13. metadata-eval 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. +-commutative 99.9%

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

      16. remove-double-neg 99.9%

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

      17. unsub-neg 99.9%

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

      18. div-sub 99.9%

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

      19. sub-neg 99.9%

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

      20. distribute-frac-neg 99.9%

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

      21. remove-double-neg 99.9%

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

      22. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 80.8%

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

   if 4.5999999999999997e67 < u

   1. Initial program 77.7%

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

      1. +-commutative 77.7%

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

      2. +-commutative 77.7%

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

      3. times-frac 98.1%

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

      4. +-commutative 98.1%

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

      5. +-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 86.5%

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

      1. mul-1-neg 86.5%

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

   6. Simplified 86.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{t1}{\frac{u}{v} \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{elif}\;u \leq 4.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u + t1}\\ \end{array} \]

Alternative 5: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+128}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(u + t1\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1e+128)
   (/ (* t1 (/ (- v) u)) u)
   (/ v (* (+ u t1) (- -1.0 (/ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e+128) {
		tmp = (t1 * (-v / u)) / u;
	} else {
		tmp = v / ((u + t1) * (-1.0 - (u / t1)));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1d+128)) then
        tmp = (t1 * (-v / u)) / u
    else
        tmp = v / ((u + t1) * ((-1.0d0) - (u / t1)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e+128) {
		tmp = (t1 * (-v / u)) / u;
	} else {
		tmp = v / ((u + t1) * (-1.0 - (u / t1)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1e+128:
		tmp = (t1 * (-v / u)) / u
	else:
		tmp = v / ((u + t1) * (-1.0 - (u / t1)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1e+128)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	else
		tmp = Float64(v / Float64(Float64(u + t1) * Float64(-1.0 - Float64(u / t1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1e+128)
		tmp = (t1 * (-v / u)) / u;
	else
		tmp = v / ((u + t1) * (-1.0 - (u / t1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1e+128], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[(N[(u + t1), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.0000000000000001e128

   1. Initial program 71.2%

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

      1. +-commutative 71.2%

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

      2. +-commutative 71.2%

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

      3. times-frac 94.9%

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

      4. +-commutative 94.9%

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

      5. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in t1 around 0 91.8%

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

      1. mul-1-neg 91.8%

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

   6. Simplified 91.8%

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

      1. distribute-neg-frac 91.8%

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

      2. associate-*l/ 96.7%

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

   8. Applied egg-rr 96.7%

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

   9. Taylor expanded in t1 around 0 96.9%

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

   if -1.0000000000000001e128 < u

   1. Initial program 67.3%

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

      1. +-commutative 67.3%

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

      2. +-commutative 67.3%

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

      3. *-commutative 67.3%

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

      4. associate-*l/ 69.1%

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

      5. distribute-rgt-neg-in 69.1%

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

      6. distribute-lft-neg-in 69.1%

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

      7. distribute-frac-neg 69.1%

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

      8. /-rgt-identity 69.1%

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

      9. metadata-eval 69.1%

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

      10. associate-/r/ 73.0%

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

      11. associate-/r* 73.0%

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

      12. times-frac 96.4%

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

      13. metadata-eval 96.4%

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

      14. /-rgt-identity 96.4%

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

      15. +-commutative 96.4%

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

      16. remove-double-neg 96.4%

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

      17. unsub-neg 96.4%

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

      18. div-sub 96.5%

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

      19. sub-neg 96.5%

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

      20. distribute-frac-neg 96.5%

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

      21. remove-double-neg 96.5%

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

      22. *-inverses 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in v around 0 96.5%

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

      1. +-commutative 96.5%

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

      2. *-commutative 96.5%

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

      3. mul-1-neg 96.5%

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

      4. distribute-neg-frac 96.5%

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

      5. associate-/r* 98.6%

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

   6. Simplified 98.6%

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

      1. frac-2neg 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. remove-double-neg 98.6%

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

      4. div-inv 98.6%

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

      5. +-commutative 98.6%

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

      6. distribute-neg-in 98.6%

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

      7. metadata-eval 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.6%

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

      2. *-rgt-identity 98.6%

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

      3. associate-/l/ 96.5%

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

      4. unsub-neg 96.5%

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

      5. +-commutative 96.5%

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

   10. Simplified 96.5%

       \[\leadsto \color{blue}{\frac{v}{\left(-1 - \frac{u}{t1}\right) \cdot \left(u + t1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+128}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(u + t1\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array} \]

