Rosa's DopplerBench

Percentage Accurate: 72.7% → 98.3%

Time: 10.1s

Alternatives: 15

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: 72.7% 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* 73.0%

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

   2. distribute-lft-neg-out 73.0%

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

   3. distribute-rgt-neg-in 73.0%

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

   4. associate-/r* 83.4%

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

   5. distribute-neg-frac2 83.4%

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

3. Simplified 83.4%

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

   1. associate-*r/ 97.9%

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

   2. +-commutative 97.9%

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

   3. distribute-neg-in 97.9%

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

   4. sub-neg 97.9%

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

   5. associate-*l/ 97.4%

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

   6. frac-2neg 97.4%

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

   7. associate-*r/ 97.5%

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

   8. remove-double-neg 97.5%

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

   9. sub-neg 97.5%

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

   10. distribute-neg-in 97.5%

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

   11. +-commutative 97.5%

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

   12. frac-2neg 97.5%

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

   13. add-sqr-sqrt 45.0%

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

   14. sqrt-unprod 42.2%

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

   15. sqr-neg 42.2%

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

   16. sqrt-unprod 19.6%

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

   17. add-sqr-sqrt 35.2%

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

   18. add-sqr-sqrt 15.8%

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

   19. sqrt-unprod 55.9%

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

6. Applied egg-rr 97.5%

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

Alternative 2: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{u \cdot \frac{v}{t1} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 10^{+122}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= t1 -2e+156)
     (/ (- (* u (/ v t1)) v) (+ t1 u))
     (if (<= t1 1e+122) (* t1 (/ (/ v (+ t1 u)) t_1)) (/ v t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -2e+156) {
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	} else if (t1 <= 1e+122) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = v / t_1;
	}
	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) :: tmp
    t_1 = -u - t1
    if (t1 <= (-2d+156)) then
        tmp = ((u * (v / t1)) - v) / (t1 + u)
    else if (t1 <= 1d+122) then
        tmp = t1 * ((v / (t1 + u)) / t_1)
    else
        tmp = v / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -2e+156) {
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	} else if (t1 <= 1e+122) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = v / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if t1 <= -2e+156:
		tmp = ((u * (v / t1)) - v) / (t1 + u)
	elif t1 <= 1e+122:
		tmp = t1 * ((v / (t1 + u)) / t_1)
	else:
		tmp = v / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (t1 <= -2e+156)
		tmp = Float64(Float64(Float64(u * Float64(v / t1)) - v) / Float64(t1 + u));
	elseif (t1 <= 1e+122)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / t_1));
	else
		tmp = Float64(v / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (t1 <= -2e+156)
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	elseif (t1 <= 1e+122)
		tmp = t1 * ((v / (t1 + u)) / t_1);
	else
		tmp = v / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[t1, -2e+156], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1e+122], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(v / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2e156

   1. Initial program 41.8%

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

      1. associate-/l* 42.7%

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

      2. distribute-lft-neg-out 42.7%

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

      3. distribute-rgt-neg-in 42.7%

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

      4. associate-/r* 66.9%

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

      5. distribute-neg-frac2 66.9%

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

   3. Simplified 66.9%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-*l/ 99.9%

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

      6. frac-2neg 99.9%

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

      7. associate-*r/ 100.0%

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

      8. remove-double-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. +-commutative 100.0%

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

      12. frac-2neg 100.0%

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

      13. add-sqr-sqrt 99.6%

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

      14. sqrt-unprod 2.1%

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

      15. sqr-neg 2.1%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 41.7%

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

      18. add-sqr-sqrt 34.3%

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

      19. sqrt-unprod 42.7%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 97.8%

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

      1. +-commutative 97.8%

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

      2. mul-1-neg 97.8%

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

      3. sub-neg 97.8%

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

      4. associate-/l* 97.8%

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

   9. Simplified 97.8%

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

   if -2e156 < t1 < 1.00000000000000001e122

   1. Initial program 81.5%

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

      1. associate-/l* 81.9%

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

      2. distribute-lft-neg-out 81.9%

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

      3. distribute-rgt-neg-in 81.9%

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

      4. associate-/r* 89.3%

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

      5. distribute-neg-frac2 89.3%

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

   3. Simplified 89.3%

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

   if 1.00000000000000001e122 < t1

   1. Initial program 48.6%

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

      1. associate-/l* 52.1%

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

      2. distribute-lft-neg-out 52.1%

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

      3. distribute-rgt-neg-in 52.1%

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

      4. associate-/r* 67.1%

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

      5. distribute-neg-frac2 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in t1 around inf 63.4%

