Rosa's DopplerBench

Percentage Accurate: 72.4% → 97.9%

Time: 9.7s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% 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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. times-frac 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \mathbf{if}\;t1 \leq -5.6 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-228}:\\ \;\;\;\;\frac{v}{\frac{-u}{\frac{t1}{u}}}\\ \mathbf{elif}\;t1 \leq 1100000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (/ (* (- t1) (/ v u)) u)))
   (if (<= t1 -5.6e+49)
     t_1
     (if (<= t1 -1.15e-176)
       t_2
       (if (<= t1 1.26e-228)
         (/ v (/ (- u) (/ t1 u)))
         (if (<= t1 1100000000.0) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = (-t1 * (v / u)) / u;
	double tmp;
	if (t1 <= -5.6e+49) {
		tmp = t_1;
	} else if (t1 <= -1.15e-176) {
		tmp = t_2;
	} else if (t1 <= 1.26e-228) {
		tmp = v / (-u / (t1 / u));
	} else if (t1 <= 1100000000.0) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    t_2 = (-t1 * (v / u)) / u
    if (t1 <= (-5.6d+49)) then
        tmp = t_1
    else if (t1 <= (-1.15d-176)) then
        tmp = t_2
    else if (t1 <= 1.26d-228) then
        tmp = v / (-u / (t1 / u))
    else if (t1 <= 1100000000.0d0) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = (-t1 * (v / u)) / u;
	double tmp;
	if (t1 <= -5.6e+49) {
		tmp = t_1;
	} else if (t1 <= -1.15e-176) {
		tmp = t_2;
	} else if (t1 <= 1.26e-228) {
		tmp = v / (-u / (t1 / u));
	} else if (t1 <= 1100000000.0) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	t_2 = (-t1 * (v / u)) / u
	tmp = 0
	if t1 <= -5.6e+49:
		tmp = t_1
	elif t1 <= -1.15e-176:
		tmp = t_2
	elif t1 <= 1.26e-228:
		tmp = v / (-u / (t1 / u))
	elif t1 <= 1100000000.0:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	t_2 = Float64(Float64(Float64(-t1) * Float64(v / u)) / u)
	tmp = 0.0
	if (t1 <= -5.6e+49)
		tmp = t_1;
	elseif (t1 <= -1.15e-176)
		tmp = t_2;
	elseif (t1 <= 1.26e-228)
		tmp = Float64(v / Float64(Float64(-u) / Float64(t1 / u)));
	elseif (t1 <= 1100000000.0)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	t_2 = (-t1 * (v / u)) / u;
	tmp = 0.0;
	if (t1 <= -5.6e+49)
		tmp = t_1;
	elseif (t1 <= -1.15e-176)
		tmp = t_2;
	elseif (t1 <= 1.26e-228)
		tmp = v / (-u / (t1 / u));
	elseif (t1 <= 1100000000.0)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t1) * N[(v / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]}, If[LessEqual[t1, -5.6e+49], t$95$1, If[LessEqual[t1, -1.15e-176], t$95$2, If[LessEqual[t1, 1.26e-228], N[(v / N[((-u) / N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1100000000.0], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.5999999999999996e49 or 1.1e9 < t1

   1. Initial program 52.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 86.1%

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

   if -5.5999999999999996e49 < t1 < -1.1500000000000001e-176 or 1.25999999999999997e-228 < t1 < 1.1e9

   1. Initial program 85.1%

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

      1. associate-/l* 82.0%

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

      2. associate-/l* 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in t1 around 0 63.8%

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

      1. unpow2 63.8%

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

   6. Simplified 63.8%

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

      1. frac-2neg 63.8%

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

      2. div-inv 62.8%

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

      3. remove-double-neg 62.8%

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

      4. associate-/l* 68.2%

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

      5. distribute-neg-frac 68.2%

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

   8. Applied egg-rr 68.2%

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

      1. associate-/r/ 68.2%

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

      2. associate-*l/ 68.3%

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

      3. *-lft-identity 68.3%

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

   10. Simplified 68.3%

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

   11. Taylor expanded in t1 around 0 63.7%

       \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 63.7%

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

       2. associate-/l* 63.8%

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

       3. associate-*r/ 69.2%

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

       4. associate-*r/ 69.2%

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

       5. neg-mul-1 69.2%

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

       6. associate-*r/ 63.8%

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

       7. associate-/l* 69.2%

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

       8. associate-/l* 70.5%

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

   13. Simplified 70.5%

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

   if -1.1500000000000001e-176 < t1 < 1.25999999999999997e-228

   1. Initial program 79.7%

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

      1. associate-/l* 77.5%

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

      2. associate-/l* 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in t1 around 0 77.5%

