Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.1%

Time: 10.2s

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: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. *-commutative 71.6%

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

   2. times-frac 98.2%

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

   3. neg-mul-1 98.2%

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

   4. associate-/l* 98.2%

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

   5. associate-*r/ 98.2%

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

   6. associate-/l* 98.2%

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

   7. associate-/l/ 98.2%

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

   8. neg-mul-1 98.2%

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

   9. *-lft-identity 98.2%

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

   10. metadata-eval 98.2%

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

   11. times-frac 98.2%

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

   12. neg-mul-1 98.2%

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

   13. remove-double-neg 98.2%

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

   14. neg-mul-1 98.2%

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

   15. sub0-neg 98.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 98.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 98.2%

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

   18. div-sub 98.3%

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

   19. distribute-frac-neg 98.3%

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

   20. *-inverses 98.3%

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

   21. metadata-eval 98.3%

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

3. Simplified 98.3%

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

   1. div-inv 98.1%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in v around 0 96.8%

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

   1. associate-*r/ 96.8%

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

   2. *-commutative 96.8%

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

   3. times-frac 98.2%

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

   4. associate-*l/ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. *-commutative 99.7%

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

   7. neg-mul-1 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 72.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{-6} \lor \neg \left(u \leq 7.6 \cdot 10^{-87} \lor \neg \left(u \leq 3.4 \cdot 10^{+34}\right) \land u \leq 4.5 \cdot 10^{+54}\right):\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.9e-6)
         (not (or (<= u 7.6e-87) (and (not (<= u 3.4e+34)) (<= u 4.5e+54)))))
   (* t1 (/ (- v) (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.9e-6) || !((u <= 7.6e-87) || (!(u <= 3.4e+34) && (u <= 4.5e+54)))) {
		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 <= (-4.9d-6)) .or. (.not. (u <= 7.6d-87) .or. (.not. (u <= 3.4d+34)) .and. (u <= 4.5d+54))) 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 <= -4.9e-6) || !((u <= 7.6e-87) || (!(u <= 3.4e+34) && (u <= 4.5e+54)))) {
		tmp = t1 * (-v / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.9e-6) or not ((u <= 7.6e-87) or (not (u <= 3.4e+34) and (u <= 4.5e+54))):
		tmp = t1 * (-v / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.9e-6) || !((u <= 7.6e-87) || (!(u <= 3.4e+34) && (u <= 4.5e+54))))
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(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 <= -4.9e-6) || ~(((u <= 7.6e-87) || (~((u <= 3.4e+34)) && (u <= 4.5e+54)))))
		tmp = t1 * (-v / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.9e-6], N[Not[Or[LessEqual[u, 7.6e-87], And[N[Not[LessEqual[u, 3.4e+34]], $MachinePrecision], LessEqual[u, 4.5e+54]]]], $MachinePrecision]], N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.89999999999999967e-6 or 7.6e-87 < u < 3.3999999999999999e34 or 4.49999999999999984e54 < u

   1. Initial program 80.4%

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

      1. distribute-lft-neg-out 80.4%

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

      2. distribute-rgt-neg-in 80.4%

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

      3. associate-*r/ 81.8%

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

      4. neg-mul-1 81.8%

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

      5. *-commutative 81.8%

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

      6. associate-/l* 81.8%

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

      7. *-rgt-identity 81.8%

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

      8. associate-*r/ 81.8%

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

      9. *-inverses 81.8%

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

      10. associate-*r* 81.8%

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

      11. times-frac 81.8%

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

      12. *-commutative 81.8%

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

      13. neg-mul-1 81.8%

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

      14. associate-/l* 81.6%

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

      15. associate-/r/ 81.8%

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

      16. neg-mul-1 81.8%

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

      17. *-commutative 81.8%

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

      18. associate-/r* 81.8%

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

      19. *-inverses 81.8%

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

      20. metadata-eval 81.8%

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

      21. neg-mul-1 81.8%

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

      22. distribute-neg-in 81.8%

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

      23. sub-neg 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in t1 around 0 75.0%

      \[\leadsto t1 \cdot \color{blue}{\left(-1 \cdot \frac{v}{{u}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 75.0%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]

      2. neg-mul-1 75.0%

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

      3. unpow2 75.0%

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

   6. Simplified 75.0%

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

   if -4.89999999999999967e-6 < u < 7.6e-87 or 3.3999999999999999e34 < u < 4.49999999999999984e54