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

   6. Taylor expanded in v around 0 85.4%

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

      1. associate-*r/ 85.4%

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

      2. mul-1-neg 85.4%

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

      3. +-commutative 85.4%

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

   8. Simplified 85.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{u \cdot \frac{v}{t1} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 10^{+122}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{-55}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 0.006:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot \left(-t1\right)}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.5e-55)
   (* t1 (/ (/ v (+ t1 u)) (- u)))
   (if (<= u 0.006) (- (/ v t1)) (/ (* (/ v u) (- t1)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.5e-55) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 0.006) {
		tmp = -(v / t1);
	} else {
		tmp = ((v / u) * -t1) / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.5d-55)) then
        tmp = t1 * ((v / (t1 + u)) / -u)
    else if (u <= 0.006d0) then
        tmp = -(v / t1)
    else
        tmp = ((v / u) * -t1) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.5e-55) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 0.006) {
		tmp = -(v / t1);
	} else {
		tmp = ((v / u) * -t1) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.5e-55:
		tmp = t1 * ((v / (t1 + u)) / -u)
	elif u <= 0.006:
		tmp = -(v / t1)
	else:
		tmp = ((v / u) * -t1) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.5e-55)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(-u)));
	elseif (u <= 0.006)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(Float64(Float64(v / u) * Float64(-t1)) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.5e-55)
		tmp = t1 * ((v / (t1 + u)) / -u);
	elseif (u <= 0.006)
		tmp = -(v / t1);
	else
		tmp = ((v / u) * -t1) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.5e-55], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 0.006], (-N[(v / t1), $MachinePrecision]), N[(N[(N[(v / u), $MachinePrecision] * (-t1)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.50000000000000008e-55

   1. Initial program 77.2%

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

      1. associate-/l* 76.0%

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

      2. distribute-lft-neg-out 76.0%

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

      3. distribute-rgt-neg-in 76.0%

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

      4. associate-/r* 86.5%

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

      5. distribute-neg-frac2 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in t1 around 0 81.3%

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

   if -1.50000000000000008e-55 < u < 0.0060000000000000001

   1. Initial program 67.0%

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

      1. associate-/l* 67.3%

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

      2. distribute-lft-neg-out 67.3%

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

      3. distribute-rgt-neg-in 67.3%

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

      4. associate-/r* 76.4%

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

      5. distribute-neg-frac2 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in t1 around inf 80.8%

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

      1. associate-*r/ 80.8%

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

      2. neg-mul-1 80.8%

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

   7. Simplified 80.8%

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

   if 0.0060000000000000001 < u

   1. Initial program 74.6%

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

      1. associate-/l* 79.5%

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

      2. distribute-lft-neg-out 79.5%

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

      3. distribute-rgt-neg-in 79.5%

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

      4. associate-/r* 92.0%

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

      5. distribute-neg-frac2 92.0%

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

   3. Simplified 92.0%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-*l/ 96.9%

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

      6. frac-2neg 96.9%

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

      7. associate-*r/ 96.9%

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

      8. remove-double-neg 96.9%

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

      9. sub-neg 96.9%

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

      10. distribute-neg-in 96.9%

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

      11. +-commutative 96.9%

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

      12. frac-2neg 96.9%

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

      13. add-sqr-sqrt 38.2%

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

      14. sqrt-unprod 56.1%

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

      15. sqr-neg 56.1%

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

      16. sqrt-unprod 28.2%

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

      17. add-sqr-sqrt 53.4%

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

      18. add-sqr-sqrt 1.6%

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

      19. sqrt-unprod 77.3%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in t1 around 0 81.6%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 81.6%

