Rosa's DopplerBench

Percentage Accurate: 73.4% → 98.0%

Time: 18.2s

Alternatives: 19

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% 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: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

   1. times-frac 98.7%

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

   2. distribute-frac-neg 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. +-commutative 98.7%

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

   5. distribute-neg-in 98.7%

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

   6. unsub-neg 98.7%

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

3. Simplified 98.7%

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

   1. frac-2neg 98.7%

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

   2. frac-2neg 98.7%

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

   3. frac-times 70.4%

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

   4. sub-neg 70.4%

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

   5. distribute-neg-in 70.4%

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

   6. +-commutative 70.4%

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

   7. remove-double-neg 70.4%

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

   8. frac-times 98.7%

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

   9. associate-*r/ 99.0%

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

   10. add-sqr-sqrt 48.2%

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

   11. sqrt-unprod 42.8%

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

   12. sqr-neg 42.8%

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

   13. sqrt-unprod 17.0%

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

   14. add-sqr-sqrt 37.2%

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

   15. add-sqr-sqrt 19.4%

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

   16. sqrt-unprod 58.4%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in t1 around inf 99.1%

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

8. Taylor expanded in v around 0 99.1%

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

   1. associate-*r/ 99.1%

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

   2. mul-1-neg 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

    \[\leadsto \frac{\frac{v}{-1 - \frac{u}{t1}}}{u + t1} \]
12. Add Preprocessing

Alternative 2: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{t\_1}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{+93}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u + t1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{u}{t1} - v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= t1 -3.7e+118)
     (/ v t_1)
     (if (<= t1 1.35e+93)
       (* t1 (/ (/ v (+ u t1)) t_1))
       (/ (- (* v (/ u t1)) v) (+ u t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -3.7e+118) {
		tmp = v / t_1;
	} else if (t1 <= 1.35e+93) {
		tmp = t1 * ((v / (u + t1)) / t_1);
	} else {
		tmp = ((v * (u / t1)) - v) / (u + t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -u - t1
    if (t1 <= (-3.7d+118)) then
        tmp = v / t_1
    else if (t1 <= 1.35d+93) then
        tmp = t1 * ((v / (u + t1)) / t_1)
    else
        tmp = ((v * (u / t1)) - v) / (u + t1)
    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 <= -3.7e+118) {
		tmp = v / t_1;
	} else if (t1 <= 1.35e+93) {
		tmp = t1 * ((v / (u + t1)) / t_1);
	} else {
		tmp = ((v * (u / t1)) - v) / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if t1 <= -3.7e+118:
		tmp = v / t_1
	elif t1 <= 1.35e+93:
		tmp = t1 * ((v / (u + t1)) / t_1)
	else:
		tmp = ((v * (u / t1)) - v) / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (t1 <= -3.7e+118)
		tmp = Float64(v / t_1);
	elseif (t1 <= 1.35e+93)
		tmp = Float64(t1 * Float64(Float64(v / Float64(u + t1)) / t_1));
	else
		tmp = Float64(Float64(Float64(v * Float64(u / t1)) - v) / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (t1 <= -3.7e+118)
		tmp = v / t_1;
	elseif (t1 <= 1.35e+93)
		tmp = t1 * ((v / (u + t1)) / t_1);
	else
		tmp = ((v * (u / t1)) - v) / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[t1, -3.7e+118], N[(v / t$95$1), $MachinePrecision], If[LessEqual[t1, 1.35e+93], N[(t1 * N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(v * N[(u / t1), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.69999999999999987e118