Alternative 6: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118} \lor \neg \left(u \leq 6 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.8e+118) (not (<= u 6e+67)))
   (* (/ v u) (/ (- t1) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.8e+118) || !(u <= 6e+67)) {
		tmp = (v / 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 <= (-3.8d+118)) .or. (.not. (u <= 6d+67))) then
        tmp = (v / 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 <= -3.8e+118) || !(u <= 6e+67)) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.8e+118) or not (u <= 6e+67):
		tmp = (v / u) * (-t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.8e+118) || !(u <= 6e+67))
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.8e+118) || ~((u <= 6e+67)))
		tmp = (v / u) * (-t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.8e+118], N[Not[LessEqual[u, 6e+67]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.80000000000000016e118 or 6.0000000000000002e67 < u

   1. Initial program 75.4%

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

      1. +-commutative 75.4%

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

      2. +-commutative 75.4%

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

      3. times-frac 96.8%

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

      4. +-commutative 96.8%

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

      5. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in t1 around 0 88.7%

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

      1. mul-1-neg 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in t1 around 0 87.8%

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

   if -3.80000000000000016e118 < u < 6.0000000000000002e67

   1. Initial program 63.6%

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

      1. +-commutative 63.6%

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

      2. +-commutative 63.6%

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

      3. *-commutative 63.6%

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

      4. associate-*l/ 65.4%

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

      5. distribute-rgt-neg-in 65.4%

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

      6. distribute-lft-neg-in 65.4%

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

      7. distribute-frac-neg 65.4%

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

      8. /-rgt-identity 65.4%

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

      9. metadata-eval 65.4%

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

      10. associate-/r/ 72.4%

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

      11. associate-/r* 72.4%

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

      12. times-frac 99.9%

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

      13. metadata-eval 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. +-commutative 99.9%

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

      16. remove-double-neg 99.9%

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

      17. unsub-neg 99.9%

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

      18. div-sub 99.9%

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

      19. sub-neg 99.9%

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

      20. distribute-frac-neg 99.9%

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

      21. remove-double-neg 99.9%

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

      22. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 78.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118} \lor \neg \left(u \leq 6 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 7: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118} \lor \neg \left(u \leq 6 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.8e+118) (not (<= u 6e+67)))
   (/ (* t1 (/ (- v) u)) u)
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.8e+118) || !(u <= 6e+67)) {
		tmp = (t1 * (-v / u)) / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.8d+118)) .or. (.not. (u <= 6d+67))) then
        tmp = (t1 * (-v / u)) / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.8e+118) || !(u <= 6e+67)) {
		tmp = (t1 * (-v / u)) / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.8e+118) or not (u <= 6e+67):
		tmp = (t1 * (-v / u)) / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.8e+118) || !(u <= 6e+67))
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.8e+118) || ~((u <= 6e+67)))
		tmp = (t1 * (-v / u)) / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.8e+118], N[Not[LessEqual[u, 6e+67]], $MachinePrecision]], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.80000000000000016e118 or 6.0000000000000002e67 < u

   1. Initial program 75.4%

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

      1. +-commutative 75.4%

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

      2. +-commutative 75.4%

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

      3. times-frac 96.8%

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

      4. +-commutative 96.8%

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

      5. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in t1 around 0 88.7%

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

      1. mul-1-neg 88.7%

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

   6. Simplified 88.7%

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

      1. distribute-neg-frac 88.7%

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

      2. associate-*l/ 91.6%

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

   8. Applied egg-rr 91.6%

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

   9. Taylor expanded in t1 around 0 90.6%

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

   if -3.80000000000000016e118 < u < 6.0000000000000002e67

   1. Initial program 63.6%

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

      1. +-commutative 63.6%

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

      2. +-commutative 63.6%

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

      3. *-commutative 63.6%

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

      4. associate-*l/ 65.4%

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

      5. distribute-rgt-neg-in 65.4%

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

      6. distribute-lft-neg-in 65.4%

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

      7. distribute-frac-neg 65.4%

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

      8. /-rgt-identity 65.4%

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

      9. metadata-eval 65.4%

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

      10. associate-/r/ 72.4%

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

      11. associate-/r* 72.4%

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

      12. times-frac 99.9%

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

      13. metadata-eval 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. +-commutative 99.9%