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

      4. associate-*l/ 85.7%

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

      5. associate-*r/ 85.7%

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

      6. *-commutative 85.7%

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

      7. associate-*l* 85.7%

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

      8. mul-1-neg 85.7%

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

   9. Simplified 85.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{-55}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 0.006:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot \left(-t1\right)}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.6 \cdot 10^{-61} \lor \neg \left(u \leq 7.2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9.6e-61) (not (<= u 7.2e-11)))
   (/ (* t1 (/ v u)) (- u))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.6e-61) || !(u <= 7.2e-11)) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-9.6d-61)) .or. (.not. (u <= 7.2d-11))) then
        tmp = (t1 * (v / u)) / -u
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.6e-61) || !(u <= 7.2e-11)) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9.6e-61) or not (u <= 7.2e-11):
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9.6e-61) || !(u <= 7.2e-11))
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9.6e-61) || ~((u <= 7.2e-11)))
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9.6e-61], N[Not[LessEqual[u, 7.2e-11]], $MachinePrecision]], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -9.6000000000000004e-61 or 7.19999999999999969e-11 < u

   1. Initial program 76.4%

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

      1. associate-*l/ 74.9%

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

      2. *-commutative 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in t1 around 0 68.3%

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

      1. distribute-frac-neg 68.3%

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

      2. distribute-rgt-neg-out 68.3%

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

      3. add-sqr-sqrt 37.1%

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

      4. sqrt-unprod 46.6%

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

      5. sqr-neg 46.6%

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

      6. sqrt-unprod 20.7%

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

      7. add-sqr-sqrt 48.8%

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

      8. *-commutative 48.8%

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

      9. associate-/r* 48.0%

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

      10. associate-*l/ 47.9%

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

   7. Applied egg-rr 79.1%

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

   8. Taylor expanded in t1 around 0 77.7%

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

      1. associate-/l* 81.5%

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

   10. Simplified 81.5%

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

   if -9.6000000000000004e-61 < u < 7.19999999999999969e-11

   1. Initial program 66.4%

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

      1. associate-/l* 66.7%

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

      2. distribute-lft-neg-out 66.7%

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

      3. distribute-rgt-neg-in 66.7%

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

      4. associate-/r* 76.0%

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

      5. distribute-neg-frac2 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in t1 around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   7. Simplified 81.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.6 \cdot 10^{-61} \lor \neg \left(u \leq 7.2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{-39} \lor \neg \left(u \leq 1.8 \cdot 10^{-6}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.6e-39) (not (<= u 1.8e-6)))
   (* t1 (/ (/ v u) (- u)))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.6e-39) || !(u <= 1.8e-6)) {
		tmp = t1 * ((v / u) / -u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.6d-39)) .or. (.not. (u <= 1.8d-6))) then
        tmp = t1 * ((v / u) / -u)
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.6e-39) || !(u <= 1.8e-6)) {
		tmp = t1 * ((v / u) / -u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.6e-39) or not (u <= 1.8e-6):
		tmp = t1 * ((v / u) / -u)
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.6e-39) || !(u <= 1.8e-6))
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.6e-39) || ~((u <= 1.8e-6)))
		tmp = t1 * ((v / u) / -u);
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.6e-39], N[Not[LessEqual[u, 1.8e-6]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -5.6000000000000003e-39 or 1.79999999999999992e-6 < u

   1. Initial program 75.6%

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

      1. associate-*l/ 74.0%

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

      2. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in t1 around 0 68.6%

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

      1. distribute-frac-neg 68.6%

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

      2. distribute-rgt-neg-out 68.6%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 46.8%

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

      5. sqr-neg 46.8%

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

      6. sqrt-unprod 21.4%

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

      7. add-sqr-sqrt 50.4%

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

      8. *-commutative 50.4%

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

      9. associate-/r* 49.6%

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

      10. associate-*l/ 49.4%

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

   7. Applied egg-rr 79.8%

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

   8. Taylor expanded in t1 around 0 78.3%

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

      1. associate-/l* 82.2%

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

      2. un-div-inv 82.2%

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

      3. associate-/l* 80.4%

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

      4. un-div-inv 80.4%

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

   10. Applied egg-rr 80.4%

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

   if -5.6000000000000003e-39 < u < 1.79999999999999992e-6

   1. Initial program 67.8%

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

      1. associate-/l* 67.3%

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

      2. distribute-lft-neg-out 67.3%

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

      3. distribute-rgt-neg-in 67.3%

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

      4. associate-/r* 76.2%

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

      5. distribute-neg-frac2 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in t1 around inf 79.7%