   1. Initial program 61.7%

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

      1. *-commutative 61.7%

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

      2. times-frac 97.5%

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

      3. neg-mul-1 97.5%

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

      4. associate-/l* 97.6%

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

      5. associate-*r/ 97.6%

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

      6. associate-/l* 97.6%

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

      7. associate-/l/ 97.6%

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

      8. neg-mul-1 97.6%

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

      9. *-lft-identity 97.6%

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

      10. metadata-eval 97.6%

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

      11. times-frac 97.6%

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

      12. neg-mul-1 97.6%

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

      13. remove-double-neg 97.6%

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

      14. neg-mul-1 97.6%

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

      15. sub0-neg 97.6%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 97.6%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 97.6%

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

      18. div-sub 97.6%

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

      19. distribute-frac-neg 97.6%

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

      20. *-inverses 97.6%

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

      21. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in t1 around inf 83.1%

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

      1. mul-1-neg 83.1%

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

   6. Simplified 83.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{-6} \lor \neg \left(u \leq 7.6 \cdot 10^{-87} \lor \neg \left(u \leq 3.4 \cdot 10^{+34}\right) \land u \leq 4.5 \cdot 10^{+54}\right):\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+23} \lor \neg \left(t1 \leq 5.2 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5e+23) (not (<= t1 5.2e+84)))
   (/ (- v) t1)
   (* (/ v u) (/ (- t1) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e+23) || !(t1 <= 5.2e+84)) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5d+23)) .or. (.not. (t1 <= 5.2d+84))) then
        tmp = -v / t1
    else
        tmp = (v / u) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e+23) || !(t1 <= 5.2e+84)) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5e+23) or not (t1 <= 5.2e+84):
		tmp = -v / t1
	else:
		tmp = (v / u) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5e+23) || !(t1 <= 5.2e+84))
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5e+23) || ~((t1 <= 5.2e+84)))
		tmp = -v / t1;
	else
		tmp = (v / u) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5e+23], N[Not[LessEqual[t1, 5.2e+84]], $MachinePrecision]], N[((-v) / t1), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.9999999999999999e23 or 5.2000000000000002e84 < t1

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 87.4%

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

      1. mul-1-neg 87.4%

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

   6. Simplified 87.4%

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

   if -4.9999999999999999e23 < t1 < 5.2000000000000002e84

   1. Initial program 82.8%

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

      1. associate-/l* 82.7%

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

   3. Simplified 82.7%

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

      1. associate-/l* 91.5%

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

      2. div-inv 91.4%

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

      3. clear-num 91.5%

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

   5. Applied egg-rr 91.5%

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

   6. Taylor expanded in t1 around 0 69.8%

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

   7. Taylor expanded in t1 around 0 66.1%

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

      1. unpow2 66.1%

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

      2. associate-/l* 71.3%

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

   9. Simplified 71.3%

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

       1. neg-mul-1 71.3%

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

       2. associate-/r/ 71.3%

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

       3. times-frac 73.7%

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

       4. metadata-eval 73.7%

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

       5. distribute-neg-frac 73.7%

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

       6. clear-num 73.6%

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

       7. distribute-neg-frac 73.6%

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

   11. Applied egg-rr 73.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{+23} \lor \neg \left(t1 \leq 5.2 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 4: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\frac{u + t1}{v}}\\ \mathbf{elif}\;t1 \leq 5.1 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.4e-32)
   (/ -1.0 (/ (+ u t1) v))
   (if (<= t1 5.1e+84) (* (/ v u) (/ (- t1) u)) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.4e-32) {
		tmp = -1.0 / ((u + t1) / v);
	} else if (t1 <= 5.1e+84) {
		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 (t1 <= (-2.4d-32)) then
        tmp = (-1.0d0) / ((u + t1) / v)
    else if (t1 <= 5.1d+84) 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 (t1 <= -2.4e-32) {
		tmp = -1.0 / ((u + t1) / v);
	} else if (t1 <= 5.1e+84) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.4e-32:
		tmp = -1.0 / ((u + t1) / v)
	elif t1 <= 5.1e+84:
		tmp = (v / u) * (-t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.4e-32)
		tmp = Float64(-1.0 / Float64(Float64(u + t1) / v));
	elseif (t1 <= 5.1e+84)
		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 (t1 <= -2.4e-32)
		tmp = -1.0 / ((u + t1) / v);
	elseif (t1 <= 5.1e+84)
		tmp = (v / u) * (-t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.4e-32], N[(-1.0 / N[(N[(u + t1), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.1e+84], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.4000000000000001e-32