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

      16. remove-double-neg 99.9%

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

      17. unsub-neg 99.9%

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

      18. div-sub 99.9%

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

      19. sub-neg 99.9%

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

      20. distribute-frac-neg 99.9%

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

      21. remove-double-neg 99.9%

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

      22. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 78.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+118} \lor \neg \left(u \leq 6 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.9 \cdot 10^{+126} \lor \neg \left(u \leq 3.8 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.9e+126) (not (<= u 3.8e+74)))
   (* (/ v u) (/ t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.9e+126) || !(u <= 3.8e+74)) {
		tmp = (v / 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 <= (-6.9d+126)) .or. (.not. (u <= 3.8d+74))) then
        tmp = (v / 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 <= -6.9e+126) || !(u <= 3.8e+74)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.9e+126) or not (u <= 3.8e+74):
		tmp = (v / u) * (t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.9e+126) || !(u <= 3.8e+74))
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.9e+126) || ~((u <= 3.8e+74)))
		tmp = (v / u) * (t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.9e+126], N[Not[LessEqual[u, 3.8e+74]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.8999999999999996e126 or 3.7999999999999998e74 < u

   1. Initial program 75.1%

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

      1. +-commutative 75.1%

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

      2. +-commutative 75.1%

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

      3. times-frac 96.7%

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

      4. +-commutative 96.7%

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

      5. +-commutative 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in t1 around 0 89.3%

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

      1. mul-1-neg 89.3%

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

   6. Simplified 89.3%

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

      1. distribute-neg-frac 89.3%

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

      2. associate-*l/ 92.3%

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

   8. Applied egg-rr 92.3%

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

   9. Taylor expanded in t1 around 0 83.8%

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

       1. *-commutative 83.8%

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

       2. associate-*r/ 83.8%

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

       3. associate-*r* 83.8%

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

       4. mul-1-neg 83.8%

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

   11. Simplified 83.8%

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

       1. associate-/l/ 72.9%

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

       2. times-frac 88.3%

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

       3. add-sqr-sqrt 42.9%

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

       4. sqrt-unprod 65.8%

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

       5. sqr-neg 65.8%

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

       6. sqrt-unprod 35.8%

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

       7. add-sqr-sqrt 65.4%

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

   13. Applied egg-rr 65.4%

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

   if -6.8999999999999996e126 < u < 3.7999999999999998e74

   1. Initial program 64.1%

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

      1. +-commutative 64.1%

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

      2. +-commutative 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l/ 65.8%

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

      5. distribute-rgt-neg-in 65.8%

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

      6. distribute-lft-neg-in 65.8%

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

      7. distribute-frac-neg 65.8%

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

      8. /-rgt-identity 65.8%

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

      9. metadata-eval 65.8%

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

      10. associate-/r/ 72.6%

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

      11. associate-/r* 72.6%

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

      12. times-frac 99.9%

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

      13. metadata-eval 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. +-commutative 99.9%

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

      16. remove-double-neg 99.9%

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

      17. unsub-neg 99.9%

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

      18. div-sub 99.9%

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

      19. sub-neg 99.9%

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

      20. distribute-frac-neg 99.9%

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

      21. remove-double-neg 99.9%

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

      22. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 77.5%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.9 \cdot 10^{+126} \lor \neg \left(u \leq 3.8 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9: 58.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{+131} \lor \neg \left(u \leq 1.65 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -8.2e+131) (not (<= u 1.65e+136))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.2e+131) || !(u <= 1.65e+136)) {
		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 <= (-8.2d+131)) .or. (.not. (u <= 1.65d+136))) 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 <= -8.2e+131) || !(u <= 1.65e+136)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -8.2e+131) or not (u <= 1.65e+136):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -8.2e+131) || !(u <= 1.65e+136))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -8.2e+131) || ~((u <= 1.65e+136)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -8.2e+131], N[Not[LessEqual[u, 1.65e+136]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -8.20000000000000015e131 or 1.64999999999999996e136 < u

   1. Initial program 74.0%

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

      1. +-commutative 74.0%

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

      2. +-commutative 74.0%

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

      3. times-frac 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in t1 around 0 94.8%