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

      1. associate-*r/ 79.7%

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

      2. neg-mul-1 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{-39} \lor \neg \left(u \leq 1.8 \cdot 10^{-6}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{-56}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 4.1 \cdot 10^{-11}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -8.5e-56)
   (* t1 (/ (/ v (+ t1 u)) (- u)))
   (if (<= u 4.1e-11) (- (/ v t1)) (/ (* t1 (/ v u)) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.5e-56) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 4.1e-11) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-8.5d-56)) then
        tmp = t1 * ((v / (t1 + u)) / -u)
    else if (u <= 4.1d-11) then
        tmp = -(v / t1)
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.5e-56) {
		tmp = t1 * ((v / (t1 + u)) / -u);
	} else if (u <= 4.1e-11) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -8.5e-56:
		tmp = t1 * ((v / (t1 + u)) / -u)
	elif u <= 4.1e-11:
		tmp = -(v / t1)
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -8.5e-56)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(-u)));
	elseif (u <= 4.1e-11)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -8.5e-56)
		tmp = t1 * ((v / (t1 + u)) / -u);
	elseif (u <= 4.1e-11)
		tmp = -(v / t1);
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -8.5e-56], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.1e-11], (-N[(v / t1), $MachinePrecision]), N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.49999999999999932e-56

   1. Initial program 77.2%

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

      1. associate-/l* 76.0%

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

      2. distribute-lft-neg-out 76.0%

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

      3. distribute-rgt-neg-in 76.0%

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

      4. associate-/r* 86.5%

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

      5. distribute-neg-frac2 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in t1 around 0 81.3%

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

   if -8.49999999999999932e-56 < u < 4.1000000000000001e-11

   1. Initial program 67.0%

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

      1. associate-/l* 67.3%

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

      2. distribute-lft-neg-out 67.3%

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

      3. distribute-rgt-neg-in 67.3%

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

      4. associate-/r* 76.4%

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

      5. distribute-neg-frac2 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in t1 around inf 80.8%

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

      1. associate-*r/ 80.8%

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

      2. neg-mul-1 80.8%

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

   7. Simplified 80.8%

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

   if 4.1000000000000001e-11 < u

   1. Initial program 74.6%

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

      1. associate-*l/ 75.0%

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

      2. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in t1 around 0 70.4%

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

      1. distribute-frac-neg 70.4%

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

      2. distribute-rgt-neg-out 70.4%

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

      3. add-sqr-sqrt 36.1%

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

      4. sqrt-unprod 52.9%

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

      5. sqr-neg 52.9%

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

      6. sqrt-unprod 25.3%

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

      7. add-sqr-sqrt 53.8%

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

      8. *-commutative 53.8%

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

      9. associate-/r* 53.6%

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

      10. associate-*l/ 53.4%

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

   7. Applied egg-rr 82.0%

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

   8. Taylor expanded in t1 around 0 81.6%

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

      1. associate-/l* 85.6%

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

   10. Simplified 85.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{-56}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq 4.1 \cdot 10^{-11}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.7e-59)
   (/ (/ t1 (/ u v)) (- u))
   (if (<= u 2.5e-11) (- (/ v t1)) (/ (* t1 (/ v u)) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.7e-59) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= 2.5e-11) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.7d-59)) then
        tmp = (t1 / (u / v)) / -u
    else if (u <= 2.5d-11) then
        tmp = -(v / t1)
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.7e-59) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= 2.5e-11) {
		tmp = -(v / t1);
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.7e-59:
		tmp = (t1 / (u / v)) / -u
	elif u <= 2.5e-11:
		tmp = -(v / t1)
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.7e-59)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(-u));
	elseif (u <= 2.5e-11)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.7e-59)
		tmp = (t1 / (u / v)) / -u;
	elseif (u <= 2.5e-11)
		tmp = -(v / t1);
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.7e-59], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, 2.5e-11], (-N[(v / t1), $MachinePrecision]), N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.70000000000000009e-59