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

      1. unpow2 77.5%

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

   6. Simplified 77.5%

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

      1. associate-/r/ 86.6%

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

      2. add-sqr-sqrt 32.5%

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

      3. sqrt-unprod 47.3%

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

      4. sqr-neg 47.3%

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

      5. sqrt-unprod 28.7%

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

      6. add-sqr-sqrt 46.8%

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

   8. Applied egg-rr 46.8%

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

      1. associate-*l/ 46.9%

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

      2. associate-/r* 47.1%

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

      3. associate-*r/ 47.1%

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

      4. add-sqr-sqrt 33.9%

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

      5. sqrt-prod 54.8%

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

      6. sqr-neg 54.8%

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

      7. sqrt-unprod 25.9%

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

      8. add-sqr-sqrt 82.9%

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

      9. neg-mul-1 82.9%

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

      10. associate-/r* 82.9%

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

   10. Applied egg-rr 82.9%

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

       1. associate-/l/ 82.9%

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

       2. associate-*r/ 85.2%

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

       3. associate-*l/ 87.5%

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

       4. *-commutative 87.5%

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

       5. associate-/l* 99.7%

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

       6. *-commutative 99.7%

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

       7. neg-mul-1 99.7%

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

   12. Simplified 99.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-228}:\\ \;\;\;\;\frac{v}{\frac{-u}{\frac{t1}{u}}}\\ \mathbf{elif}\;t1 \leq 1100000000:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1250000000 \lor \neg \left(t1 \leq 15500000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1250000000.0) (not (<= t1 15500000.0)))
   (/ (- v) (+ t1 u))
   (* t1 (/ (- v) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1250000000.0) || !(t1 <= 15500000.0)) {
		tmp = -v / (t1 + u);
	} 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 ((t1 <= (-1250000000.0d0)) .or. (.not. (t1 <= 15500000.0d0))) then
        tmp = -v / (t1 + u)
    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 ((t1 <= -1250000000.0) || !(t1 <= 15500000.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 * (-v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1250000000.0) or not (t1 <= 15500000.0):
		tmp = -v / (t1 + u)
	else:
		tmp = t1 * (-v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1250000000.0) || !(t1 <= 15500000.0))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1250000000.0) || ~((t1 <= 15500000.0)))
		tmp = -v / (t1 + u);
	else
		tmp = t1 * (-v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1250000000.0], N[Not[LessEqual[t1, 15500000.0]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.25e9 or 1.55e7 < t1

   1. Initial program 54.2%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 84.3%

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

   if -1.25e9 < t1 < 1.55e7

   1. Initial program 83.9%

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

      1. associate-/l* 80.8%

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

      2. neg-mul-1 80.8%

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

      3. *-commutative 80.8%

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

      4. associate-*r/ 79.1%

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

      5. associate-/l* 79.2%

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

      6. neg-mul-1 79.2%

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

      7. associate-/r* 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in t1 around 0 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

      3. unpow2 68.2%

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

   6. Simplified 68.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1250000000 \lor \neg \left(t1 \leq 15500000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 4: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+53} \lor \neg \left(t1 \leq 800000000000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.6e+53) (not (<= t1 800000000000.0)))
   (/ (- v) (+ t1 u))
   (* t1 (/ (/ v u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e+53) || !(t1 <= 800000000000.0)) {
		tmp = -v / (t1 + u);
	} 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 ((t1 <= (-1.6d+53)) .or. (.not. (t1 <= 800000000000.0d0))) then
        tmp = -v / (t1 + u)
    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 ((t1 <= -1.6e+53) || !(t1 <= 800000000000.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 * ((v / u) / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.6e+53) or not (t1 <= 800000000000.0):
		tmp = -v / (t1 + u)
	else:
		tmp = t1 * ((v / u) / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.6e+53) || !(t1 <= 800000000000.0))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.6e+53) || ~((t1 <= 800000000000.0)))
		tmp = -v / (t1 + u);
	else
		tmp = t1 * ((v / u) / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.6e+53], N[Not[LessEqual[t1, 800000000000.0]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.6e53 or 8e11 < t1

