Rosa's DopplerBench

Percentage Accurate: 72.2% → 98.0%

Time: 7.4s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.2% 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, 1.0× 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 69.8%

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

   1. times-frac 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{if}\;t1 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.5 \cdot 10^{-240}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{t1}{u}}{u} \cdot \left(-v\right)\\ \mathbf{elif}\;t1 \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (* t1 (/ (/ v u) (- u)))))
   (if (<= t1 -1e-12)
     t_1
     (if (<= t1 -1.5e-240)
       t_2
       (if (<= t1 1.5e-186)
         (* (/ (/ t1 u) u) (- v))
         (if (<= t1 2.55e+33) 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 <= -1e-12) {
		tmp = t_1;
	} else if (t1 <= -1.5e-240) {
		tmp = t_2;
	} else if (t1 <= 1.5e-186) {
		tmp = ((t1 / u) / u) * -v;
	} else if (t1 <= 2.55e+33) {
		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 <= (-1d-12)) then
        tmp = t_1
    else if (t1 <= (-1.5d-240)) then
        tmp = t_2
    else if (t1 <= 1.5d-186) then
        tmp = ((t1 / u) / u) * -v
    else if (t1 <= 2.55d+33) 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 <= -1e-12) {
		tmp = t_1;
	} else if (t1 <= -1.5e-240) {
		tmp = t_2;
	} else if (t1 <= 1.5e-186) {
		tmp = ((t1 / u) / u) * -v;
	} else if (t1 <= 2.55e+33) {
		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 <= -1e-12:
		tmp = t_1
	elif t1 <= -1.5e-240:
		tmp = t_2
	elif t1 <= 1.5e-186:
		tmp = ((t1 / u) / u) * -v
	elif t1 <= 2.55e+33:
		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(t1 * Float64(Float64(v / u) / Float64(-u)))
	tmp = 0.0
	if (t1 <= -1e-12)
		tmp = t_1;
	elseif (t1 <= -1.5e-240)
		tmp = t_2;
	elseif (t1 <= 1.5e-186)
		tmp = Float64(Float64(Float64(t1 / u) / u) * Float64(-v));
	elseif (t1 <= 2.55e+33)
		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 <= -1e-12)
		tmp = t_1;
	elseif (t1 <= -1.5e-240)
		tmp = t_2;
	elseif (t1 <= 1.5e-186)
		tmp = ((t1 / u) / u) * -v;
	elseif (t1 <= 2.55e+33)
		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[(t1 * N[(N[(v / u), $MachinePrecision] / (-u)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1e-12], t$95$1, If[LessEqual[t1, -1.5e-240], t$95$2, If[LessEqual[t1, 1.5e-186], N[(N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision] * (-v)), $MachinePrecision], If[LessEqual[t1, 2.55e+33], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.9999999999999998e-13 or 2.5499999999999999e33 < t1

   1. Initial program 60.0%

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

      1. times-frac 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around inf 87.4%

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

   if -9.9999999999999998e-13 < t1 < -1.49999999999999995e-240 or 1.5000000000000001e-186 < t1 < 2.5499999999999999e33

   1. Initial program 84.3%

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

   2. Taylor expanded in t1 around 0 67.8%

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

      1. unpow2 67.8%

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

   4. Simplified 67.8%

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

      1. frac-2neg 67.8%

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

      2. div-inv 67.9%

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

      3. distribute-lft-neg-out 67.9%

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

      4. remove-double-neg 67.9%

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

      5. distribute-rgt-neg-in 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. associate-*l* 70.8%

