Rosa's DopplerBench

Percentage Accurate: 72.3% → 98.3%

Time: 10.8s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% 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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-*l/ 75.5%

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

   2. *-commutative 75.5%

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

3. Simplified 75.5%

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

   1. associate-*r/ 72.1%

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

   2. *-commutative 72.1%

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

   3. times-frac 97.4%

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

   4. frac-2neg 97.4%

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

   5. +-commutative 97.4%

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

   6. distribute-neg-in 97.4%

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

   7. sub-neg 97.4%

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

   8. associate-*r/ 97.8%

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

   9. add-sqr-sqrt 47.0%

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

   10. sqrt-unprod 43.5%

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

   11. sqr-neg 43.5%

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

   12. sqrt-unprod 19.6%

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

   13. add-sqr-sqrt 33.6%

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

   14. sub-neg 33.6%

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

   15. +-commutative 33.6%

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

   16. add-sqr-sqrt 14.0%

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

   17. sqrt-unprod 44.6%

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

   18. sqr-neg 44.6%

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

   19. sqrt-unprod 39.6%

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

   20. add-sqr-sqrt 18.8%

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

   21. sqrt-unprod 43.1%

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

   22. sqr-neg 43.1%

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

6. Applied egg-rr 97.8%

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

Alternative 2: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := \frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{if}\;t1 \leq -1.95 \cdot 10^{+148}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -9 \cdot 10^{-152}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 6.4 \cdot 10^{+59}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (/ v (- (- t1) (* u 2.0)))))
   (if (<= t1 -1.95e+148)
     t_2
     (if (<= t1 -9e-152)
       (* v (/ t1 (* (+ t1 u) t_1)))
       (if (<= t1 4.8e-230)
         (/ v (* (- u) (/ u t1)))
         (if (<= t1 6.4e+59) (* t1 (/ (/ v (+ t1 u)) t_1)) t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / (-t1 - (u * 2.0));
	double tmp;
	if (t1 <= -1.95e+148) {
		tmp = t_2;
	} else if (t1 <= -9e-152) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else if (t1 <= 4.8e-230) {
		tmp = v / (-u * (u / t1));
	} else if (t1 <= 6.4e+59) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -u - t1
    t_2 = v / (-t1 - (u * 2.0d0))
    if (t1 <= (-1.95d+148)) then
        tmp = t_2
    else if (t1 <= (-9d-152)) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else if (t1 <= 4.8d-230) then
        tmp = v / (-u * (u / t1))
    else if (t1 <= 6.4d+59) then
        tmp = t1 * ((v / (t1 + u)) / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / (-t1 - (u * 2.0));
	double tmp;
	if (t1 <= -1.95e+148) {
		tmp = t_2;
	} else if (t1 <= -9e-152) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else if (t1 <= 4.8e-230) {
		tmp = v / (-u * (u / t1));
	} else if (t1 <= 6.4e+59) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	t_2 = v / (-t1 - (u * 2.0))
	tmp = 0
	if t1 <= -1.95e+148:
		tmp = t_2
	elif t1 <= -9e-152:
		tmp = v * (t1 / ((t1 + u) * t_1))
	elif t1 <= 4.8e-230:
		tmp = v / (-u * (u / t1))
	elif t1 <= 6.4e+59:
		tmp = t1 * ((v / (t1 + u)) / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -1.95e+148)
		tmp = t_2;
	elseif (t1 <= -9e-152)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	elseif (t1 <= 4.8e-230)
		tmp = Float64(v / Float64(Float64(-u) * Float64(u / t1)));
	elseif (t1 <= 6.4e+59)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	t_2 = v / (-t1 - (u * 2.0));
	tmp = 0.0;
	if (t1 <= -1.95e+148)
		tmp = t_2;
	elseif (t1 <= -9e-152)
		tmp = v * (t1 / ((t1 + u) * t_1));
	elseif (t1 <= 4.8e-230)
		tmp = v / (-u * (u / t1));
	elseif (t1 <= 6.4e+59)
		tmp = t1 * ((v / (t1 + u)) / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.95e+148], t$95$2, If[LessEqual[t1, -9e-152], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.8e-230], N[(v / N[((-u) * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 6.4e+59], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.95000000000000001e148 or 6.39999999999999964e59 < t1