   1. Initial program 45.8%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 45.8%

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

      4. sub-neg 45.8%

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

      5. distribute-neg-in 45.8%

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

      6. +-commutative 45.8%

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

      7. remove-double-neg 45.8%

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

      8. frac-times 99.9%

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

      9. associate-*r/ 99.9%

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

      10. add-sqr-sqrt 99.2%

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

      11. sqrt-unprod 12.6%

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

      12. sqr-neg 12.6%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 41.3%

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

      15. add-sqr-sqrt 38.6%

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

      16. sqrt-unprod 42.2%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 92.8%

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

      1. mul-1-neg 92.8%

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

   9. Simplified 92.8%

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

   if -3.69999999999999987e118 < t1 < 1.35e93

   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* 83.8%

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

      2. distribute-lft-neg-out 83.8%

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

      3. distribute-rgt-neg-in 83.8%

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

      4. associate-/r* 90.3%

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

      5. distribute-neg-frac2 90.3%

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

   3. Simplified 90.3%

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

   if 1.35e93 < t1

   1. Initial program 44.6%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

      1. frac-2neg 99.9%

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

      2. frac-2neg 99.9%

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

      3. frac-times 44.6%

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

      4. sub-neg 44.6%

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

      5. distribute-neg-in 44.6%

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

      6. +-commutative 44.6%

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

      7. remove-double-neg 44.6%

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

      8. frac-times 99.9%

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

      9. associate-*r/ 99.9%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 8.4%

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

      12. sqr-neg 8.4%

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

      13. sqrt-unprod 31.9%

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

      14. add-sqr-sqrt 31.9%

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

      15. add-sqr-sqrt 0.1%

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

      16. sqrt-unprod 47.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 78.0%

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

      1. +-commutative 78.0%

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

      2. mul-1-neg 78.0%

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

      3. sub-neg 78.0%

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

      4. *-commutative 78.0%

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

      5. *-lft-identity 78.0%

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

      6. times-frac 87.8%

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

      7. /-rgt-identity 87.8%

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

   9. Simplified 87.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.7 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{+93}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u + t1}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{u}{t1} - v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u + t1}\\ \mathbf{if}\;u \leq -1.25 \cdot 10^{-11} \lor \neg \left(u \leq 7.2 \cdot 10^{-68}\right):\\ \;\;\;\;t\_1 \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ u t1))))
   (if (or (<= u -1.25e-11) (not (<= u 7.2e-68)))
     (* t_1 (/ (- t1) u))
     (* t_1 (+ (/ u t1) -1.0)))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (u + t1);
	double tmp;
	if ((u <= -1.25e-11) || !(u <= 7.2e-68)) {
		tmp = t_1 * (-t1 / u);
	} else {
		tmp = t_1 * ((u / t1) + -1.0);
	}
	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 = v / (u + t1)
    if ((u <= (-1.25d-11)) .or. (.not. (u <= 7.2d-68))) then
        tmp = t_1 * (-t1 / u)
    else
        tmp = t_1 * ((u / t1) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (u + t1);
	double tmp;
	if ((u <= -1.25e-11) || !(u <= 7.2e-68)) {
		tmp = t_1 * (-t1 / u);
	} else {
		tmp = t_1 * ((u / t1) + -1.0);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (u + t1)
	tmp = 0
	if (u <= -1.25e-11) or not (u <= 7.2e-68):
		tmp = t_1 * (-t1 / u)
	else:
		tmp = t_1 * ((u / t1) + -1.0)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(u + t1))
	tmp = 0.0
	if ((u <= -1.25e-11) || !(u <= 7.2e-68))
		tmp = Float64(t_1 * Float64(Float64(-t1) / u));
	else
		tmp = Float64(t_1 * Float64(Float64(u / t1) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (u + t1);
	tmp = 0.0;
	if ((u <= -1.25e-11) || ~((u <= 7.2e-68)))
		tmp = t_1 * (-t1 / u);
	else
		tmp = t_1 * ((u / t1) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[u, -1.25e-11], N[Not[LessEqual[u, 7.2e-68]], $MachinePrecision]], N[(t$95$1 * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.25000000000000005e-11 or 7.20000000000000015e-68 < u