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

      2. associate-/r* 71.8%

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

      3. associate-*r/ 77.1%

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

      4. associate-*r/ 77.0%

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

      5. associate-*l/ 77.0%

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

      6. *-rgt-identity 77.0%

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

   8. Simplified 77.0%

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

   if -1.49999999999999995e-240 < t1 < 1.5000000000000001e-186

   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. associate-/l* 76.0%

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

      2. neg-mul-1 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-*r/ 76.0%

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

      5. associate-/l* 76.0%

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

      6. neg-mul-1 76.0%

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

      7. associate-/r* 81.3%

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

   3. Simplified 81.3%

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

      1. associate-*r/ 88.9%

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

      2. add-sqr-sqrt 31.1%

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

      3. sqrt-unprod 50.7%

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

      4. sqr-neg 50.7%

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

      5. sqrt-unprod 33.1%

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

      6. add-sqr-sqrt 47.6%

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

      7. frac-2neg 47.6%

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

      8. add-sqr-sqrt 14.5%

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

      9. sqrt-unprod 61.4%

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

      10. sqr-neg 61.4%

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

      11. sqrt-unprod 57.6%

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

      12. add-sqr-sqrt 88.9%

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

      13. distribute-neg-in 88.9%

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

      14. add-sqr-sqrt 27.9%

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

      15. sqrt-unprod 84.6%

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

      16. sqr-neg 84.6%

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

      17. sqrt-unprod 58.3%

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

      18. add-sqr-sqrt 84.5%

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

      19. sub-neg 84.5%

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

   5. Applied egg-rr 84.5%

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

   6. Taylor expanded in t1 around 0 68.4%

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

      1. mul-1-neg 68.4%

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

      2. *-commutative 68.4%

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

      3. associate-*r/ 81.3%

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

      4. unpow2 81.3%

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

      5. associate-/r* 93.0%

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

   8. Simplified 93.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -1.5 \cdot 10^{-240}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{t1}{u}}{u} \cdot \left(-v\right)\\ \mathbf{elif}\;t1 \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 3: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-74} \lor \neg \left(t1 \leq 9.2 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u}}{u} \cdot \left(-v\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5e-74) (not (<= t1 9.2e+32)))
   (/ (- v) (+ t1 u))
   (* (/ (/ t1 u) u) (- v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5e-74) || !(t1 <= 9.2e+32)) {
		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 <= (-5d-74)) .or. (.not. (t1 <= 9.2d+32))) 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 <= -5e-74) || !(t1 <= 9.2e+32)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = ((t1 / u) / u) * -v;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5e-74) or not (t1 <= 9.2e+32):
		tmp = -v / (t1 + u)
	else:
		tmp = ((t1 / u) / u) * -v
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5e-74) || !(t1 <= 9.2e+32))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(Float64(t1 / u) / u) * Float64(-v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5e-74) || ~((t1 <= 9.2e+32)))
		tmp = -v / (t1 + u);
	else
		tmp = ((t1 / u) / u) * -v;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5e-74], N[Not[LessEqual[t1, 9.2e+32]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision] * (-v)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.99999999999999998e-74 or 9.1999999999999998e32 < t1

   1. Initial program 63.0%

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

      1. 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 83.5%

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

   if -4.99999999999999998e-74 < t1 < 9.1999999999999998e32

   1. Initial program 78.7%

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

      1. associate-/l* 81.8%

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

      2. neg-mul-1 81.8%

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

      3. *-commutative 81.8%

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

      4. associate-*r/ 81.8%

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

      5. associate-/l* 81.9%

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

      6. neg-mul-1 81.9%

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

      7. associate-/r* 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r/ 95.6%

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

      2. add-sqr-sqrt 45.1%

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

      3. sqrt-unprod 57.0%

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

      4. sqr-neg 57.0%

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

      5. sqrt-unprod 24.3%

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

      6. add-sqr-sqrt 42.4%

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

      7. frac-2neg 42.4%

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

      8. add-sqr-sqrt 18.1%

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

      9. sqrt-unprod 54.6%

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

      10. sqr-neg 54.6%

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

      11. sqrt-unprod 50.4%

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

      12. add-sqr-sqrt 95.6%

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

      13. distribute-neg-in 95.6%

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

      14. add-sqr-sqrt 46.1%

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

      15. sqrt-unprod 84.4%

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

      16. sqr-neg 84.4%

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

      17. sqrt-unprod 42.4%

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

      18. add-sqr-sqrt 78.2%

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

      19. sub-neg 78.2%

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

   5. Applied egg-rr 78.2%

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

   6. Taylor expanded in t1 around 0 70.3%

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

      1. mul-1-neg 70.3%

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

      2. *-commutative 70.3%

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

      3. associate-*r/ 74.8%

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

      4. unpow2 74.8%

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

      5. associate-/r* 78.7%

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

   8. Simplified 78.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5 \cdot 10^{-74} \lor \neg \left(t1 \leq 9.2 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u}}{u} \cdot \left(-v\right)\\ \end{array} \]