   1. Initial program 42.4%

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

      1. associate-*l/ 46.1%

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

      2. *-commutative 46.1%

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

   3. Simplified 46.1%

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

      1. associate-*r/ 42.4%

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

      2. times-frac 100.0%

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

      3. *-commutative 100.0%

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

      4. frac-2neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. sub-neg 100.0%

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

      9. clear-num 100.0%

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

      10. frac-2neg 100.0%

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

      11. frac-times 97.6%

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

      12. *-un-lft-identity 97.6%

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

      13. +-commutative 97.6%

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

      14. distribute-neg-in 97.6%

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

      15. sub-neg 97.6%

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

      16. sub-neg 97.6%

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

      17. +-commutative 97.6%

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

      18. add-sqr-sqrt 39.3%

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

      19. sqrt-unprod 35.8%

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

      20. sqr-neg 35.8%

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

      21. sqrt-unprod 21.1%

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

      22. add-sqr-sqrt 9.0%

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

      23. sqrt-unprod 21.1%

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

      24. sqr-neg 21.1%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in u around 0 96.4%

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

      1. *-commutative 96.4%

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

   9. Simplified 96.4%

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

   if -1.95000000000000001e148 < t1 < -9.0000000000000008e-152

   1. Initial program 91.1%

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

      1. associate-*l/ 97.6%

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

      2. *-commutative 97.6%

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

   3. Simplified 97.6%

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

   if -9.0000000000000008e-152 < t1 < 4.8000000000000004e-230

   1. Initial program 78.8%

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

      1. times-frac 89.4%

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

      2. distribute-frac-neg 89.4%

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

      3. distribute-neg-frac2 89.4%

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

      4. +-commutative 89.4%

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

      5. distribute-neg-in 89.4%

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

      6. unsub-neg 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in t1 around 0 85.3%

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

   6. Taylor expanded in t1 around 0 85.4%

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

      1. associate-*r/ 85.4%

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

      2. mul-1-neg 85.4%

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

   8. Simplified 85.4%

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

      1. clear-num 85.4%

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

      2. frac-2neg 85.4%

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

      3. frac-times 90.8%

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

      4. *-un-lft-identity 90.8%

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

      5. add-sqr-sqrt 49.0%

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

      6. sqrt-unprod 52.1%

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

      7. sqr-neg 52.1%

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

      8. sqrt-unprod 12.8%

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

      9. add-sqr-sqrt 32.2%

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

      10. add-sqr-sqrt 19.8%

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

      11. sqrt-unprod 33.2%

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

      12. sqr-neg 33.2%

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

      13. sqrt-unprod 37.8%

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

      14. add-sqr-sqrt 90.8%

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

   10. Applied egg-rr 90.8%

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

   if 4.8000000000000004e-230 < t1 < 6.39999999999999964e59

   1. Initial program 86.1%

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

      1. associate-/l* 86.5%

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

   3. Simplified 86.5%

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

      1. associate-/r* 94.2%

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

      2. div-inv 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. associate-*r/ 94.2%

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

      2. *-rgt-identity 94.2%

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

   8. Simplified 94.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.95 \cdot 10^{+148}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{elif}\;t1 \leq -9 \cdot 10^{-152}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 6.4 \cdot 10^{+59}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{+153} \lor \neg \left(t1 \leq 6.8 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1e+153) (not (<= t1 6.8e+126)))
   (/ v (- (- t1) (* u 2.0)))
   (* v (/ t1 (* (+ t1 u) (- (- u) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e+153) || !(t1 <= 6.8e+126)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1d+153)) .or. (.not. (t1 <= 6.8d+126))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = v * (t1 / ((t1 + u) * (-u - t1)))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1e+153) || !(t1 <= 6.8e+126)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1e+153) or not (t1 <= 6.8e+126):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = v * (t1 / ((t1 + u) * (-u - t1)))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1e+153) || !(t1 <= 6.8e+126))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) - t1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1e+153) || ~((t1 <= 6.8e+126)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1e+153], N[Not[LessEqual[t1, 6.8e+126]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1e153 or 6.79999999999999979e126 < t1