   1. Initial program 66.4%

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

      1. associate-/l* 67.4%

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

   3. Simplified 67.4%

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

      1. associate-/l* 79.5%

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

      2. div-inv 79.4%

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

      3. clear-num 79.3%

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

   5. Applied egg-rr 79.3%

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

      1. neg-mul-1 79.3%

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

      2. *-commutative 79.3%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. mul-1-neg 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in t1 around inf 86.5%

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

   if -2.4000000000000001e-32 < t1 < 5.1000000000000001e84

   1. Initial program 81.2%

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

      1. associate-/l* 81.7%

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

   3. Simplified 81.7%

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

      1. associate-/l* 91.4%

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

      2. div-inv 91.3%

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

      3. clear-num 91.3%

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

   5. Applied egg-rr 91.3%

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

   6. Taylor expanded in t1 around 0 71.8%

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

   7. Taylor expanded in t1 around 0 67.8%

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

      1. unpow2 67.8%

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

      2. associate-/l* 73.4%

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

   9. Simplified 73.4%

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

       1. neg-mul-1 73.4%

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

       2. associate-/r/ 73.4%

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

       3. times-frac 75.5%

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

       4. metadata-eval 75.5%

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

       5. distribute-neg-frac 75.5%

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

       6. clear-num 75.5%

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

       7. distribute-neg-frac 75.5%

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

   11. Applied egg-rr 75.5%

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

   if 5.1000000000000001e84 < t1

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 94.4%

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

      1. mul-1-neg 94.4%

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

   6. Simplified 94.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\frac{u + t1}{v}}\\ \mathbf{elif}\;t1 \leq 5.1 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 5: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\frac{u + t1}{v}}\\ \mathbf{elif}\;t1 \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.15e-34)
   (/ -1.0 (/ (+ u t1) v))
   (if (<= t1 1.05e+85) (/ (* t1 (/ v u)) (- u)) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.15e-34) {
		tmp = -1.0 / ((u + t1) / v);
	} else if (t1 <= 1.05e+85) {
		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 (t1 <= (-1.15d-34)) then
        tmp = (-1.0d0) / ((u + t1) / v)
    else if (t1 <= 1.05d+85) 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 (t1 <= -1.15e-34) {
		tmp = -1.0 / ((u + t1) / v);
	} else if (t1 <= 1.05e+85) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.15e-34:
		tmp = -1.0 / ((u + t1) / v)
	elif t1 <= 1.05e+85:
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.15e-34)
		tmp = Float64(-1.0 / Float64(Float64(u + t1) / v));
	elseif (t1 <= 1.05e+85)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.15e-34)
		tmp = -1.0 / ((u + t1) / v);
	elseif (t1 <= 1.05e+85)
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.15e-34], N[(-1.0 / N[(N[(u + t1), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.05e+85], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.15000000000000006e-34

   1. Initial program 66.4%

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

      1. associate-/l* 67.4%

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

   3. Simplified 67.4%

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

      1. associate-/l* 79.5%

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

      2. div-inv 79.4%

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

      3. clear-num 79.3%

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

   5. Applied egg-rr 79.3%

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

      1. neg-mul-1 79.3%

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

      2. *-commutative 79.3%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. mul-1-neg 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in t1 around inf 86.5%

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

   if -1.15000000000000006e-34 < t1 < 1.05000000000000005e85

   1. Initial program 81.2%

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

   2. Taylor expanded in t1 around 0 67.9%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{{u}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 67.9%