   1. Initial program 76.3%

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

      1. times-frac 99.1%

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

      2. distribute-frac-neg 99.1%

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

      3. distribute-neg-frac2 99.1%

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

      4. +-commutative 99.1%

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

      5. distribute-neg-in 99.1%

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

      6. unsub-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around 0 83.2%

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

      1. associate-*r/ 83.2%

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

      2. mul-1-neg 83.2%

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

   7. Simplified 83.2%

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

   if -1.25000000000000005e-11 < u < 7.20000000000000015e-68

   1. Initial program 64.1%

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

      1. 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 82.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{-11} \lor \neg \left(u \leq 7.2 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{v}{u + t1} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u + t1} \cdot \left(\frac{u}{t1} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{-10} \lor \neg \left(u \leq 3.25 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{v}{u + t1} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{u}{t1} \cdot 2 + -1}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.7e-10) (not (<= u 3.25e-62)))
   (* (/ v (+ u t1)) (/ (- t1) u))
   (* v (/ (+ (* (/ u t1) 2.0) -1.0) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.7e-10) || !(u <= 3.25e-62)) {
		tmp = (v / (u + t1)) * (-t1 / u);
	} else {
		tmp = v * ((((u / t1) * 2.0) + -1.0) / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.7d-10)) .or. (.not. (u <= 3.25d-62))) then
        tmp = (v / (u + t1)) * (-t1 / u)
    else
        tmp = v * ((((u / t1) * 2.0d0) + (-1.0d0)) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.7e-10) || !(u <= 3.25e-62)) {
		tmp = (v / (u + t1)) * (-t1 / u);
	} else {
		tmp = v * ((((u / t1) * 2.0) + -1.0) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.7e-10) or not (u <= 3.25e-62):
		tmp = (v / (u + t1)) * (-t1 / u)
	else:
		tmp = v * ((((u / t1) * 2.0) + -1.0) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.7e-10) || !(u <= 3.25e-62))
		tmp = Float64(Float64(v / Float64(u + t1)) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(v * Float64(Float64(Float64(Float64(u / t1) * 2.0) + -1.0) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.7e-10) || ~((u <= 3.25e-62)))
		tmp = (v / (u + t1)) * (-t1 / u);
	else
		tmp = v * ((((u / t1) * 2.0) + -1.0) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.7e-10], N[Not[LessEqual[u, 3.25e-62]], $MachinePrecision]], N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(N[(N[(u / t1), $MachinePrecision] * 2.0), $MachinePrecision] + -1.0), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.70000000000000015e-10 or 3.25000000000000013e-62 < u

   1. Initial program 76.0%

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

      1. times-frac 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-neg-frac2 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. unsub-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in t1 around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

   7. Simplified 83.7%

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

   if -3.70000000000000015e-10 < u < 3.25000000000000013e-62

   1. Initial program 64.6%

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

      1. associate-*l/ 72.6%

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

      2. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in t1 around inf 82.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{-10} \lor \neg \left(u \leq 3.25 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{v}{u + t1} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{u}{t1} \cdot 2 + -1}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -0.0031 \lor \neg \left(u \leq 1.25 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{v}{u + t1} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -0.0031) (not (<= u 1.25e-62)))
   (* (/ v (+ u t1)) (/ (- t1) u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -0.0031) || !(u <= 1.25e-62)) {
		tmp = (v / (u + t1)) * (-t1 / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-0.0031d0)) .or. (.not. (u <= 1.25d-62))) then
        tmp = (v / (u + t1)) * (-t1 / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -0.0031) || !(u <= 1.25e-62)) {
		tmp = (v / (u + t1)) * (-t1 / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -0.0031) or not (u <= 1.25e-62):
		tmp = (v / (u + t1)) * (-t1 / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -0.0031) || !(u <= 1.25e-62))
		tmp = Float64(Float64(v / Float64(u + t1)) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -0.0031) || ~((u <= 1.25e-62)))
		tmp = (v / (u + t1)) * (-t1 / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -0.0031], N[Not[LessEqual[u, 1.25e-62]], $MachinePrecision]], N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -0.00309999999999999989 or 1.25e-62 < u

   1. Initial program 76.0%

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

      1. times-frac 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-neg-frac2 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. unsub-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in t1 around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. mul-1-neg 83.7%

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

   7. Simplified 83.7%

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

   if -0.00309999999999999989 < u < 1.25e-62

   1. Initial program 64.6%

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

      1. times-frac 98.4%

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

      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 81.7%

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

      1. associate-*r/ 81.7%

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

      2. neg-mul-1 81.7%

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

   7. Simplified 81.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -0.0031 \lor \neg \left(u \leq 1.25 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{v}{u + t1} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u + t1}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.3e-11)
   (/ (/ (- t1) u) (/ u v))
   (if (<= u 3.3e-62) (/ v (- t1)) (* t1 (/ (/ v (+ u t1)) (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e-11) {
		tmp = (-t1 / u) / (u / v);
	} else if (u <= 3.3e-62) {
		tmp = v / -t1;
	} else {
		tmp = t1 * ((v / (u + t1)) / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.3d-11)) then
        tmp = (-t1 / u) / (u / v)
    else if (u <= 3.3d-62) then
        tmp = v / -t1
    else
        tmp = t1 * ((v / (u + t1)) / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e-11) {
		tmp = (-t1 / u) / (u / v);
	} else if (u <= 3.3e-62) {
		tmp = v / -t1;
	} else {
		tmp = t1 * ((v / (u + t1)) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.3e-11:
		tmp = (-t1 / u) / (u / v)
	elif u <= 3.3e-62:
		tmp = v / -t1
	else:
		tmp = t1 * ((v / (u + t1)) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.3e-11)
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	elseif (u <= 3.3e-62)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(u + t1)) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.3e-11)
		tmp = (-t1 / u) / (u / v);
	elseif (u <= 3.3e-62)
		tmp = v / -t1;
	else
		tmp = t1 * ((v / (u + t1)) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.3e-11], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.3e-62], N[(v / (-t1)), $MachinePrecision], N[(t1 * N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.3e-11