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

      1. mul-1-neg 94.8%

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

   6. Simplified 94.8%

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

      1. distribute-neg-frac 94.8%

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

      2. associate-*l/ 96.0%

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

   8. Applied egg-rr 96.0%

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

      1. frac-2neg 96.0%

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

      2. associate-*r/ 87.9%

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

      3. add-sqr-sqrt 45.7%

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

      4. sqrt-unprod 72.3%

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

      5. sqr-neg 72.3%

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

      6. sqrt-unprod 39.4%

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

      7. add-sqr-sqrt 72.5%

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

      8. *-commutative 72.5%

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

      9. add-sqr-sqrt 48.8%

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

      10. sqrt-unprod 76.3%

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

      11. sqr-neg 76.3%

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

      12. sqrt-unprod 31.9%

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

      13. add-sqr-sqrt 87.9%

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

      14. distribute-neg-in 87.9%

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

      15. add-sqr-sqrt 55.9%

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

      16. sqrt-unprod 87.2%

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

      17. sqr-neg 87.2%

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

      18. sqrt-unprod 32.0%

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

      19. add-sqr-sqrt 87.9%

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

      20. sub-neg 87.9%

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

   10. Applied egg-rr 87.9%

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

       1. associate-/l* 94.8%

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

   12. Simplified 94.8%

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

   13. Taylor expanded in t1 around inf 39.7%

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

   if -8.20000000000000015e131 < u < 1.64999999999999996e136

   1. Initial program 65.6%

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

      1. +-commutative 65.6%

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

      2. +-commutative 65.6%

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

      3. *-commutative 65.6%

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

      4. associate-*l/ 67.7%

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

      5. distribute-rgt-neg-in 67.7%

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

      6. distribute-lft-neg-in 67.7%

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

      7. distribute-frac-neg 67.7%

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

      8. /-rgt-identity 67.7%

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

      9. metadata-eval 67.7%

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

      10. associate-/r/ 71.8%

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

      11. associate-/r* 71.8%

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

      12. times-frac 97.9%

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

      13. metadata-eval 97.9%

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

      14. /-rgt-identity 97.9%

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

      15. +-commutative 97.9%

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

      16. remove-double-neg 97.9%

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

      17. unsub-neg 97.9%

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

      18. div-sub 97.9%

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

      19. sub-neg 97.9%

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

      20. distribute-frac-neg 97.9%

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

      21. remove-double-neg 97.9%

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

      22. *-inverses 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in t1 around inf 73.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{+131} \lor \neg \left(u \leq 1.65 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10: 23.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{+163} \lor \neg \left(t1 \leq 1.5 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -6e+163) (not (<= t1 1.5e+119))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6e+163) || !(t1 <= 1.5e+119)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-6d+163)) .or. (.not. (t1 <= 1.5d+119))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6e+163) || !(t1 <= 1.5e+119)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -6e+163) or not (t1 <= 1.5e+119):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -6e+163) || !(t1 <= 1.5e+119))
		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 <= -6e+163) || ~((t1 <= 1.5e+119)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -6e+163], N[Not[LessEqual[t1, 1.5e+119]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.00000000000000027e163 or 1.50000000000000001e119 < t1

   1. Initial program 42.3%

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

      1. +-commutative 42.3%

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

      2. +-commutative 42.3%

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

      3. times-frac 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. frac-2neg 100.0%

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

      3. frac-times 98.7%

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

      4. *-un-lft-identity 98.7%

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

      5. add-sqr-sqrt 49.0%

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

      6. sqrt-unprod 49.9%

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

      7. sqr-neg 49.9%

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

      8. sqrt-unprod 19.3%

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

      9. add-sqr-sqrt 37.6%

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

      10. add-sqr-sqrt 21.5%

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

      11. sqrt-unprod 9.9%

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

      12. sqr-neg 9.9%

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

      13. sqrt-unprod 47.8%

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

      14. add-sqr-sqrt 98.7%

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

      15. distribute-neg-in 98.7%

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

      16. add-sqr-sqrt 50.5%

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

      17. sqrt-unprod 38.5%

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

      18. sqr-neg 38.5%

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

      19. sqrt-unprod 17.1%

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

      20. add-sqr-sqrt 38.7%

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

      21. sub-neg 38.7%

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

   5. Applied egg-rr 38.7%

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

   6. Taylor expanded in t1 around inf 34.9%

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

   if -6.00000000000000027e163 < t1 < 1.50000000000000001e119

   1. Initial program 78.0%

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

      1. +-commutative 78.0%

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

      2. +-commutative 78.0%

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

      3. times-frac 97.3%

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

      4. +-commutative 97.3%

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

      5. +-commutative 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in t1 around 0 63.4%