   1. Initial program 52.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 86.1%

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

   if -1.6e53 < t1 < 8e11

   1. Initial program 83.4%

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

      1. associate-/l* 80.6%

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

      2. associate-/l* 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in t1 around 0 68.0%

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

      1. unpow2 68.0%

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

   6. Simplified 68.0%

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

      1. frac-2neg 68.0%

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

      2. div-inv 66.6%

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

      3. remove-double-neg 66.6%

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

      4. associate-/l* 72.6%

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

      5. distribute-neg-frac 72.6%

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

   8. Applied egg-rr 72.6%

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

      1. associate-/r/ 72.7%

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

      2. associate-*l/ 72.7%

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

      3. *-lft-identity 72.7%

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

   10. Simplified 72.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+53} \lor \neg \left(t1 \leq 800000000000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]

Alternative 5: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+56} \lor \neg \left(t1 \leq 27000000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.7e+56) (not (<= t1 27000000.0)))
   (/ (- v) (+ t1 u))
   (/ (- t1) (* u (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.7e+56) || !(t1 <= 27000000.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -t1 / (u * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.7d+56)) .or. (.not. (t1 <= 27000000.0d0))) then
        tmp = -v / (t1 + u)
    else
        tmp = -t1 / (u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.7e+56) || !(t1 <= 27000000.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.7e+56) or not (t1 <= 27000000.0):
		tmp = -v / (t1 + u)
	else:
		tmp = -t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.7e+56) || !(t1 <= 27000000.0))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-t1) / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.7e+56) || ~((t1 <= 27000000.0)))
		tmp = -v / (t1 + u);
	else
		tmp = -t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.7e+56], N[Not[LessEqual[t1, 27000000.0]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.7000000000000001e56 or 2.7e7 < t1

   1. Initial program 52.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 86.1%

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

   if -2.7000000000000001e56 < t1 < 2.7e7

   1. Initial program 83.4%

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

      1. associate-/l* 80.6%

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

      2. associate-/l* 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in t1 around 0 68.0%

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

      1. unpow2 68.0%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in u around 0 68.0%

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

      1. unpow2 68.0%

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

      2. associate-*r/ 74.1%

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

   9. Simplified 74.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+56} \lor \neg \left(t1 \leq 27000000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 6: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.7 \cdot 10^{+50} \lor \neg \left(t1 \leq 200000000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{-\frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.7e+50) (not (<= t1 200000000.0)))
   (/ (- v) (+ t1 u))
   (/ (/ v u) (- (/ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.7e+50) || !(t1 <= 200000000.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (v / u) / -(u / t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.7d+50)) .or. (.not. (t1 <= 200000000.0d0))) then
        tmp = -v / (t1 + u)
    else
        tmp = (v / u) / -(u / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.7e+50) || !(t1 <= 200000000.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (v / u) / -(u / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.7e+50) or not (t1 <= 200000000.0):
		tmp = -v / (t1 + u)
	else:
		tmp = (v / u) / -(u / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.7e+50) || !(t1 <= 200000000.0))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(v / u) / Float64(-Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.7e+50) || ~((t1 <= 200000000.0)))
		tmp = -v / (t1 + u);
	else
		tmp = (v / u) / -(u / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.7e+50], N[Not[LessEqual[t1, 200000000.0]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] / (-N[(u / t1), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.7000000000000001e50 or 2e8 < t1