   1. Initial program 38.8%

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

      1. associate-*l/ 41.5%

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

      2. *-commutative 41.5%

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

   3. Simplified 41.5%

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

      1. associate-*r/ 38.8%

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

      2. times-frac 100.0%

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

      3. *-commutative 100.0%

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

      4. frac-2neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. sub-neg 100.0%

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

      9. clear-num 100.0%

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

      10. frac-2neg 100.0%

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

      11. frac-times 98.7%

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

      12. *-un-lft-identity 98.7%

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

      13. +-commutative 98.7%

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

      14. distribute-neg-in 98.7%

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

      15. sub-neg 98.7%

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

      16. sub-neg 98.7%

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

      17. +-commutative 98.7%

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

      18. add-sqr-sqrt 43.6%

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

      19. sqrt-unprod 37.7%

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

      20. sqr-neg 37.7%

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

      21. sqrt-unprod 21.4%

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

      22. add-sqr-sqrt 8.2%

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

      23. sqrt-unprod 21.4%

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

      24. sqr-neg 21.4%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in u around 0 98.7%

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

      1. *-commutative 98.7%

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

   9. Simplified 98.7%

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

   if -1e153 < t1 < 6.79999999999999979e126

   1. Initial program 85.3%

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

      1. associate-*l/ 89.0%

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

      2. *-commutative 89.0%

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

   3. Simplified 89.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{+153} \lor \neg \left(t1 \leq 6.8 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.6 \cdot 10^{-50} \lor \neg \left(t1 \leq 1.02 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.6e-50) (not (<= t1 1.02e-53)))
   (/ v (- (- t1) (* u 2.0)))
   (/ (* v (/ t1 u)) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.6e-50) || !(t1 <= 1.02e-53)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.6d-50)) .or. (.not. (t1 <= 1.02d-53))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (v * (t1 / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.6e-50) || !(t1 <= 1.02e-53)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.6e-50) or not (t1 <= 1.02e-53):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (v * (t1 / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.6e-50) || !(t1 <= 1.02e-53))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.6e-50) || ~((t1 <= 1.02e-53)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (v * (t1 / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.6e-50], N[Not[LessEqual[t1, 1.02e-53]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.5999999999999996e-50 or 1.02000000000000002e-53 < t1

   1. Initial program 63.0%

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

      1. associate-*l/ 69.0%

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

      2. *-commutative 69.0%

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

   3. Simplified 69.0%

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

      1. associate-*r/ 63.0%

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

      2. times-frac 100.0%

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

      3. *-commutative 100.0%

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

      4. frac-2neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. sub-neg 100.0%

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

      9. clear-num 99.9%

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

      10. frac-2neg 99.9%

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

      11. frac-times 97.9%

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

      12. *-un-lft-identity 97.9%

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

      13. +-commutative 97.9%

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

      14. distribute-neg-in 97.9%

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

      15. sub-neg 97.9%

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

      16. sub-neg 97.9%

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

      17. +-commutative 97.9%

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

      18. add-sqr-sqrt 50.1%

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

      19. sqrt-unprod 56.1%

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

      20. sqr-neg 56.1%

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

      21. sqrt-unprod 19.3%

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

      22. add-sqr-sqrt 8.4%

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

      23. sqrt-unprod 18.0%

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

      24. sqr-neg 18.0%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u around 0 88.1%

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

      1. *-commutative 88.1%

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

   9. Simplified 88.1%

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

   if -5.5999999999999996e-50 < t1 < 1.02000000000000002e-53

   1. Initial program 84.5%

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

      1. times-frac 93.8%

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

      2. distribute-frac-neg 93.8%

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

      3. distribute-neg-frac2 93.8%

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

      4. +-commutative 93.8%

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

      5. distribute-neg-in 93.8%

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

      6. unsub-neg 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in t1 around 0 78.2%