   1. Initial program 77.7%

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

      1. associate-*l/ 74.9%

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

      2. *-commutative 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in t1 around 0 66.8%

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

      1. distribute-frac-neg 66.8%

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

      2. distribute-rgt-neg-out 66.8%

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

      3. add-sqr-sqrt 37.9%

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

      4. sqrt-unprod 41.8%

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

      5. sqr-neg 41.8%

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

      6. sqrt-unprod 17.1%

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

      7. add-sqr-sqrt 45.0%

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

      8. *-commutative 45.0%

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

      9. associate-/r* 43.7%

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

      10. associate-*l/ 43.6%

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

   7. Applied egg-rr 76.9%

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

   8. Taylor expanded in t1 around 0 74.8%

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

      1. associate-/l* 78.4%

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

   10. Applied egg-rr 78.4%

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

       1. *-rgt-identity 78.4%

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

       2. associate-*r/ 78.3%

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

       3. *-commutative 78.3%

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

       4. associate-/r/ 79.5%

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

       5. associate-*r/ 79.5%

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

       6. *-rgt-identity 79.5%

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

   12. Simplified 79.5%

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

   if -1.70000000000000009e-59 < u < 2.50000000000000009e-11

   1. Initial program 66.4%

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

      1. associate-/l* 66.7%

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

      2. distribute-lft-neg-out 66.7%

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

      3. distribute-rgt-neg-in 66.7%

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

      4. associate-/r* 76.0%

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

      5. distribute-neg-frac2 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in t1 around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   7. Simplified 81.3%

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

   if 2.50000000000000009e-11 < u

   1. Initial program 74.6%

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

      1. associate-*l/ 75.0%

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

      2. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in t1 around 0 70.4%

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

      1. distribute-frac-neg 70.4%

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

      2. distribute-rgt-neg-out 70.4%

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

      3. add-sqr-sqrt 36.1%

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

      4. sqrt-unprod 52.9%

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

      5. sqr-neg 52.9%

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

      6. sqrt-unprod 25.3%

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

      7. add-sqr-sqrt 53.8%

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

      8. *-commutative 53.8%

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

      9. associate-/r* 53.6%

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

      10. associate-*l/ 53.4%

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

   7. Applied egg-rr 82.0%

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

   8. Taylor expanded in t1 around 0 81.6%

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

      1. associate-/l* 85.6%

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

   10. Simplified 85.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{-12}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.7e-59)
   (* (/ v u) (/ t1 (- u)))
   (if (<= u 5e-12) (- (/ v t1)) (* t1 (/ (/ v u) (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.7e-59) {
		tmp = (v / u) * (t1 / -u);
	} else if (u <= 5e-12) {
		tmp = -(v / t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.7d-59)) then
        tmp = (v / u) * (t1 / -u)
    else if (u <= 5d-12) then
        tmp = -(v / t1)
    else
        tmp = t1 * ((v / u) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.7e-59) {
		tmp = (v / u) * (t1 / -u);
	} else if (u <= 5e-12) {
		tmp = -(v / t1);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.7e-59:
		tmp = (v / u) * (t1 / -u)
	elif u <= 5e-12:
		tmp = -(v / t1)
	else:
		tmp = t1 * ((v / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.7e-59)
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	elseif (u <= 5e-12)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.7e-59)
		tmp = (v / u) * (t1 / -u);
	elseif (u <= 5e-12)
		tmp = -(v / t1);
	else
		tmp = t1 * ((v / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.7e-59], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 5e-12], (-N[(v / t1), $MachinePrecision]), N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.70000000000000009e-59

   1. Initial program 77.7%

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

      1. associate-*l/ 74.9%

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

      2. *-commutative 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in t1 around 0 66.8%