   1. Initial program 52.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 86.1%

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

   if -3.7000000000000001e50 < t1 < 2e8

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. times-frac 98.2%

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

      3. neg-mul-1 98.2%

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

      4. associate-/l* 97.5%

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

      5. associate-*r/ 97.5%

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

      6. associate-/l* 97.5%

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

      7. associate-/l/ 97.5%

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

      8. neg-mul-1 97.5%

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

      9. *-lft-identity 97.5%

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

      10. metadata-eval 97.5%

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

      11. times-frac 97.5%

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

      12. neg-mul-1 97.5%

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

      13. remove-double-neg 97.5%

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

      14. neg-mul-1 97.5%

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

      15. sub0-neg 97.5%

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

      16. associate--r+ 97.5%

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

      17. neg-sub0 97.5%

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

      18. div-sub 97.5%

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

      19. distribute-frac-neg 97.5%

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

      20. *-inverses 97.5%

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

      21. metadata-eval 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around 0 75.9%

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

   5. Taylor expanded in u around inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. distribute-neg-frac 77.2%

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

   7. Simplified 77.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.7 \cdot 10^{+50} \lor \neg \left(t1 \leq 200000000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{-\frac{u}{t1}}\\ \end{array} \]

Alternative 7: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 120000000:\\ \;\;\;\;\frac{\frac{v}{u}}{-\frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.6e+51)
   (/ (/ v t1) (- -1.0 (/ u t1)))
   (if (<= t1 120000000.0) (/ (/ v u) (- (/ u t1))) (/ (- v) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.6e+51) {
		tmp = (v / t1) / (-1.0 - (u / t1));
	} else if (t1 <= 120000000.0) {
		tmp = (v / u) / -(u / t1);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.6d+51)) then
        tmp = (v / t1) / ((-1.0d0) - (u / t1))
    else if (t1 <= 120000000.0d0) then
        tmp = (v / u) / -(u / t1)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.6e+51) {
		tmp = (v / t1) / (-1.0 - (u / t1));
	} else if (t1 <= 120000000.0) {
		tmp = (v / u) / -(u / t1);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.6e+51:
		tmp = (v / t1) / (-1.0 - (u / t1))
	elif t1 <= 120000000.0:
		tmp = (v / u) / -(u / t1)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.6e+51)
		tmp = Float64(Float64(v / t1) / Float64(-1.0 - Float64(u / t1)));
	elseif (t1 <= 120000000.0)
		tmp = Float64(Float64(v / u) / Float64(-Float64(u / t1)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.6e+51)
		tmp = (v / t1) / (-1.0 - (u / t1));
	elseif (t1 <= 120000000.0)
		tmp = (v / u) / -(u / t1);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.6e+51], N[(N[(v / t1), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 120000000.0], N[(N[(v / u), $MachinePrecision] / (-N[(u / t1), $MachinePrecision])), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.6000000000000001e51

   1. Initial program 44.4%

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

      1. *-commutative 44.4%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around inf 88.3%

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

   if -2.6000000000000001e51 < t1 < 1.2e8

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. times-frac 98.2%

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

      3. neg-mul-1 98.2%

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

      4. associate-/l* 97.5%

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

      5. associate-*r/ 97.5%

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

      6. associate-/l* 97.5%

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

      7. associate-/l/ 97.5%

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

      8. neg-mul-1 97.5%

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

      9. *-lft-identity 97.5%

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

      10. metadata-eval 97.5%

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

      11. times-frac 97.5%

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

      12. neg-mul-1 97.5%

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

      13. remove-double-neg 97.5%

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

      14. neg-mul-1 97.5%

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

      15. sub0-neg 97.5%

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

      16. associate--r+ 97.5%

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

      17. neg-sub0 97.5%

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

      18. div-sub 97.5%

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

      19. distribute-frac-neg 97.5%

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

      20. *-inverses 97.5%

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

      21. metadata-eval 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around 0 75.9%

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

   5. Taylor expanded in u around inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. distribute-neg-frac 77.2%

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

   7. Simplified 77.2%

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

   if 1.2e8 < t1

   1. Initial program 59.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 84.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 120000000:\\ \;\;\;\;\frac{\frac{v}{u}}{-\frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 8: 68.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+161} \lor \neg \left(u \leq 1.9 \cdot 10^{+123}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.1e+161) (not (<= u 1.9e+123)))
   (* v (/ t1 (* u u)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.1e+161) || !(u <= 1.9e+123)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / (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 <= (-4.1d+161)) .or. (.not. (u <= 1.9d+123))) then
        tmp = v * (t1 / (u * u))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.1e+161) || !(u <= 1.9e+123)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.1e+161) or not (u <= 1.9e+123):
		tmp = v * (t1 / (u * u))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.1e+161) || !(u <= 1.9e+123))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.1e+161) || ~((u <= 1.9e+123)))
		tmp = v * (t1 / (u * u));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.1e+161], N[Not[LessEqual[u, 1.9e+123]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.1000000000000001e161 or 1.89999999999999997e123 < u