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

   6. Taylor expanded in t1 around 0 80.4%

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

      1. associate-*r/ 80.4%

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

      2. mul-1-neg 80.4%

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

   8. Simplified 80.4%

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

      1. *-commutative 80.4%

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

      2. frac-2neg 80.4%

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

      3. associate-*l/ 81.4%

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

      4. add-sqr-sqrt 43.8%

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

      5. sqrt-unprod 49.0%

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

      6. sqr-neg 49.0%

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

      7. sqrt-unprod 16.5%

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

      8. add-sqr-sqrt 36.1%

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

      9. add-sqr-sqrt 14.2%

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

      10. sqrt-unprod 44.5%

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

      11. sqr-neg 44.5%

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

      12. sqrt-unprod 44.1%

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

      13. add-sqr-sqrt 81.4%

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

   10. Applied egg-rr 81.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.6 \cdot 10^{-50} \lor \neg \left(t1 \leq 1.02 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{-54} \lor \neg \left(t1 \leq 7.8 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.22e-54) (not (<= t1 7.8e-54)))
   (/ v (- (- t1) (* u 2.0)))
   (* (/ t1 u) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.22e-54) || !(t1 <= 7.8e-54)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (t1 / u) * (v / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.22d-54)) .or. (.not. (t1 <= 7.8d-54))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = (t1 / u) * (v / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.22e-54) || !(t1 <= 7.8e-54)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = (t1 / u) * (v / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.22e-54) or not (t1 <= 7.8e-54):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = (t1 / u) * (v / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.22e-54) || !(t1 <= 7.8e-54))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(Float64(t1 / u) * Float64(v / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.22e-54) || ~((t1 <= 7.8e-54)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = (t1 / u) * (v / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.22e-54], N[Not[LessEqual[t1, 7.8e-54]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[(v / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.22e-54 or 7.8e-54 < t1

   1. Initial program 63.0%

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

      1. associate-*l/ 69.0%

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

      2. *-commutative 69.0%

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

   3. Simplified 69.0%

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

      1. associate-*r/ 63.0%

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

      2. times-frac 100.0%

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

      3. *-commutative 100.0%

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

      4. frac-2neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. sub-neg 100.0%

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

      9. clear-num 99.9%

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

      10. frac-2neg 99.9%

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

      11. frac-times 97.9%

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

      12. *-un-lft-identity 97.9%

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

      13. +-commutative 97.9%

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

      14. distribute-neg-in 97.9%

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

      15. sub-neg 97.9%

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

      16. sub-neg 97.9%

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

      17. +-commutative 97.9%

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

      18. add-sqr-sqrt 50.1%

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

      19. sqrt-unprod 56.1%

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

      20. sqr-neg 56.1%

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

      21. sqrt-unprod 19.3%

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

      22. add-sqr-sqrt 8.4%

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

      23. sqrt-unprod 18.0%

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

      24. sqr-neg 18.0%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u around 0 88.1%

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

      1. *-commutative 88.1%

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

   9. Simplified 88.1%

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

   if -1.22e-54 < t1 < 7.8e-54

   1. Initial program 84.5%

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

      1. times-frac 93.8%

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

      2. distribute-frac-neg 93.8%

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

      3. distribute-neg-frac2 93.8%

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

      4. +-commutative 93.8%

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

      5. distribute-neg-in 93.8%

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

      6. unsub-neg 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in t1 around 0 78.2%

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

   6. Taylor expanded in t1 around 0 80.4%

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

      1. associate-*r/ 80.4%

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

      2. mul-1-neg 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{-54} \lor \neg \left(t1 \leq 7.8 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-55} \lor \neg \left(t1 \leq 1.4 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.6e-55) (not (<= t1 1.4e-54)))
   (/ v (- (- u) t1))
   (* (/ t1 u) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-55) || !(t1 <= 1.4e-54)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / u) * (v / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.6d-55)) .or. (.not. (t1 <= 1.4d-54))) then
        tmp = v / (-u - t1)
    else
        tmp = (t1 / u) * (v / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-55) || !(t1 <= 1.4e-54)) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / u) * (v / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.6e-55) or not (t1 <= 1.4e-54):
		tmp = v / (-u - t1)
	else:
		tmp = (t1 / u) * (v / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.6e-55) || !(t1 <= 1.4e-54))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(t1 / u) * Float64(v / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.6e-55) || ~((t1 <= 1.4e-54)))
		tmp = v / (-u - t1);
	else
		tmp = (t1 / u) * (v / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.6e-55], N[Not[LessEqual[t1, 1.4e-54]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[(v / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6000000000000001e-55 or 1.4000000000000001e-54 < t1