Alternative 4: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-6} \lor \neg \left(t1 \leq 8 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.8e-6) (not (<= t1 8e+30)))
   (/ (- v) (+ t1 u))
   (* (/ t1 u) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e-6) || !(t1 <= 8e+30)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 / u) * (v / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.8d-6)) .or. (.not. (t1 <= 8d+30))) then
        tmp = -v / (t1 + u)
    else
        tmp = (t1 / u) * (v / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e-6) || !(t1 <= 8e+30)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 / u) * (v / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.8e-6) or not (t1 <= 8e+30):
		tmp = -v / (t1 + u)
	else:
		tmp = (t1 / u) * (v / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.8e-6) || !(t1 <= 8e+30))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(t1 / u) * Float64(v / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.8e-6) || ~((t1 <= 8e+30)))
		tmp = -v / (t1 + u);
	else
		tmp = (t1 / u) * (v / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.8e-6], N[Not[LessEqual[t1, 8e+30]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[(v / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.79999999999999987e-6 or 8.0000000000000002e30 < t1

   1. Initial program 60.0%

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

      1. times-frac 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around inf 87.4%

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

   if -2.79999999999999987e-6 < t1 < 8.0000000000000002e30

   1. Initial program 79.8%

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

      1. associate-/l* 82.6%

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

      2. neg-mul-1 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-*r/ 82.5%

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

      5. associate-/l* 82.6%

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

      6. neg-mul-1 82.6%

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

      7. associate-/r* 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in t1 around 0 72.3%

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

      1. associate-*r/ 72.3%

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

      2. neg-mul-1 72.3%

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

      3. unpow2 72.3%

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

   6. Simplified 72.3%

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

      1. frac-2neg 72.3%

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

      2. remove-double-neg 72.3%

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

      3. associate-*r/ 68.0%

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

      4. distribute-rgt-neg-in 68.0%

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

   8. Applied egg-rr 68.0%

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

      1. times-frac 79.5%

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

   10. Simplified 79.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-6} \lor \neg \left(t1 \leq 8 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \]

Alternative 5: 67.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+115} \lor \neg \left(u \leq 5.4 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.2e+115) (not (<= u 5.4e+110)))
   (* (/ t1 u) (/ v u))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.2e+115) || !(u <= 5.4e+110)) {
		tmp = (t1 / u) * (v / 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 <= (-6.2d+115)) .or. (.not. (u <= 5.4d+110))) then
        tmp = (t1 / u) * (v / 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 <= -6.2e+115) || !(u <= 5.4e+110)) {
		tmp = (t1 / u) * (v / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.2e+115) or not (u <= 5.4e+110):
		tmp = (t1 / u) * (v / u)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.2e+115) || !(u <= 5.4e+110))
		tmp = Float64(Float64(t1 / u) * Float64(v / 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 <= -6.2e+115) || ~((u <= 5.4e+110)))
		tmp = (t1 / u) * (v / u);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.2e+115], N[Not[LessEqual[u, 5.4e+110]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.2000000000000001e115 or 5.40000000000000019e110 < u