   1. Initial program 80.8%

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

      1. associate-/l* 78.0%

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

      2. distribute-lft-neg-out 78.0%

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

      3. distribute-rgt-neg-in 78.0%

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

      4. associate-/r* 90.3%

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

      5. distribute-neg-frac2 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t1 around 0 83.5%

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

   7. Applied egg-rr 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-/r* 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in t1 around 0 84.6%

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

       1. associate-*r/ 84.6%

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

       2. mul-1-neg 84.6%

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

   12. Simplified 84.6%

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

   if -1.3e-11 < u < 3.30000000000000004e-62

   1. Initial program 64.6%

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

      1. times-frac 98.4%

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

      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 81.7%

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

      1. associate-*r/ 81.7%

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

      2. neg-mul-1 81.7%

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

   7. Simplified 81.7%

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

   if 3.30000000000000004e-62 < u

   1. Initial program 71.6%

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

      1. associate-/l* 74.4%

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

      2. distribute-lft-neg-out 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-/r* 88.4%

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

      5. distribute-neg-frac2 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in t1 around 0 78.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u + t1}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{-11} \lor \neg \left(u \leq 5.8 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2e-11) (not (<= u 5.8e-51)))
   (/ (/ (- t1) u) (/ u v))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e-11) || !(u <= 5.8e-51)) {
		tmp = (-t1 / u) / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2d-11)) .or. (.not. (u <= 5.8d-51))) then
        tmp = (-t1 / u) / (u / v)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e-11) || !(u <= 5.8e-51)) {
		tmp = (-t1 / u) / (u / v);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2e-11) or not (u <= 5.8e-51):
		tmp = (-t1 / u) / (u / v)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2e-11) || !(u <= 5.8e-51))
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2e-11) || ~((u <= 5.8e-51)))
		tmp = (-t1 / u) / (u / v);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2e-11], N[Not[LessEqual[u, 5.8e-51]], $MachinePrecision]], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.99999999999999988e-11 or 5.79999999999999945e-51 < u

   1. Initial program 76.3%

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

      1. associate-/l* 75.7%

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

      2. distribute-lft-neg-out 75.7%

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

      3. distribute-rgt-neg-in 75.7%

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

      4. associate-/r* 89.1%

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

      5. distribute-neg-frac2 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in t1 around 0 80.6%

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

   7. Applied egg-rr 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r* 81.7%

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

   9. Simplified 81.7%

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

   10. Taylor expanded in t1 around 0 81.7%

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

       1. associate-*r/ 81.7%

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

       2. mul-1-neg 81.7%

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

   12. Simplified 81.7%

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

   if -1.99999999999999988e-11 < u < 5.79999999999999945e-51

   1. Initial program 64.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 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 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{-11} \lor \neg \left(u \leq 5.8 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{-10} \lor \neg \left(u \leq 5.2 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.15e-10) (not (<= u 5.2e-51)))
   (* (/ (- t1) u) (/ v u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.15e-10) || !(u <= 5.2e-51)) {
		tmp = (-t1 / u) * (v / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.15d-10)) .or. (.not. (u <= 5.2d-51))) then
        tmp = (-t1 / u) * (v / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.15e-10) || !(u <= 5.2e-51)) {
		tmp = (-t1 / u) * (v / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.15e-10) or not (u <= 5.2e-51):
		tmp = (-t1 / u) * (v / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.15e-10) || !(u <= 5.2e-51))
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.15e-10) || ~((u <= 5.2e-51)))
		tmp = (-t1 / u) * (v / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.15e-10], N[Not[LessEqual[u, 5.2e-51]], $MachinePrecision]], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.15000000000000004e-10 or 5.2e-51 < u

   1. Initial program 76.3%

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

      1. times-frac 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-neg-frac2 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. unsub-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in t1 around 0 81.2%