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

   4. Simplified 67.9%

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

      1. *-commutative 67.9%

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

      2. times-frac 75.5%

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

      3. add-sqr-sqrt 29.8%

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

      4. sqrt-unprod 51.2%

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

      5. sqr-neg 51.2%

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

      6. sqrt-unprod 28.1%

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

      7. add-sqr-sqrt 42.4%

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

   6. Applied egg-rr 42.4%

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

      1. *-commutative 42.4%

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

      2. frac-2neg 42.4%

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

      3. associate-*l/ 42.4%

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

      4. add-sqr-sqrt 14.3%

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

      5. sqrt-unprod 49.4%

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

      6. sqr-neg 49.4%

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

      7. sqrt-unprod 45.6%

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

      8. add-sqr-sqrt 75.9%

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

   8. Applied egg-rr 75.9%

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

   if 1.05000000000000005e85 < t1

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 94.4%

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

      1. mul-1-neg 94.4%

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

   6. Simplified 94.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\frac{u + t1}{v}}\\ \mathbf{elif}\;t1 \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 6: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.15e-83)
   (/ (/ v t1) (- -1.0 (/ u t1)))
   (if (<= t1 7.2e+84) (/ (* t1 (/ v u)) (- u)) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.15e-83) {
		tmp = (v / t1) / (-1.0 - (u / t1));
	} else if (t1 <= 7.2e+84) {
		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 (t1 <= (-1.15d-83)) then
        tmp = (v / t1) / ((-1.0d0) - (u / t1))
    else if (t1 <= 7.2d+84) 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 (t1 <= -1.15e-83) {
		tmp = (v / t1) / (-1.0 - (u / t1));
	} else if (t1 <= 7.2e+84) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.15e-83:
		tmp = (v / t1) / (-1.0 - (u / t1))
	elif t1 <= 7.2e+84:
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.15e-83)
		tmp = Float64(Float64(v / t1) / Float64(-1.0 - Float64(u / t1)));
	elseif (t1 <= 7.2e+84)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.15e-83)
		tmp = (v / t1) / (-1.0 - (u / t1));
	elseif (t1 <= 7.2e+84)
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.15e-83], N[(N[(v / t1), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 7.2e+84], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.14999999999999995e-83

   1. Initial program 69.0%

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

      1. *-commutative 69.0%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around inf 84.1%

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

   if -1.14999999999999995e-83 < t1 < 7.1999999999999999e84

   1. Initial program 80.4%

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

   2. Taylor expanded in t1 around 0 68.6%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{{u}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 68.6%

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

   4. Simplified 68.6%

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

      1. *-commutative 68.6%

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

      2. times-frac 76.5%

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

      3. add-sqr-sqrt 28.8%

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

      4. sqrt-unprod 51.2%

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

      5. sqr-neg 51.2%

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

      6. sqrt-unprod 29.3%

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

      7. add-sqr-sqrt 43.5%

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

   6. Applied egg-rr 43.5%

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

      1. *-commutative 43.5%

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

      2. frac-2neg 43.5%

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

      3. associate-*l/ 43.5%

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

      4. add-sqr-sqrt 14.1%

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

      5. sqrt-unprod 50.8%

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

      6. sqr-neg 50.8%

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

      7. sqrt-unprod 47.7%

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

      8. add-sqr-sqrt 77.0%

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

   8. Applied egg-rr 77.0%

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

   if 7.1999999999999999e84 < t1

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 94.4%

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

      1. mul-1-neg 94.4%

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

   6. Simplified 94.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 7: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{+64} \lor \neg \left(u \leq 9.5 \cdot 10^{+136}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.05e+64) (not (<= u 9.5e+136)))
   (* t1 (/ v (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.05e+64) || !(u <= 9.5e+136)) {
		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 <= (-1.05d+64)) .or. (.not. (u <= 9.5d+136))) 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 <= -1.05e+64) || !(u <= 9.5e+136)) {
		tmp = t1 * (v / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.05e+64) or not (u <= 9.5e+136):
		tmp = t1 * (v / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.05e+64) || !(u <= 9.5e+136))
		tmp = Float64(t1 * Float64(v / Float64(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 <= -1.05e+64) || ~((u <= 9.5e+136)))
		tmp = t1 * (v / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.05e+64], N[Not[LessEqual[u, 9.5e+136]], $MachinePrecision]], N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.05e64 or 9.49999999999999907e136 < u

   1. Initial program 83.3%

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

   2. Taylor expanded in t1 around 0 83.4%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{{u}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 83.4%

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

   4. Simplified 83.4%

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

      1. expm1-log1p-u 83.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-t1\right) \cdot v}{u \cdot u}\right)\right)} \]

      2. expm1-udef 74.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-t1\right) \cdot v}{u \cdot u}\right)} - 1} \]

      3. div-inv 74.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-t1\right) \cdot v\right) \cdot \frac{1}{u \cdot u}}\right)} - 1 \]

      4. associate-*l* 74.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-t1\right) \cdot \left(v \cdot \frac{1}{u \cdot u}\right)}\right)} - 1 \]