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

      1. distribute-frac-neg 66.8%

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

      2. distribute-rgt-neg-out 66.8%

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

      3. add-sqr-sqrt 37.9%

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

      4. sqrt-unprod 41.8%

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

      5. sqr-neg 41.8%

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

      6. sqrt-unprod 17.1%

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

      7. add-sqr-sqrt 45.0%

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

      8. *-commutative 45.0%

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

      9. associate-/r* 43.7%

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

      10. associate-*l/ 43.6%

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

   7. Applied egg-rr 76.9%

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

   8. Taylor expanded in t1 around 0 74.8%

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

      1. associate-/l/ 66.9%

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

      2. *-commutative 66.9%

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

      3. times-frac 76.3%

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

   10. Applied egg-rr 76.3%

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

   if -1.70000000000000009e-59 < u < 4.9999999999999997e-12

   1. Initial program 66.4%

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

      1. associate-/l* 66.7%

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

      2. distribute-lft-neg-out 66.7%

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

      3. distribute-rgt-neg-in 66.7%

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

      4. associate-/r* 76.0%

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

      5. distribute-neg-frac2 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in t1 around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   7. Simplified 81.3%

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

   if 4.9999999999999997e-12 < u

   1. Initial program 74.6%

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

      1. associate-*l/ 75.0%

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

      2. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in t1 around 0 70.4%

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

      1. distribute-frac-neg 70.4%

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

      2. distribute-rgt-neg-out 70.4%

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

      3. add-sqr-sqrt 36.1%

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

      4. sqrt-unprod 52.9%

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

      5. sqr-neg 52.9%

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

      6. sqrt-unprod 25.3%

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

      7. add-sqr-sqrt 53.8%

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

      8. *-commutative 53.8%

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

      9. associate-/r* 53.6%

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

      10. associate-*l/ 53.4%

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

   7. Applied egg-rr 82.0%

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

   8. Taylor expanded in t1 around 0 81.6%

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

      1. associate-/l* 85.6%

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

      2. un-div-inv 85.6%

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

      3. associate-/l* 83.4%

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

      4. un-div-inv 83.5%

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

   10. Applied egg-rr 83.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{-12}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+111} \lor \neg \left(u \leq 9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.15e+111) (not (<= u 9e+43))) (/ v (- u)) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.15e+111) || !(u <= 9e+43)) {
		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 <= (-1.15d+111)) .or. (.not. (u <= 9d+43))) 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 <= -1.15e+111) || !(u <= 9e+43)) {
		tmp = v / -u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.15e+111) or not (u <= 9e+43):
		tmp = v / -u
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.15e+111) || !(u <= 9e+43))
		tmp = Float64(v / Float64(-u));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.15e+111) || ~((u <= 9e+43)))
		tmp = v / -u;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.15e+111], N[Not[LessEqual[u, 9e+43]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -1.15000000000000001e111 or 9e43 < u

   1. Initial program 72.5%

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

      1. associate-/l* 73.9%

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

      2. distribute-lft-neg-out 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

      4. associate-/r* 88.0%

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

      5. distribute-neg-frac2 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in t1 around inf 45.2%

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

   6. Taylor expanded in t1 around 0 40.5%

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

      1. associate-*r/ 40.5%

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

      2. mul-1-neg 40.5%

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

   8. Simplified 40.5%

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

   if -1.15000000000000001e111 < u < 9e43

   1. Initial program 71.8%

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

      1. associate-/l* 72.4%

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

      2. distribute-lft-neg-out 72.4%

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

      3. distribute-rgt-neg-in 72.4%

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

      4. associate-/r* 80.6%

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

      5. distribute-neg-frac2 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in t1 around inf 68.5%

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

      1. associate-*r/ 68.5%

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

      2. neg-mul-1 68.5%

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

   7. Simplified 68.5%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+111} \lor \neg \left(u \leq 9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+111} \lor \neg \left(u \leq 9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.15e+111) (not (<= u 9e+43))) (/ v u) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.15e+111) || !(u <= 9e+43)) {
		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 <= (-1.15d+111)) .or. (.not. (u <= 9d+43))) 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 <= -1.15e+111) || !(u <= 9e+43)) {
		tmp = v / u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.15e+111) or not (u <= 9e+43):
		tmp = v / u
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.15e+111) || !(u <= 9e+43))
		tmp = Float64(v / u);
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.15e+111) || ~((u <= 9e+43)))
		tmp = v / u;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.15e+111], N[Not[LessEqual[u, 9e+43]], $MachinePrecision]], N[(v / u), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -1.15000000000000001e111 or 9e43 < u