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

      1. mul-1-neg 63.4%

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

   6. Simplified 63.4%

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

      1. distribute-neg-frac 63.4%

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

      2. associate-*l/ 63.6%

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

   8. Applied egg-rr 63.6%

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

      1. frac-2neg 63.6%

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

      2. associate-*r/ 59.6%

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

      3. add-sqr-sqrt 28.2%

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

      4. sqrt-unprod 42.0%

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

      5. sqr-neg 42.0%

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

      6. sqrt-unprod 20.3%

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

      7. add-sqr-sqrt 37.3%

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

      8. *-commutative 37.3%

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

      9. add-sqr-sqrt 23.1%

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

      10. sqrt-unprod 43.8%

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

      11. sqr-neg 43.8%

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

      12. sqrt-unprod 23.0%

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

      13. add-sqr-sqrt 59.6%

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

      14. distribute-neg-in 59.6%

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

      15. add-sqr-sqrt 36.5%

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

      16. sqrt-unprod 59.4%

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

      17. sqr-neg 59.4%

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

      18. sqrt-unprod 23.1%

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

      19. add-sqr-sqrt 59.4%

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

      20. sub-neg 59.4%

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

   10. Applied egg-rr 59.4%

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

       1. associate-/l* 62.5%

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

   12. Simplified 62.5%

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

   13. Taylor expanded in t1 around inf 18.6%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{+163} \lor \neg \left(t1 \leq 1.5 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 11: 61.8% accurate, 2.0× 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(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 67.9%

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

   1. +-commutative 67.9%

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

   2. +-commutative 67.9%

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

   3. times-frac 98.1%

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

   4. distribute-frac-neg 98.1%

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

   5. distribute-lft-neg-in 98.1%

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

   6. distribute-rgt-neg-in 98.1%

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

   7. distribute-frac-neg 98.1%

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

   8. associate-/r/ 85.4%

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

   9. associate-/r/ 85.4%

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

   10. *-commutative 85.4%

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

   11. +-commutative 85.4%

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

   12. +-commutative 85.4%

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

3. Simplified 85.4%

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

4. Taylor expanded in t1 around inf 54.7%

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

   1. mul-1-neg 54.7%

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

6. Simplified 54.7%

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

7. Taylor expanded in v around 0 64.3%

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

   1. +-commutative 64.3%

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

   2. associate-*r/ 64.3%

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

   3. mul-1-neg 64.3%

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

9. Simplified 64.3%

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

10. Final simplification 64.3%

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

Alternative 12: 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 67.9%

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

   1. +-commutative 67.9%

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

   2. +-commutative 67.9%

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

   3. times-frac 98.1%

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

   4. +-commutative 98.1%

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

   5. +-commutative 98.1%

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

3. Simplified 98.1%

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

   1. clear-num 98.0%

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

   2. frac-2neg 98.0%

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

   3. frac-times 94.9%

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

   4. *-un-lft-identity 94.9%

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

   5. add-sqr-sqrt 43.8%

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

   6. sqrt-unprod 48.2%

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

   7. sqr-neg 48.2%

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

   8. sqrt-unprod 19.7%

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

   9. add-sqr-sqrt 37.3%

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

   10. add-sqr-sqrt 23.2%

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

   11. sqrt-unprod 44.0%

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

   12. sqr-neg 44.0%

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

   13. sqrt-unprod 40.9%

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

   14. add-sqr-sqrt 94.9%

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

   15. distribute-neg-in 94.9%

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

   16. add-sqr-sqrt 53.6%

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

   17. sqrt-unprod 64.1%

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

   18. sqr-neg 64.1%

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

   19. sqrt-unprod 21.3%

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

   20. add-sqr-sqrt 52.9%

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

   21. sub-neg 52.9%

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

5. Applied egg-rr 52.9%

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

6. Taylor expanded in t1 around inf 12.6%

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

7. Final simplification 12.6%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))