   1. Initial program 73.3%

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

      1. associate-/l* 75.2%

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

      2. associate-/l* 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in t1 around 0 75.2%

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

      1. unpow2 75.2%

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

   6. Simplified 75.2%

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

      1. associate-/r/ 73.9%

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

      2. add-sqr-sqrt 36.8%

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

      3. sqrt-unprod 61.4%

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

      4. sqr-neg 61.4%

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

      5. sqrt-unprod 37.2%

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

      6. add-sqr-sqrt 72.6%

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

   8. Applied egg-rr 72.6%

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

   if -4.1000000000000001e161 < u < 1.89999999999999997e123

   1. Initial program 65.8%

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

      1. times-frac 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in t1 around inf 72.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+161} \lor \neg \left(u \leq 1.9 \cdot 10^{+123}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 9: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-commutative 67.9%

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

   2. times-frac 99.0%

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

   3. neg-mul-1 99.0%

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

   4. associate-/l* 98.7%

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

   5. associate-*r/ 98.7%

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

   6. associate-/l* 98.7%

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

   7. associate-/l/ 98.7%

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

   8. neg-mul-1 98.7%

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

   9. *-lft-identity 98.7%

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

   10. metadata-eval 98.7%

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

   11. times-frac 98.7%

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

   12. neg-mul-1 98.7%

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

   13. remove-double-neg 98.7%

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

   14. neg-mul-1 98.7%

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

   15. sub0-neg 98.7%

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

   16. associate--r+ 98.7%

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

   17. neg-sub0 98.7%

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

   18. div-sub 98.7%

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

   19. distribute-frac-neg 98.7%

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

   20. *-inverses 98.7%

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

   21. metadata-eval 98.7%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 10: 58.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+209}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.6e+209) (/ v u) (if (<= u 2.05e+125) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e+209) {
		tmp = v / u;
	} else if (u <= 2.05e+125) {
		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 <= (-2.6d+209)) then
        tmp = v / u
    else if (u <= 2.05d+125) 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 <= -2.6e+209) {
		tmp = v / u;
	} else if (u <= 2.05e+125) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.6e+209:
		tmp = v / u
	elif u <= 2.05e+125:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.6e+209)
		tmp = Float64(v / u);
	elseif (u <= 2.05e+125)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.6e+209)
		tmp = v / u;
	elseif (u <= 2.05e+125)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.6e+209], N[(v / u), $MachinePrecision], If[LessEqual[u, 2.05e+125], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.6e209 or 2.04999999999999996e125 < u

   1. Initial program 76.2%

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

      1. associate-/l* 78.1%

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

      2. associate-/l* 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in t1 around 0 91.1%

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

      1. frac-2neg 91.1%

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

      2. div-inv 91.1%

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

      3. remove-double-neg 91.1%

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

      4. div-inv 91.1%

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

      5. distribute-lft-neg-in 91.1%

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

      6. distribute-neg-in 91.1%

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

      7. add-sqr-sqrt 47.6%

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

      8. sqrt-unprod 88.6%

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

      9. sqr-neg 88.6%

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

      10. sqrt-unprod 43.4%

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

      11. add-sqr-sqrt 91.0%

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

      12. sub-neg 91.0%

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

      13. clear-num 91.0%

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

   6. Applied egg-rr 91.0%

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

      1. associate-*r/ 91.0%

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

      2. *-rgt-identity 91.0%

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

      3. *-commutative 91.0%

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

      4. associate-/r* 96.1%

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

      5. associate-/r/ 96.2%

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

      6. associate-*l/ 86.1%

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

      7. *-commutative 86.1%

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

      8. associate-*r/ 96.2%

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

   8. Simplified 96.2%

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

   9. Taylor expanded in t1 around inf 42.3%

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

   if -2.6e209 < u < 2.04999999999999996e125

   1. Initial program 65.2%

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

      1. times-frac 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in t1 around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