   1. Initial program 63.0%

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

      1. associate-*l/ 69.0%

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

      2. *-commutative 69.0%

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

   3. Simplified 69.0%

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

      1. associate-*r/ 63.0%

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

      2. *-commutative 63.0%

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

      3. times-frac 100.0%

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

      4. frac-2neg 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*r/ 100.0%

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

      9. add-sqr-sqrt 50.3%

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

      10. sqrt-unprod 37.9%

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

      11. sqr-neg 37.9%

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

      12. sqrt-unprod 17.9%

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

      13. add-sqr-sqrt 31.8%

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

      14. sub-neg 31.8%

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

      15. +-commutative 31.8%

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

      16. add-sqr-sqrt 13.9%

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

      17. sqrt-unprod 45.3%

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

      18. sqr-neg 45.3%

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

      19. sqrt-unprod 45.4%

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

      20. add-sqr-sqrt 20.2%

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

      21. sqrt-unprod 45.4%

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

      22. sqr-neg 45.4%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 87.8%

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

      1. mul-1-neg 87.8%

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

   9. Simplified 87.8%

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

   if -1.6000000000000001e-55 < t1 < 1.4000000000000001e-54

   1. Initial program 84.5%

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

      1. times-frac 93.8%

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

      2. distribute-frac-neg 93.8%

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

      3. distribute-neg-frac2 93.8%

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

      4. +-commutative 93.8%

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

      5. distribute-neg-in 93.8%

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

      6. unsub-neg 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in t1 around 0 78.2%

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

   6. Taylor expanded in t1 around 0 80.4%

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

      1. associate-*r/ 80.4%

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

      2. mul-1-neg 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-55} \lor \neg \left(t1 \leq 1.4 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+108} \lor \neg \left(u \leq 4.8 \cdot 10^{+147}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.5e+108) (not (<= u 4.8e+147)))
   (* v (/ (/ t1 u) u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.5e+108) || !(u <= 4.8e+147)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.5d+108)) .or. (.not. (u <= 4.8d+147))) then
        tmp = v * ((t1 / u) / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.5e+108) || !(u <= 4.8e+147)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.5e+108) or not (u <= 4.8e+147):
		tmp = v * ((t1 / u) / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.5e+108) || !(u <= 4.8e+147))
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.5e+108) || ~((u <= 4.8e+147)))
		tmp = v * ((t1 / u) / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.5e+108], N[Not[LessEqual[u, 4.8e+147]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.5000000000000002e108 or 4.80000000000000004e147 < u

   1. Initial program 86.6%

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

      1. times-frac 98.2%

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

      2. distribute-frac-neg 98.2%

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

      3. distribute-neg-frac2 98.2%

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

      4. +-commutative 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. unsub-neg 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around 0 91.4%