   1. Initial program 76.9%

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

   2. Taylor expanded in t1 around 0 74.6%

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

      1. unpow2 74.6%

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

   4. Simplified 74.6%

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

      1. *-commutative 74.6%

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

      2. times-frac 91.2%

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

      3. add-sqr-sqrt 44.6%

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

      4. sqrt-unprod 67.6%

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

      5. sqr-neg 67.6%

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

      6. sqrt-prod 33.6%

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

      7. add-sqr-sqrt 66.0%

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

   6. Applied egg-rr 66.0%

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

   if -6.2000000000000001e115 < u < 5.40000000000000019e110

   1. Initial program 66.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 68.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+115} \lor \neg \left(u \leq 5.4 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 6: 67.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+112} \lor \neg \left(u \leq 4.7 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.8e+112) (not (<= u 4.7e+101)))
   (/ t1 (/ u (/ v u)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.8e+112) || !(u <= 4.7e+101)) {
		tmp = t1 / (u / (v / 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 <= (-3.8d+112)) .or. (.not. (u <= 4.7d+101))) then
        tmp = t1 / (u / (v / 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 <= -3.8e+112) || !(u <= 4.7e+101)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.8e+112) or not (u <= 4.7e+101):
		tmp = t1 / (u / (v / u))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.8e+112) || !(u <= 4.7e+101))
		tmp = Float64(t1 / Float64(u / Float64(v / 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 <= -3.8e+112) || ~((u <= 4.7e+101)))
		tmp = t1 / (u / (v / u));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.8e+112], N[Not[LessEqual[u, 4.7e+101]], $MachinePrecision]], N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.80000000000000008e112 or 4.69999999999999971e101 < u

   1. Initial program 77.5%

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

      1. associate-/l* 78.9%

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

      2. neg-mul-1 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*r/ 78.9%

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

      5. associate-/l* 78.9%

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

      6. neg-mul-1 78.9%

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

      7. associate-/r* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in t1 around 0 77.8%

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

      1. associate-*r/ 77.8%

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

      2. neg-mul-1 77.8%

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

      3. unpow2 77.8%

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

   6. Simplified 77.8%

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

      1. clear-num 77.9%

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

      2. un-div-inv 77.9%

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

      3. associate-/l* 88.8%

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

      4. add-sqr-sqrt 46.5%

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

      5. sqrt-unprod 67.0%

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

      6. sqr-neg 67.0%

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

      7. sqrt-unprod 34.5%

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

      8. add-sqr-sqrt 68.1%

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

   8. Applied egg-rr 68.1%

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

   if -3.80000000000000008e112 < u < 4.69999999999999971e101

   1. Initial program 65.9%

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

      1. times-frac 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in t1 around inf 69.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+112} \lor \neg \left(u \leq 4.7 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7: 97.8% 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 69.8%

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

   1. *-commutative 69.8%

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

   2. times-frac 98.5%

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

   3. neg-mul-1 98.5%

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

   4. associate-/l* 98.4%

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

   5. associate-*r/ 98.4%

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

   6. associate-/l* 98.4%

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

   7. associate-/l/ 98.4%

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

   8. neg-mul-1 98.4%

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

   9. *-lft-identity 98.4%

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

   10. metadata-eval 98.4%

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

   11. times-frac 98.4%

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

   12. neg-mul-1 98.4%

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

   13. remove-double-neg 98.4%

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

   14. neg-mul-1 98.4%

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

   15. sub0-neg 98.4%

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

   16. associate--r+ 98.4%

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

   17. neg-sub0 98.4%

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

   18. div-sub 98.5%

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

   19. distribute-frac-neg 98.5%

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

   20. *-inverses 98.5%

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

   21. metadata-eval 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 8: 57.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.2e+112)
   (/ (- v) u)
   (if (<= u 5e+137) (/ (- v) t1) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.2e+112) {
		tmp = -v / u;
	} else if (u <= 5e+137) {
		tmp = -v / 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 (u <= (-2.2d+112)) then
        tmp = -v / u
    else if (u <= 5d+137) then
        tmp = -v / 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 (u <= -2.2e+112) {
		tmp = -v / u;
	} else if (u <= 5e+137) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.2e+112:
		tmp = -v / u
	elif u <= 5e+137:
		tmp = -v / t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.2e+112)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 5e+137)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.2e+112)
		tmp = -v / u;
	elseif (u <= 5e+137)
		tmp = -v / t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.2e+112], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 5e+137], N[((-v) / t1), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.1999999999999999e112