   1. Initial program 72.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around 0 69.0%

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

      1. frac-2neg 69.0%

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

      2. div-inv 69.1%

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

      3. remove-double-neg 69.1%

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

      4. *-commutative 69.1%

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

      5. distribute-rgt-neg-in 69.1%

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

      6. distribute-neg-in 69.1%

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

      7. add-sqr-sqrt 31.4%

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

      8. sqrt-unprod 69.1%

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

      9. sqr-neg 69.1%

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

      10. sqrt-unprod 37.6%

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

      11. add-sqr-sqrt 69.1%

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

      12. sub-neg 69.1%

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

   7. Applied egg-rr 69.1%

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

      1. associate-*r/ 69.0%

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

      2. *-rgt-identity 69.0%

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

      3. associate-/r* 72.3%

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

   9. Simplified 72.3%

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

   10. Taylor expanded in t1 around inf 39.6%

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

   if -1.15000000000000001e111 < u < 9e43

   1. Initial program 71.8%

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

      1. associate-/l* 72.4%

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

      2. distribute-lft-neg-out 72.4%

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

      3. distribute-rgt-neg-in 72.4%

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

      4. associate-/r* 80.6%

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

      5. distribute-neg-frac2 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in t1 around inf 68.5%

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

      1. associate-*r/ 68.5%

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

      2. neg-mul-1 68.5%

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

   7. Simplified 68.5%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+111} \lor \neg \left(u \leq 9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 22.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{+139} \lor \neg \left(t1 \leq 1.45 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.1e+139) (not (<= t1 1.45e+86))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.1e+139) || !(t1 <= 1.45e+86)) {
		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 <= (-3.1d+139)) .or. (.not. (t1 <= 1.45d+86))) 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 <= -3.1e+139) || !(t1 <= 1.45e+86)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.1e+139) or not (t1 <= 1.45e+86):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.1e+139) || !(t1 <= 1.45e+86))
		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 <= -3.1e+139) || ~((t1 <= 1.45e+86)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.1e+139], N[Not[LessEqual[t1, 1.45e+86]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.1e139 or 1.44999999999999995e86 < t1

   1. Initial program 50.4%

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

      1. associate-/l* 53.8%

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

      2. distribute-lft-neg-out 53.8%

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

      3. distribute-rgt-neg-in 53.8%

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

      4. associate-/r* 69.7%

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

      5. distribute-neg-frac2 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in t1 around inf 66.6%

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

      1. clear-num 65.0%

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

      2. un-div-inv 65.1%

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

      3. div-inv 65.0%

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

      4. add-sqr-sqrt 22.9%

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

      5. sqrt-unprod 42.8%

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

      6. sqr-neg 42.8%

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

      7. sqrt-unprod 26.9%

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

      8. add-sqr-sqrt 42.5%

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

      9. clear-num 42.5%

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

   7. Applied egg-rr 42.5%

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

      1. associate-/r* 39.8%

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

      2. *-inverses 39.8%

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

      3. +-commutative 39.8%

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

   9. Simplified 39.8%

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

   10. Taylor expanded in u around 0 38.4%

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

   if -3.1e139 < t1 < 1.44999999999999995e86

   1. Initial program 81.9%

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

      1. associate-*l/ 84.6%

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

      2. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in t1 around 0 58.6%

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

      1. frac-2neg 58.6%

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

      2. div-inv 58.5%

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

      3. remove-double-neg 58.5%

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

      4. *-commutative 58.5%

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

      5. distribute-rgt-neg-in 58.5%

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

      6. distribute-neg-in 58.5%

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

      7. add-sqr-sqrt 29.8%

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

      8. sqrt-unprod 58.5%

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

      9. sqr-neg 58.5%

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

      10. sqrt-unprod 28.0%

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

      11. add-sqr-sqrt 56.9%

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

      12. sub-neg 56.9%

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

   7. Applied egg-rr 56.9%

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

      1. associate-*r/ 56.9%

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

      2. *-rgt-identity 56.9%

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

      3. associate-/r* 61.7%

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

   9. Simplified 61.7%

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

   10. Taylor expanded in t1 around inf 17.8%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{+139} \lor \neg \left(t1 \leq 1.45 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u_, v_, t1_] := N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / N[((-u) - t1), $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. Final simplification 97.4%