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

   6. Taylor expanded in t1 around 0 81.2%

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

      1. associate-*r/ 83.4%

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

      2. mul-1-neg 83.4%

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

   8. Simplified 81.2%

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

   if -1.15000000000000004e-10 < u < 5.2e-51

   1. Initial program 64.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 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 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{-10} \lor \neg \left(u \leq 5.2 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.56 \cdot 10^{-9} \lor \neg \left(u \leq 6.2 \cdot 10^{-52}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.56e-9) (not (<= u 6.2e-52)))
   (* t1 (/ (/ v u) (- u)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.56e-9) || !(u <= 6.2e-52)) {
		tmp = t1 * ((v / u) / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.56d-9)) .or. (.not. (u <= 6.2d-52))) then
        tmp = t1 * ((v / u) / -u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.56e-9) || !(u <= 6.2e-52)) {
		tmp = t1 * ((v / u) / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.56e-9) or not (u <= 6.2e-52):
		tmp = t1 * ((v / u) / -u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.56e-9) || !(u <= 6.2e-52))
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.56e-9) || ~((u <= 6.2e-52)))
		tmp = t1 * ((v / u) / -u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.56e-9], N[Not[LessEqual[u, 6.2e-52]], $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 < -1.56e-9 or 6.1999999999999998e-52 < u

   1. Initial program 76.3%

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

      1. associate-/l* 75.7%

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

      2. distribute-lft-neg-out 75.7%

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

      3. distribute-rgt-neg-in 75.7%

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

      4. associate-/r* 89.1%

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

      5. distribute-neg-frac2 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in t1 around 0 80.6%

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

   6. Taylor expanded in t1 around 0 78.6%

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

   if -1.56e-9 < u < 6.1999999999999998e-52

   1. Initial program 64.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 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 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.56 \cdot 10^{-9} \lor \neg \left(u \leq 6.2 \cdot 10^{-52}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+63} \lor \neg \left(u \leq 1.35 \cdot 10^{+59}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.2e+63) (not (<= u 1.35e+59)))
   (* t1 (/ (/ v u) u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e+63) || !(u <= 1.35e+59)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.2d+63)) .or. (.not. (u <= 1.35d+59))) then
        tmp = t1 * ((v / u) / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e+63) || !(u <= 1.35e+59)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.2e+63) or not (u <= 1.35e+59):
		tmp = t1 * ((v / u) / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.2e+63) || !(u <= 1.35e+59))
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.2e+63) || ~((u <= 1.35e+59)))
		tmp = t1 * ((v / u) / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.2e+63], N[Not[LessEqual[u, 1.35e+59]], $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 < -3.20000000000000011e63 or 1.3500000000000001e59 < 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* 75.9%

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

      2. distribute-lft-neg-out 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

      4. associate-/r* 91.3%

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

      5. distribute-neg-frac2 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in t1 around 0 83.6%

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

   6. Taylor expanded in t1 around 0 83.1%

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

      1. associate-*r/ 86.2%

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

      2. add-sqr-sqrt 45.0%

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

      3. sqrt-unprod 69.6%

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

      4. sqr-neg 69.6%

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

      5. sqrt-unprod 28.6%

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

      6. add-sqr-sqrt 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. associate-/l* 64.0%

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

   10. Simplified 64.0%

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

   if -3.20000000000000011e63 < u < 1.3500000000000001e59

   1. Initial program 66.2%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around inf 75.3%

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

      1. associate-*r/ 75.3%

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

      2. neg-mul-1 75.3%

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

   7. Simplified 75.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+63} \lor \neg \left(u \leq 1.35 \cdot 10^{+59}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 9.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.8e+56)
   (/ t1 (* u (/ u v)))
   (if (<= u 9.4e+58) (/ v (- t1)) (* t1 (/ (/ v u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.8e+56) {
		tmp = t1 / (u * (u / v));
	} else if (u <= 9.4e+58) {
		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 <= (-6.8d+56)) then
        tmp = t1 / (u * (u / v))
    else if (u <= 9.4d+58) 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 <= -6.8e+56) {
		tmp = t1 / (u * (u / v));
	} else if (u <= 9.4e+58) {
		tmp = v / -t1;
	} else {
		tmp = t1 * ((v / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.8e+56:
		tmp = t1 / (u * (u / v))
	elif u <= 9.4e+58:
		tmp = v / -t1
	else:
		tmp = t1 * ((v / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.8e+56)
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	elseif (u <= 9.4e+58)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.8e+56)
		tmp = t1 / (u * (u / v));
	elseif (u <= 9.4e+58)
		tmp = v / -t1;
	else
		tmp = t1 * ((v / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.8e+56], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 9.4e+58], N[(v / (-t1)), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -6.80000000000000002e56

   1. Initial program 79.9%

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

      1. associate-/l* 78.3%

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

      2. distribute-lft-neg-out 78.3%

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

      3. distribute-rgt-neg-in 78.3%

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

      4. associate-/r* 92.4%

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

      5. distribute-neg-frac2 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in t1 around 0 84.7%