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

   6. Taylor expanded in t1 around 0 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   8. Simplified 91.4%

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

      1. clear-num 91.4%

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

      2. un-div-inv 91.5%

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

      3. add-sqr-sqrt 33.7%

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

      4. sqrt-unprod 73.4%

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

      5. sqr-neg 73.4%

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

      6. sqrt-unprod 48.6%

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

      7. add-sqr-sqrt 80.3%

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

   10. Applied egg-rr 80.3%

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

       1. associate-/r/ 79.1%

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

   12. Simplified 79.1%

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

   if -3.5000000000000002e108 < u < 4.80000000000000004e147

   1. Initial program 66.7%

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

      1. associate-*l/ 72.3%

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

      2. *-commutative 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   7. Simplified 71.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.5 \cdot 10^{+108} \lor \neg \left(u \leq 4.8 \cdot 10^{+147}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.15 \cdot 10^{+111}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.15e+111)
   (/ v u)
   (if (<= u 3e+168) (/ v (- t1)) (/ 1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.15e+111) {
		tmp = v / u;
	} else if (u <= 3e+168) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.15d+111)) then
        tmp = v / u
    else if (u <= 3d+168) then
        tmp = v / -t1
    else
        tmp = 1.0d0 / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.15e+111) {
		tmp = v / u;
	} else if (u <= 3e+168) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.15e+111:
		tmp = v / u
	elif u <= 3e+168:
		tmp = v / -t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.15e+111)
		tmp = Float64(v / u);
	elseif (u <= 3e+168)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.15e+111)
		tmp = v / u;
	elseif (u <= 3e+168)
		tmp = v / -t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.15e+111], N[(v / u), $MachinePrecision], If[LessEqual[u, 3e+168], N[(v / (-t1)), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.14999999999999997e111

   1. Initial program 88.2%

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

      1. times-frac 97.1%

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

      2. distribute-frac-neg 97.1%

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

      3. distribute-neg-frac2 97.1%

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

      4. +-commutative 97.1%

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

      5. distribute-neg-in 97.1%

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

      6. unsub-neg 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t1 around 0 86.5%

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

   6. Taylor expanded in t1 around inf 44.6%

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

      1. associate-*r/ 44.6%

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

      2. mul-1-neg 44.6%

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

   8. Simplified 44.6%

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

      1. div-inv 44.6%

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

      2. add-sqr-sqrt 13.3%

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

      3. sqrt-unprod 46.4%

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

      4. sqr-neg 46.4%

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

      5. sqrt-unprod 31.2%

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

      6. add-sqr-sqrt 44.7%

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

   10. Applied egg-rr 44.7%

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

       1. associate-*r/ 44.7%

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

       2. *-rgt-identity 44.7%

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

   12. Simplified 44.7%

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

   if -2.14999999999999997e111 < u < 2.9999999999999998e168

   1. Initial program 66.6%

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

      1. associate-*l/ 71.6%

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

      2. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in t1 around inf 70.9%

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

      1. associate-*r/ 70.9%

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

      2. neg-mul-1 70.9%

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

   7. Simplified 70.9%

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

   if 2.9999999999999998e168 < u

   1. Initial program 86.4%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around 0 100.0%

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

   6. Taylor expanded in t1 around inf 43.4%

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

      1. associate-*r/ 43.4%

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

      2. mul-1-neg 43.4%

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

   8. Simplified 43.4%

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

      1. clear-num 46.3%

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

      2. inv-pow 46.3%

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

      3. add-sqr-sqrt 23.0%

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

      4. sqrt-unprod 45.6%

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

      5. sqr-neg 45.6%

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

      6. sqrt-unprod 23.2%

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

      7. add-sqr-sqrt 46.1%

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

   10. Applied egg-rr 46.1%

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

       1. unpow-1 46.1%

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

   12. Simplified 46.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.15 \cdot 10^{+111}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 3 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+112} \lor \neg \left(u \leq 2.25 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.3e+112) (not (<= u 2.25e+171))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.3e+112) || !(u <= 2.25e+171)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.3d+112)) .or. (.not. (u <= 2.25d+171))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.3e+112) || !(u <= 2.25e+171)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.3e+112) or not (u <= 2.25e+171):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.3e+112) || !(u <= 2.25e+171))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.3e+112) || ~((u <= 2.25e+171)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.3e+112], N[Not[LessEqual[u, 2.25e+171]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.3e112 or 2.24999999999999984e171 < u

   1. Initial program 87.6%

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

      1. times-frac 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. distribute-neg-frac2 98.1%

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

      4. +-commutative 98.1%

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

      5. distribute-neg-in 98.1%

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

      6. unsub-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in t1 around 0 91.1%