      5. add-sqr-sqrt 31.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \left(v \cdot \frac{1}{u \cdot u}\right)\right)} - 1 \]

      6. sqrt-unprod 64.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \left(v \cdot \frac{1}{u \cdot u}\right)\right)} - 1 \]

      7. sqr-neg 64.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{t1 \cdot t1}} \cdot \left(v \cdot \frac{1}{u \cdot u}\right)\right)} - 1 \]

      8. sqrt-unprod 42.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \left(v \cdot \frac{1}{u \cdot u}\right)\right)} - 1 \]

      9. add-sqr-sqrt 73.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{t1} \cdot \left(v \cdot \frac{1}{u \cdot u}\right)\right)} - 1 \]

      10. pow2 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \frac{1}{\color{blue}{{u}^{2}}}\right)\right)} - 1 \]

      11. pow-flip 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot \color{blue}{{u}^{\left(-2\right)}}\right)\right)} - 1 \]

      12. metadata-eval 73.6%

          \[\leadsto e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{\color{blue}{-2}}\right)\right)} - 1 \]

   6. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 73.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t1 \cdot \left(v \cdot {u}^{-2}\right)\right)\right)} \]

      2. expm1-log1p 73.5%

         \[\leadsto \color{blue}{t1 \cdot \left(v \cdot {u}^{-2}\right)} \]

      3. associate-*r* 73.2%

         \[\leadsto \color{blue}{\left(t1 \cdot v\right) \cdot {u}^{-2}} \]

      4. *-commutative 73.2%

         \[\leadsto \color{blue}{{u}^{-2} \cdot \left(t1 \cdot v\right)} \]

      5. metadata-eval 73.2%

         \[\leadsto {u}^{\color{blue}{\left(2 \cdot -1\right)}} \cdot \left(t1 \cdot v\right) \]

      6. pow-sqr 73.2%

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

      7. unpow-1 73.2%

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

      8. unpow-1 73.2%

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

      9. associate-*l* 70.8%

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

      10. associate-*l/ 70.8%

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

      11. associate-*r/ 70.8%

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

      12. *-lft-identity 70.8%

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

      13. associate-*l/ 70.8%

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

      14. *-lft-identity 70.8%

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

      15. associate-*r/ 72.1%

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

      16. associate-*r/ 73.3%

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

      17. associate-/r* 73.5%

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

   8. Simplified 73.5%

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

   if -1.05e64 < u < 9.49999999999999907e136

   1. Initial program 65.4%

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

      1. *-commutative 65.4%

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

      2. times-frac 97.6%

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

      3. neg-mul-1 97.6%

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

      4. associate-/l* 97.6%

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

      5. associate-*r/ 97.6%

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

      6. associate-/l* 97.6%

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

      7. associate-/l/ 97.6%

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

      8. neg-mul-1 97.6%

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

      9. *-lft-identity 97.6%

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

      10. metadata-eval 97.6%

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

      11. times-frac 97.6%

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

      12. neg-mul-1 97.6%

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

      13. remove-double-neg 97.6%

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

      14. neg-mul-1 97.6%

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

      15. sub0-neg 97.6%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 97.6%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 97.6%

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

      18. div-sub 97.6%

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

      19. distribute-frac-neg 97.6%

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

      20. *-inverses 97.6%

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

      21. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in t1 around inf 71.6%

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

      1. mul-1-neg 71.6%

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

   6. Simplified 71.6%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.05 \cdot 10^{+64} \lor \neg \left(u \leq 9.5 \cdot 10^{+136}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 94.8% accurate, 1.1× speedup

Math

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

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(v / Float64(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[(v / N[(N[(u + t1), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

   1. *-commutative 71.6%

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

   2. times-frac 98.2%

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

   3. neg-mul-1 98.2%

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

   4. associate-/l* 98.2%

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

   5. associate-*r/ 98.2%

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

   6. associate-/l* 98.2%

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

   7. associate-/l/ 98.2%

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

   8. neg-mul-1 98.2%

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

   9. *-lft-identity 98.2%

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

   10. metadata-eval 98.2%

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

   11. times-frac 98.2%

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

   12. neg-mul-1 98.2%

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

   13. remove-double-neg 98.2%

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

   14. neg-mul-1 98.2%

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

   15. sub0-neg 98.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 98.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 98.2%

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

   18. div-sub 98.3%

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

   19. distribute-frac-neg 98.3%

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

   20. *-inverses 98.3%

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

   21. metadata-eval 98.3%

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

3. Simplified 98.3%

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

   1. expm1-log1p-u 86.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\right)\right)} \]