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

   6. Taylor expanded in t1 around 0 83.7%

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

      1. clear-num 85.0%

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

      2. un-div-inv 85.0%

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

      3. div-inv 85.0%

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

      4. clear-num 85.0%

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

      5. add-sqr-sqrt 85.0%

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

      6. sqrt-unprod 76.6%

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

      7. sqr-neg 76.6%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 67.4%

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

   8. Applied egg-rr 67.4%

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

   if -6.80000000000000002e56 < u < 9.39999999999999944e58

   1. Initial program 66.2%

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

      1. times-frac 98.0%

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

      2. distribute-frac-neg 98.0%

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

      3. distribute-neg-frac2 98.0%

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

      4. +-commutative 98.0%

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

      5. distribute-neg-in 98.0%

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

      6. unsub-neg 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around inf 75.3%

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

      1. associate-*r/ 75.3%

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

      2. neg-mul-1 75.3%

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

   7. Simplified 75.3%

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

   if 9.39999999999999944e58 < u

   1. Initial program 72.8%

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

      1. associate-/l* 73.3%

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

      2. distribute-lft-neg-out 73.3%

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

      3. distribute-rgt-neg-in 73.3%

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

      4. associate-/r* 90.2%

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

      5. distribute-neg-frac2 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in t1 around 0 82.4%

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

   6. Taylor expanded in t1 around 0 82.4%

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

      1. associate-*r/ 84.8%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 61.8%

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

      4. sqr-neg 61.8%

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

      5. sqrt-unprod 58.9%

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

      6. add-sqr-sqrt 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-/l* 61.8%

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

   10. Simplified 61.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 9.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.22 \cdot 10^{+91}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.22e+91)
   (/ v u)
   (if (<= u 9.5e+160) (/ v (- t1)) (/ v (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.22e+91) {
		tmp = v / u;
	} else if (u <= 9.5e+160) {
		tmp = v / -t1;
	} else {
		tmp = v / (u + t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.22d+91)) then
        tmp = v / u
    else if (u <= 9.5d+160) then
        tmp = v / -t1
    else
        tmp = v / (u + t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.22e+91) {
		tmp = v / u;
	} else if (u <= 9.5e+160) {
		tmp = v / -t1;
	} else {
		tmp = v / (u + t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.22e+91:
		tmp = v / u
	elif u <= 9.5e+160:
		tmp = v / -t1
	else:
		tmp = v / (u + t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.22e+91)
		tmp = Float64(v / u);
	elseif (u <= 9.5e+160)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(u + t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.22e+91)
		tmp = v / u;
	elseif (u <= 9.5e+160)
		tmp = v / -t1;
	else
		tmp = v / (u + t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.22e+91], N[(v / u), $MachinePrecision], If[LessEqual[u, 9.5e+160], N[(v / (-t1)), $MachinePrecision], N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.2199999999999999e91

   1. Initial program 78.3%

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

      1. associate-/l* 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in t1 around inf 33.6%

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

      1. associate-*r/ 32.9%

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

      2. *-commutative 32.9%

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

      3. times-frac 38.7%

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

      4. add-sqr-sqrt 16.6%

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

      5. sqrt-unprod 44.6%

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

      6. sqr-neg 44.6%

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

      7. sqrt-unprod 15.9%

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

      8. add-sqr-sqrt 32.1%

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

      9. *-inverses 32.1%

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

      10. *-un-lft-identity 32.1%

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

      11. clear-num 32.1%

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

   7. Applied egg-rr 32.1%

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

      1. associate-/r/ 32.1%

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

   9. Simplified 32.1%

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

   10. Taylor expanded in t1 around 0 32.7%

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

   if -1.2199999999999999e91 < u < 9.5000000000000006e160

   1. Initial program 68.0%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   7. Simplified 68.2%

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

   if 9.5000000000000006e160 < u

   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%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in t1 around inf 45.4%

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

      1. associate-*r/ 44.8%

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

      2. *-commutative 44.8%

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

      3. times-frac 46.7%

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

      4. add-sqr-sqrt 29.3%

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

      5. sqrt-unprod 49.2%

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

      6. sqr-neg 49.2%

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

      7. sqrt-unprod 14.6%

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

      8. add-sqr-sqrt 44.2%

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

      9. *-inverses 44.2%

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

      10. *-un-lft-identity 44.2%

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

   7. Applied egg-rr 44.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.22 \cdot 10^{+91}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+91} \lor \neg \left(u \leq 7.2 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.2e+91) (not (<= u 7.2e+164))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+91) || !(u <= 7.2e+164)) {
		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 <= (-1.2d+91)) .or. (.not. (u <= 7.2d+164))) 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 <= -1.2e+91) || !(u <= 7.2e+164)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.2e+91) or not (u <= 7.2e+164):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.2e+91) || !(u <= 7.2e+164))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.2e+91) || ~((u <= 7.2e+164)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.2e+91], N[Not[LessEqual[u, 7.2e+164]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.19999999999999991e91 or 7.19999999999999981e164 < u