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

Alternative 13: 62.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 4.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 4.4e+146) (/ v (- u t1)) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 4.4e+146) {
		tmp = v / (u - t1);
	} 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 (v <= 4.4d+146) then
        tmp = v / (u - t1)
    else
        tmp = -(v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 4.4e+146) {
		tmp = v / (u - t1);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 4.4e+146:
		tmp = v / (u - t1)
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 4.4e+146)
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 4.4e+146)
		tmp = v / (u - t1);
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 4.4e+146], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if v < 4.3999999999999996e146

   1. Initial program 74.4%

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

      1. times-frac 98.4%

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

      2. distribute-frac-neg 98.4%

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

      3. distribute-neg-frac2 98.4%

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

      4. +-commutative 98.4%

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

      5. distribute-neg-in 98.4%

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

      6. unsub-neg 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in t1 around inf 65.3%

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

      1. clear-num 64.8%

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

      2. frac-times 52.8%

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

      3. *-commutative 52.8%

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

      4. *-un-lft-identity 52.8%

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

      5. add-sqr-sqrt 25.5%

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

      6. sqrt-unprod 55.5%

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

      7. sqr-neg 55.5%

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

      8. sqrt-unprod 27.5%

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

      9. add-sqr-sqrt 52.5%

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

   7. Applied egg-rr 52.5%

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

      1. associate-/l/ 62.3%

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

      2. associate-/r/ 62.9%

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

      3. associate-*l/ 60.6%

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

      4. *-commutative 60.6%

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

      5. associate-/l* 62.9%

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

      6. *-inverses 62.9%

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

      7. *-rgt-identity 62.9%

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

   9. Simplified 62.9%

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

   if 4.3999999999999996e146 < v

   1. Initial program 61.3%

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

      1. associate-/l* 64.3%

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

      2. distribute-lft-neg-out 64.3%

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

      3. distribute-rgt-neg-in 64.3%

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

      4. associate-/r* 85.4%

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

      5. distribute-neg-frac2 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in t1 around inf 50.2%

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

      1. associate-*r/ 50.2%

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

      2. neg-mul-1 50.2%

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

   7. Simplified 50.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 4.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.0% 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* 73.0%

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

   2. distribute-lft-neg-out 73.0%

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

   3. distribute-rgt-neg-in 73.0%

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

   4. associate-/r* 83.4%

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

   5. distribute-neg-frac2 83.4%

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

3. Simplified 83.4%

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

5. Taylor expanded in t1 around inf 51.4%

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

6. Taylor expanded in v around 0 61.0%

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

   1. associate-*r/ 61.0%

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

   2. mul-1-neg 61.0%

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

   3. +-commutative 61.0%

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

8. Simplified 61.0%

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

9. Final simplification 61.0%

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

Alternative 15: 13.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 73.0%

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

   2. distribute-lft-neg-out 73.0%

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

   3. distribute-rgt-neg-in 73.0%

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

   4. associate-/r* 83.4%

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

   5. distribute-neg-frac2 83.4%

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

3. Simplified 83.4%

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

5. Taylor expanded in t1 around inf 51.4%

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

   1. clear-num 50.8%

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

   2. un-div-inv 51.3%

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

   3. div-inv 51.4%

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

   4. add-sqr-sqrt 23.6%

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

   5. sqrt-unprod 32.7%

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

   6. sqr-neg 32.7%

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

   7. sqrt-unprod 14.7%

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

   8. add-sqr-sqrt 25.3%

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

   9. clear-num 25.3%

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

7. Applied egg-rr 25.3%

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

   1. associate-/r* 24.2%

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

   2. *-inverses 24.2%

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

   3. +-commutative 24.2%

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

9. Simplified 24.2%

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

10. Taylor expanded in u around 0 14.3%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))