   2. expm1-udef 51.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\right)} - 1} \]

5. Applied egg-rr 51.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 86.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\right)\right)} \]

   2. expm1-log1p 98.3%

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

   3. associate-/l/ 96.8%

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

   4. *-commutative 96.8%

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

7. Simplified 96.8%

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

8. Final simplification 96.8%

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

Alternative 9: 98.1% 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 71.6%

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

   1. *-commutative 71.6%

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

   2. times-frac 98.2%

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

   3. neg-mul-1 98.2%

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

   4. associate-/l* 98.2%

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

   5. associate-*r/ 98.2%

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

   6. associate-/l* 98.2%

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

   7. associate-/l/ 98.2%

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

   8. neg-mul-1 98.2%

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

   9. *-lft-identity 98.2%

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

   10. metadata-eval 98.2%

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

   11. times-frac 98.2%

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

   12. neg-mul-1 98.2%

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

   13. remove-double-neg 98.2%

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

   14. neg-mul-1 98.2%

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

   15. sub0-neg 98.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 98.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 98.2%

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

   18. div-sub 98.3%

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

   19. distribute-frac-neg 98.3%

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

   20. *-inverses 98.3%

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

   21. metadata-eval 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 10: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.8 \cdot 10^{+99} \lor \neg \left(u \leq 4 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.8e+99) (not (<= u 4e+137))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.8e+99) || !(u <= 4e+137)) {
		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 <= (-6.8d+99)) .or. (.not. (u <= 4d+137))) 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 <= -6.8e+99) || !(u <= 4e+137)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.8e+99) or not (u <= 4e+137):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.8e+99) || !(u <= 4e+137))
		tmp = Float64(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 <= -6.8e+99) || ~((u <= 4e+137)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.8e+99], N[Not[LessEqual[u, 4e+137]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.79999999999999968e99 or 4.0000000000000001e137 < u

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. times-frac 99.4%

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

      3. neg-mul-1 99.4%

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

      4. associate-/l* 99.4%

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

      5. associate-*r/ 99.4%

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

      6. associate-/l* 99.4%

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

      7. associate-/l/ 99.4%

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

      8. neg-mul-1 99.4%

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

      9. *-lft-identity 99.4%

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

      10. metadata-eval 99.4%

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

      11. times-frac 99.4%

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

      12. neg-mul-1 99.4%

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

      13. remove-double-neg 99.4%

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

      14. neg-mul-1 99.4%

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

      15. sub0-neg 99.4%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.4%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.4%

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

      18. div-sub 99.4%

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

      19. distribute-frac-neg 99.4%

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

      20. *-inverses 99.4%

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

      21. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around 0 91.6%

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

   5. Taylor expanded in u around 0 41.8%

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

      1. mul-1-neg 41.8%

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

      2. distribute-frac-neg 41.8%

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

   7. Simplified 41.8%

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

   if -6.79999999999999968e99 < u < 4.0000000000000001e137

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

      2. times-frac 97.6%

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

      3. neg-mul-1 97.6%

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

      4. associate-/l* 97.7%

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

      5. associate-*r/ 97.7%

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

      6. associate-/l* 97.7%

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

      7. associate-/l/ 97.7%

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

      8. neg-mul-1 97.7%

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

      9. *-lft-identity 97.7%

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

      10. metadata-eval 97.7%

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

      11. times-frac 97.7%

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

      12. neg-mul-1 97.7%

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

      13. remove-double-neg 97.7%

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

      14. neg-mul-1 97.7%

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

      15. sub0-neg 97.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 97.7%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 97.7%

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

      18. div-sub 97.7%

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

      19. distribute-frac-neg 97.7%

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

      20. *-inverses 97.7%

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

      21. metadata-eval 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in t1 around inf 69.5%

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

      1. mul-1-neg 69.5%

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

   6. Simplified 69.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.8 \cdot 10^{+99} \lor \neg \left(u \leq 4 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 54.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(u, v, t1)
	return Float64(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 71.6%