   1. Initial program 76.1%

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

      1. associate-/l* 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in t1 around inf 37.6%

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

      1. associate-*r/ 36.9%

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

      2. *-commutative 36.9%

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

      3. times-frac 41.3%

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

      4. add-sqr-sqrt 20.8%

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

      5. sqrt-unprod 46.1%

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

      6. sqr-neg 46.1%

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

      7. sqrt-unprod 15.5%

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

      8. add-sqr-sqrt 36.1%

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

      9. *-inverses 36.1%

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

      10. *-un-lft-identity 36.1%

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

      11. clear-num 36.1%

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

   7. Applied egg-rr 36.1%

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

      1. associate-/r/ 36.1%

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

   9. Simplified 36.1%

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

   10. Taylor expanded in t1 around 0 34.8%

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

   if -1.19999999999999991e91 < u < 7.19999999999999981e164

   1. Initial program 68.0%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   7. Simplified 68.2%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+91} \lor \neg \left(u \leq 7.2 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.22 \cdot 10^{+91}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.22e+91) (/ v u) (if (<= u 9.2e+149) (/ v (- t1)) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.22e+91) {
		tmp = v / u;
	} else if (u <= 9.2e+149) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.22d+91)) then
        tmp = v / u
    else if (u <= 9.2d+149) then
        tmp = v / -t1
    else
        tmp = v / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.22e+91) {
		tmp = v / u;
	} else if (u <= 9.2e+149) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.22e+91:
		tmp = v / u
	elif u <= 9.2e+149:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.22e+91)
		tmp = Float64(v / u);
	elseif (u <= 9.2e+149)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.22e+91)
		tmp = v / u;
	elseif (u <= 9.2e+149)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.22e+91], N[(v / u), $MachinePrecision], If[LessEqual[u, 9.2e+149], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.2199999999999999e91

   1. Initial program 78.3%

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

      1. associate-/l* 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in t1 around inf 33.6%

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

      1. associate-*r/ 32.9%

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

      2. *-commutative 32.9%

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

      3. times-frac 38.7%

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

      4. add-sqr-sqrt 16.6%

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

      5. sqrt-unprod 44.6%

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

      6. sqr-neg 44.6%

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

      7. sqrt-unprod 15.9%

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

      8. add-sqr-sqrt 32.1%

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

      9. *-inverses 32.1%

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

      10. *-un-lft-identity 32.1%

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

      11. clear-num 32.1%

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

   7. Applied egg-rr 32.1%

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

      1. associate-/r/ 32.1%

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

   9. Simplified 32.1%

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

   10. Taylor expanded in t1 around 0 32.7%

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

   if -1.2199999999999999e91 < u < 9.1999999999999993e149

   1. Initial program 68.4%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 69.6%

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

      1. associate-*r/ 69.6%

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

      2. neg-mul-1 69.6%

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

   7. Simplified 69.6%

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

   if 9.1999999999999993e149 < u

   1. Initial program 68.9%

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

      1. associate-/l* 69.6%

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

      2. distribute-lft-neg-out 69.6%

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

      3. distribute-rgt-neg-in 69.6%

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

      4. associate-/r* 89.2%

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

      5. distribute-neg-frac2 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in t1 around 0 85.6%

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

   6. Taylor expanded in t1 around inf 34.9%

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

      1. associate-*r/ 34.9%

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

      2. mul-1-neg 34.9%

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

   8. Simplified 34.9%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.22 \cdot 10^{+91}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 23.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{+130} \lor \neg \left(t1 \leq 2.5 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.9e+130) (not (<= t1 2.5e+50))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.9e+130) || !(t1 <= 2.5e+50)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.9d+130)) .or. (.not. (t1 <= 2.5d+50))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.9e+130) || !(t1 <= 2.5e+50)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.9e+130) or not (t1 <= 2.5e+50):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.9e+130) || !(t1 <= 2.5e+50))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.9e+130) || ~((t1 <= 2.5e+50)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.9e+130], N[Not[LessEqual[t1, 2.5e+50]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.9000000000000001e130 or 2.5e50 < t1