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

   6. Taylor expanded in t1 around inf 44.2%

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

      1. associate-*r/ 44.2%

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

      2. mul-1-neg 44.2%

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

   8. Simplified 44.2%

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

      1. div-inv 44.2%

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

      2. add-sqr-sqrt 16.6%

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

      3. sqrt-unprod 45.1%

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

      4. sqr-neg 45.1%

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

      5. sqrt-unprod 27.5%

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

      6. add-sqr-sqrt 44.2%

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

   10. Applied egg-rr 44.2%

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

       1. associate-*r/ 44.2%

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

       2. *-rgt-identity 44.2%

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

   12. Simplified 44.2%

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

   if -1.3e112 < u < 2.24999999999999984e171

   1. Initial program 66.6%

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

      1. associate-*l/ 71.6%

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

      2. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in t1 around inf 70.9%

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

      1. associate-*r/ 70.9%

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

      2. neg-mul-1 70.9%

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

   7. Simplified 70.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+112} \lor \neg \left(u \leq 2.25 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.36 \cdot 10^{+112}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.36e+112) (/ v u) (if (<= u 6.2e+153) (/ v (- t1)) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.36e+112) {
		tmp = v / u;
	} else if (u <= 6.2e+153) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.36d+112)) then
        tmp = v / u
    else if (u <= 6.2d+153) then
        tmp = v / -t1
    else
        tmp = v / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.36e+112) {
		tmp = v / u;
	} else if (u <= 6.2e+153) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.36e+112:
		tmp = v / u
	elif u <= 6.2e+153:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.36e+112)
		tmp = Float64(v / u);
	elseif (u <= 6.2e+153)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.36e+112)
		tmp = v / u;
	elseif (u <= 6.2e+153)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.36e+112], N[(v / u), $MachinePrecision], If[LessEqual[u, 6.2e+153], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.3600000000000001e112

   1. Initial program 88.2%

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

      1. times-frac 97.1%

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

      2. distribute-frac-neg 97.1%

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

      3. distribute-neg-frac2 97.1%

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

      4. +-commutative 97.1%

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

      5. distribute-neg-in 97.1%

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

      6. unsub-neg 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t1 around 0 86.5%

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

   6. Taylor expanded in t1 around inf 44.6%

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

      1. associate-*r/ 44.6%

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

      2. mul-1-neg 44.6%

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

   8. Simplified 44.6%

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

      1. div-inv 44.6%

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

      2. add-sqr-sqrt 13.3%

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

      3. sqrt-unprod 46.4%

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

      4. sqr-neg 46.4%

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

      5. sqrt-unprod 31.2%

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

      6. add-sqr-sqrt 44.7%

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

   10. Applied egg-rr 44.7%

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

       1. associate-*r/ 44.7%

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

       2. *-rgt-identity 44.7%

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

   12. Simplified 44.7%

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

   if -1.3600000000000001e112 < u < 6.2e153

   1. Initial program 66.7%

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

      1. associate-*l/ 72.3%

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

      2. *-commutative 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   7. Simplified 71.6%

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

   if 6.2e153 < u

   1. Initial program 83.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 99.9%

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

   6. Taylor expanded in t1 around inf 40.3%

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

      1. associate-*r/ 40.3%

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

      2. mul-1-neg 40.3%

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

   8. Simplified 40.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.36 \cdot 10^{+112}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+179} \lor \neg \left(t1 \leq 1.2 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.4e+179) (not (<= t1 1.2e+170))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.4e+179) || !(t1 <= 1.2e+170)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.4d+179)) .or. (.not. (t1 <= 1.2d+170))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.4e+179) || !(t1 <= 1.2e+170)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.4e+179) or not (t1 <= 1.2e+170):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.4e+179) || !(t1 <= 1.2e+170))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.4e+179) || ~((t1 <= 1.2e+170)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.4e+179], N[Not[LessEqual[t1, 1.2e+170]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.4e179 or 1.2e170 < t1

   1. Initial program 34.8%

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

      1. associate-*l/ 36.4%

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

      2. *-commutative 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in t1 around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. neg-mul-1 99.8%

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

   7. Simplified 99.8%

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

      1. neg-sub0 99.8%

         \[\leadsto \frac{\color{blue}{0 - v}}{t1} \]

      2. sub-neg 99.8%

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

      3. add-sqr-sqrt 57.6%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

      4. sqrt-unprod 55.1%

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

      5. sqr-neg 55.1%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{v \cdot v}}}{t1} \]

      6. sqrt-unprod 12.6%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

      7. add-sqr-sqrt 35.3%

         \[\leadsto \frac{0 + \color{blue}{v}}{t1} \]