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

   1. *-commutative 71.6%

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

   2. times-frac 98.2%

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

   3. neg-mul-1 98.2%

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

   4. associate-/l* 98.2%

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

   5. associate-*r/ 98.2%

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

   6. associate-/l* 98.2%

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

   7. associate-/l/ 98.2%

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

   8. neg-mul-1 98.2%

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

   9. *-lft-identity 98.2%

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

   10. metadata-eval 98.2%

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

   11. times-frac 98.2%

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

   12. neg-mul-1 98.2%

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

   13. remove-double-neg 98.2%

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

   14. neg-mul-1 98.2%

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

   15. sub0-neg 98.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 98.2%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 98.2%

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

   18. div-sub 98.3%

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

   19. distribute-frac-neg 98.3%

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

   20. *-inverses 98.3%

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

   21. metadata-eval 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in t1 around inf 53.6%

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

   1. mul-1-neg 53.6%

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

6. Simplified 53.6%

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

7. Final simplification 53.6%

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

Alternative 12: 14.7% 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 71.6%

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

2. Taylor expanded in t1 around inf 34.9%

   \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{{t1}^{2}}} \]
3. Step-by-step derivation

   1. unpow2 34.9%

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

4. Simplified 34.9%

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

   1. expm1-log1p-u 28.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-t1\right) \cdot v}{t1 \cdot t1}\right)\right)} \]

   2. expm1-udef 26.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-t1\right) \cdot v}{t1 \cdot t1}\right)} - 1} \]

   3. frac-2neg 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\left(-t1\right) \cdot v}{-t1 \cdot t1}}\right)} - 1 \]

   4. div-inv 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\left(-t1\right) \cdot v\right) \cdot \frac{1}{-t1 \cdot t1}}\right)} - 1 \]

   5. distribute-lft-neg-out 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\left(-t1 \cdot v\right)}\right) \cdot \frac{1}{-t1 \cdot t1}\right)} - 1 \]

   6. remove-double-neg 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(t1 \cdot v\right)} \cdot \frac{1}{-t1 \cdot t1}\right)} - 1 \]

   7. distribute-lft-neg-in 26.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot \frac{1}{\color{blue}{\left(-t1\right) \cdot t1}}\right)} - 1 \]

   8. add-sqr-sqrt 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot \frac{1}{\color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot t1}\right)} - 1 \]

   9. sqrt-unprod 25.2%

      \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot \frac{1}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot t1}\right)} - 1 \]

   10. sqr-neg 25.2%

       \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot \frac{1}{\sqrt{\color{blue}{t1 \cdot t1}} \cdot t1}\right)} - 1 \]

   11. sqrt-prod 10.5%

       \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot \frac{1}{\color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot t1}\right)} - 1 \]

   12. add-sqr-sqrt 20.7%

       \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot \frac{1}{\color{blue}{t1} \cdot t1}\right)} - 1 \]

   13. pow2 20.7%

       \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot \frac{1}{\color{blue}{{t1}^{2}}}\right)} - 1 \]

   14. pow-flip 20.7%

       \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot \color{blue}{{t1}^{\left(-2\right)}}\right)} - 1 \]

   15. metadata-eval 20.7%

       \[\leadsto e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot {t1}^{\color{blue}{-2}}\right)} - 1 \]

6. Applied egg-rr 20.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot {t1}^{-2}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 13.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(t1 \cdot v\right) \cdot {t1}^{-2}\right)\right)} \]

   2. expm1-log1p 13.3%

      \[\leadsto \color{blue}{\left(t1 \cdot v\right) \cdot {t1}^{-2}} \]

   3. *-commutative 13.3%

      \[\leadsto \color{blue}{\left(v \cdot t1\right)} \cdot {t1}^{-2} \]

   4. associate-*r* 12.3%

      \[\leadsto \color{blue}{v \cdot \left(t1 \cdot {t1}^{-2}\right)} \]

   5. *-commutative 12.3%

      \[\leadsto v \cdot \color{blue}{\left({t1}^{-2} \cdot t1\right)} \]

   6. pow-plus 11.9%

      \[\leadsto v \cdot \color{blue}{{t1}^{\left(-2 + 1\right)}} \]

   7. metadata-eval 11.9%

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

   8. unpow-1 11.9%

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

   9. *-inverses 11.9%

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

   10. associate-/r* 12.3%

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

   11. associate-*r/ 13.3%

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

   12. times-frac 11.9%

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

   13. *-inverses 11.9%

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

   14. *-rgt-identity 11.9%

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

8. Simplified 11.9%

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

9. Final simplification 11.9%

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

Reproduce

herbie shell --seed 2023200 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))