   1. Initial program 47.6%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   7. Simplified 84.2%

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

      1. add-sqr-sqrt 39.4%

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

      2. sqrt-unprod 50.5%

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

      3. sqr-neg 50.5%

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

      4. sqrt-unprod 16.4%

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

      5. add-sqr-sqrt 33.6%

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

      6. div-inv 33.6%

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

   9. Applied egg-rr 33.6%

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

       1. associate-*r/ 33.6%

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

       2. *-rgt-identity 33.6%

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

   11. Simplified 33.6%

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

   if -1.9000000000000001e130 < t1 < 2.5e50

   1. Initial program 82.6%

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

      1. associate-/l* 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in t1 around inf 42.0%

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

      1. associate-*r/ 44.9%

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

      2. *-commutative 44.9%

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

      3. times-frac 47.3%

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

      4. add-sqr-sqrt 23.3%

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

      5. sqrt-unprod 42.7%

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

      6. sqr-neg 42.7%

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

      7. sqrt-unprod 7.2%

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

      8. add-sqr-sqrt 15.3%

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

      9. *-inverses 15.3%

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

      10. *-un-lft-identity 15.3%

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

      11. clear-num 15.3%

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

   7. Applied egg-rr 15.3%

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

      1. associate-/r/ 15.3%

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

   9. Simplified 15.3%

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

   10. Taylor expanded in t1 around 0 17.9%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9 \cdot 10^{+130} \lor \neg \left(t1 \leq 2.5 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

   1. times-frac 98.7%

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

   2. distribute-frac-neg 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. +-commutative 98.7%

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

   5. distribute-neg-in 98.7%

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

   6. unsub-neg 98.7%

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

3. Simplified 98.7%

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

   1. frac-2neg 98.7%

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

   2. frac-2neg 98.7%

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

   3. frac-times 70.4%

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

   4. sub-neg 70.4%

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

   5. distribute-neg-in 70.4%

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

   6. +-commutative 70.4%

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

   7. remove-double-neg 70.4%

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

   8. frac-times 98.7%

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

   9. associate-*r/ 99.0%

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

   10. add-sqr-sqrt 48.2%

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

   11. sqrt-unprod 42.8%

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

   12. sqr-neg 42.8%

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

   13. sqrt-unprod 17.0%

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

   14. add-sqr-sqrt 37.2%

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

   15. add-sqr-sqrt 19.4%

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

   16. sqrt-unprod 58.4%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 17: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

   1. times-frac 98.7%

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

   2. distribute-frac-neg 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. +-commutative 98.7%

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

   5. distribute-neg-in 98.7%

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

   6. unsub-neg 98.7%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 18: 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 70.4%

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

   1. times-frac 98.7%

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

   2. distribute-frac-neg 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. +-commutative 98.7%

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

   5. distribute-neg-in 98.7%

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

   6. unsub-neg 98.7%

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

3. Simplified 98.7%

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

   1. frac-2neg 98.7%

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

   2. frac-2neg 98.7%

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

   3. frac-times 70.4%

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

   4. sub-neg 70.4%

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

   5. distribute-neg-in 70.4%

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

   6. +-commutative 70.4%

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

   7. remove-double-neg 70.4%

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

   8. frac-times 98.7%

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

   9. associate-*r/ 99.0%

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

   10. add-sqr-sqrt 48.2%

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

   11. sqrt-unprod 42.8%

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

   12. sqr-neg 42.8%

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

   13. sqrt-unprod 17.0%

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

   14. add-sqr-sqrt 37.2%

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

   15. add-sqr-sqrt 19.4%

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

   16. sqrt-unprod 58.4%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in t1 around inf 60.3%

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

   1. mul-1-neg 60.3%

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

9. Simplified 60.3%

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

10. Final simplification 60.3%

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

Alternative 19: 14.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

   1. times-frac 98.7%

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

   2. distribute-frac-neg 98.7%

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

   3. distribute-neg-frac2 98.7%

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

   4. +-commutative 98.7%

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

   5. distribute-neg-in 98.7%

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

   6. unsub-neg 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in t1 around inf 52.7%

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

   1. associate-*r/ 52.7%

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

   2. neg-mul-1 52.7%

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

7. Simplified 52.7%

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

   1. add-sqr-sqrt 22.9%

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

   2. sqrt-unprod 34.5%

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

   3. sqr-neg 34.5%

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

   4. sqrt-unprod 6.8%

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

   5. add-sqr-sqrt 13.4%

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

   6. div-inv 13.4%

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

9. Applied egg-rr 13.4%

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

    1. associate-*r/ 13.4%

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

    2. *-rgt-identity 13.4%

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

11. Simplified 13.4%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))