   9. Applied egg-rr 35.3%

      \[\leadsto \frac{\color{blue}{0 + v}}{t1} \]
   10. Step-by-step derivation

       1. +-lft-identity 35.3%

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

   11. Simplified 35.3%

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

   if -1.4e179 < t1 < 1.2e170

   1. Initial program 84.5%

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

      1. times-frac 96.5%

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

      2. distribute-frac-neg 96.5%

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

      3. distribute-neg-frac2 96.5%

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

      4. +-commutative 96.5%

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

      5. distribute-neg-in 96.5%

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

      6. unsub-neg 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in t1 around 0 60.7%

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

   6. Taylor expanded in t1 around inf 19.8%

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

      1. associate-*r/ 19.8%

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

      2. mul-1-neg 19.8%

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

   8. Simplified 19.8%

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

      1. div-inv 19.8%

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

      2. add-sqr-sqrt 8.7%

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

      3. sqrt-unprod 22.5%

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

      4. sqr-neg 22.5%

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

      5. sqrt-unprod 11.3%

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

      6. add-sqr-sqrt 18.2%

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

   10. Applied egg-rr 18.2%

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

       1. associate-*r/ 18.2%

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

       2. *-rgt-identity 18.2%

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

   12. Simplified 18.2%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+179} \lor \neg \left(t1 \leq 1.2 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. times-frac 97.4%

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

   2. distribute-frac-neg 97.4%

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

   3. distribute-neg-frac2 97.4%

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

   4. +-commutative 97.4%

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

   5. distribute-neg-in 97.4%

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

   6. unsub-neg 97.4%

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

3. Simplified 97.4%

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

Alternative 13: 61.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-*l/ 75.5%

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

   2. *-commutative 75.5%

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

3. Simplified 75.5%

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

   1. associate-*r/ 72.1%

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

   2. *-commutative 72.1%

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

   3. times-frac 97.4%

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

   4. frac-2neg 97.4%

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

   5. +-commutative 97.4%

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

   6. distribute-neg-in 97.4%

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

   7. sub-neg 97.4%

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

   8. associate-*r/ 97.8%

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

   9. add-sqr-sqrt 47.0%

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

   10. sqrt-unprod 43.5%

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

   11. sqr-neg 43.5%

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

   12. sqrt-unprod 19.6%

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

   13. add-sqr-sqrt 33.6%

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

   14. sub-neg 33.6%

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

   15. +-commutative 33.6%

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

   16. add-sqr-sqrt 14.0%

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

   17. sqrt-unprod 44.6%

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

   18. sqr-neg 44.6%

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

   19. sqrt-unprod 39.6%

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

   20. add-sqr-sqrt 18.8%

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

   21. sqrt-unprod 43.1%

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

   22. sqr-neg 43.1%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in t1 around inf 65.6%

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

   1. mul-1-neg 65.6%

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

9. Simplified 65.6%

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

10. Final simplification 65.6%

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

Alternative 14: 14.1% 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 72.1%

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

   1. associate-*l/ 75.5%

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

   2. *-commutative 75.5%

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

3. Simplified 75.5%

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

5. Taylor expanded in t1 around inf 56.5%

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

   1. associate-*r/ 56.5%

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

   2. neg-mul-1 56.5%

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

7. Simplified 56.5%

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

   1. neg-sub0 56.5%

      \[\leadsto \frac{\color{blue}{0 - v}}{t1} \]

   2. sub-neg 56.5%

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

   3. add-sqr-sqrt 29.9%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{t1} \]

   4. sqrt-unprod 33.9%

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

   5. sqr-neg 33.9%

      \[\leadsto \frac{0 + \sqrt{\color{blue}{v \cdot v}}}{t1} \]

   6. sqrt-unprod 4.8%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{v} \cdot \sqrt{v}}}{t1} \]

   7. add-sqr-sqrt 11.6%

      \[\leadsto \frac{0 + \color{blue}{v}}{t1} \]

9. Applied egg-rr 11.6%

   \[\leadsto \frac{\color{blue}{0 + v}}{t1} \]
10. Step-by-step derivation

    1. +-lft-identity 11.6%

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

11. Simplified 11.6%

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

Reproduce

herbie shell --seed 2024182 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))