   6. Simplified 68.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+209}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 11: 57.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.08 \cdot 10^{+222}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.08e+222) (/ (- v) u) (if (<= u 2.1e+125) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.08e+222) {
		tmp = -v / u;
	} else if (u <= 2.1e+125) {
		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 <= (-1.08d+222)) then
        tmp = -v / u
    else if (u <= 2.1d+125) 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 <= -1.08e+222) {
		tmp = -v / u;
	} else if (u <= 2.1e+125) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.08e+222:
		tmp = -v / u
	elif u <= 2.1e+125:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.08e+222)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 2.1e+125)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.08e+222)
		tmp = -v / u;
	elseif (u <= 2.1e+125)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.08e+222], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 2.1e+125], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.08e222

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

      2. times-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

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

      16. associate--r+ 100.0%

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

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around 0 93.5%

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

   5. Taylor expanded in u around 0 50.7%

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

      1. mul-1-neg 50.7%

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

      2. distribute-frac-neg 50.7%

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

   7. Simplified 50.7%

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

   if -1.08e222 < u < 2.1000000000000001e125

   1. Initial program 64.7%

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

      1. times-frac 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in t1 around inf 67.9%

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

      1. associate-*r/ 67.9%

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

      2. neg-mul-1 67.9%

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

   6. Simplified 67.9%

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

   if 2.1000000000000001e125 < u

   1. Initial program 75.9%

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

      1. associate-/l* 78.4%

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

      2. associate-/l* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in t1 around 0 91.4%

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

      1. frac-2neg 91.4%

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

      2. div-inv 91.4%

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

      3. remove-double-neg 91.4%

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

      4. div-inv 91.3%

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

      5. distribute-lft-neg-in 91.3%

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

      6. distribute-neg-in 91.3%

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

      7. add-sqr-sqrt 41.1%

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

      8. sqrt-unprod 88.7%

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

      9. sqr-neg 88.7%

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

      10. sqrt-unprod 50.2%

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

      11. add-sqr-sqrt 91.3%

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

      12. sub-neg 91.3%

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

      13. clear-num 91.4%

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

   6. Applied egg-rr 91.4%

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

      1. associate-*r/ 91.3%

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

      2. *-rgt-identity 91.3%

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

      3. *-commutative 91.3%

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

      4. associate-/r* 98.3%

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

      5. associate-/r/ 98.3%

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

      6. associate-*l/ 87.2%

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

      7. *-commutative 87.2%

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

      8. associate-*r/ 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in t1 around inf 40.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.08 \cdot 10^{+222}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 12: 61.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. times-frac 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in t1 around inf 63.4%

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

5. Final simplification 63.4%

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

Alternative 13: 17.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return v / 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 = v / u
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. associate-/l* 68.6%

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

   2. associate-/l* 81.9%

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

3. Simplified 81.9%

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

4. Taylor expanded in t1 around 0 53.0%

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

   1. frac-2neg 53.0%

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

   2. div-inv 52.3%

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

   3. remove-double-neg 52.3%

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

   4. div-inv 52.3%

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

   5. distribute-lft-neg-in 52.3%

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

   6. distribute-neg-in 52.3%

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

   7. add-sqr-sqrt 23.9%

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

   8. sqrt-unprod 54.0%

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

   9. sqr-neg 54.0%

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

   10. sqrt-unprod 26.9%

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

   11. add-sqr-sqrt 51.2%

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

   12. sub-neg 51.2%

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

   13. clear-num 51.2%

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

6. Applied egg-rr 51.2%

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

   1. associate-*r/ 51.9%

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

   2. *-rgt-identity 51.9%

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

   3. *-commutative 51.9%

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

   4. associate-/r* 48.1%

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

   5. associate-/r/ 48.7%

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

   6. associate-*l/ 45.0%

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

   7. *-commutative 45.0%

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

   8. associate-*r/ 48.7%

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

8. Simplified 48.7%

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

9. Taylor expanded in t1 around inf 15.7%

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

10. Final simplification 15.7%

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

Reproduce

herbie shell --seed 2023261 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))