   1. Initial program 78.9%

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

      1. *-commutative 78.9%

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

      2. times-frac 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-*r/ 99.8%

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

      6. associate-/l* 99.8%

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

      7. associate-/l/ 99.8%

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

      8. neg-mul-1 99.8%

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

      9. *-lft-identity 99.8%

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

      10. metadata-eval 99.8%

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

      11. times-frac 99.8%

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

      12. neg-mul-1 99.8%

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

      13. remove-double-neg 99.8%

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

      14. neg-mul-1 99.8%

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

      15. sub0-neg 99.8%

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

      16. associate--r+ 99.8%

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

      17. neg-sub0 99.8%

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

      18. div-sub 99.8%

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

      19. distribute-frac-neg 99.8%

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

      20. *-inverses 99.8%

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

      21. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 94.0%

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

   5. Taylor expanded in u around 0 38.3%

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

      1. associate-*r/ 38.3%

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

      2. neg-mul-1 38.3%

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

   7. Simplified 38.3%

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

   if -2.1999999999999999e112 < u < 5.0000000000000002e137

   1. Initial program 67.1%

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

      1. associate-/l* 71.9%

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

      2. neg-mul-1 71.9%

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

      3. *-commutative 71.9%

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

      4. associate-*r/ 71.8%

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

      5. associate-/l* 72.2%

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

      6. neg-mul-1 72.2%

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

      7. associate-/r* 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in t1 around inf 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   6. Simplified 65.8%

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

   if 5.0000000000000002e137 < u

   1. Initial program 74.5%

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

      1. associate-/l* 74.9%

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

      2. neg-mul-1 74.9%

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

      3. *-commutative 74.9%

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

      4. associate-*r/ 74.9%

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

      5. associate-/l* 74.8%

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

      6. neg-mul-1 74.8%

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

      7. associate-/r* 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r/ 99.9%

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

      2. add-sqr-sqrt 56.9%

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

      3. sqrt-unprod 74.3%

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

      4. sqr-neg 74.3%

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

      5. sqrt-unprod 34.0%

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

      6. add-sqr-sqrt 67.9%

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

      7. frac-2neg 67.9%

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

      8. add-sqr-sqrt 33.9%

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

      9. sqrt-unprod 64.8%

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

      10. sqr-neg 64.8%

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

      11. sqrt-unprod 42.8%

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

      12. add-sqr-sqrt 99.9%

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

      13. distribute-neg-in 99.9%

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

      14. add-sqr-sqrt 47.5%

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

      15. sqrt-unprod 92.7%

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

      16. sqr-neg 92.7%

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

      17. sqrt-unprod 50.1%

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

      18. add-sqr-sqrt 97.6%

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

      19. sub-neg 97.6%

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

   5. Applied egg-rr 97.6%

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

   6. Taylor expanded in t1 around inf 47.7%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]

Alternative 9: 57.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+112} \lor \neg \left(u \leq 2.5 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.9e+112) (not (<= u 2.5e+142))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+112) || !(u <= 2.5e+142)) {
		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 <= (-1.9d+112)) .or. (.not. (u <= 2.5d+142))) 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 <= -1.9e+112) || !(u <= 2.5e+142)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.9e+112) or not (u <= 2.5e+142):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.9e+112) || !(u <= 2.5e+142))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.9e+112) || ~((u <= 2.5e+142)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.9e+112], N[Not[LessEqual[u, 2.5e+142]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.90000000000000004e112 or 2.5000000000000001e142 < u

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

      2. times-frac 98.6%

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

      3. neg-mul-1 98.6%

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

      4. associate-/l* 98.6%

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

      5. associate-*r/ 98.6%

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

      6. associate-/l* 98.6%

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

      7. associate-/l/ 98.6%

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

      8. neg-mul-1 98.6%

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

      9. *-lft-identity 98.6%

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

      10. metadata-eval 98.6%

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

      11. times-frac 98.6%

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

      12. neg-mul-1 98.6%

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

      13. remove-double-neg 98.6%

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

      14. neg-mul-1 98.6%

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

      15. sub0-neg 98.6%

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

      16. associate--r+ 98.6%

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

      17. neg-sub0 98.6%

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

      18. div-sub 98.6%

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

      19. distribute-frac-neg 98.6%

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

      20. *-inverses 98.6%

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

      21. metadata-eval 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in t1 around 0 93.4%

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

   5. Taylor expanded in u around 0 44.2%

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

      1. associate-*r/ 44.2%

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

      2. neg-mul-1 44.2%

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

   7. Simplified 44.2%

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

   if -1.90000000000000004e112 < u < 2.5000000000000001e142

   1. Initial program 67.6%

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

      1. associate-/l* 72.3%

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

      2. neg-mul-1 72.3%

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

      3. *-commutative 72.3%

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

      4. associate-*r/ 72.3%

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

      5. associate-/l* 72.6%

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

      6. neg-mul-1 72.6%

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

      7. associate-/r* 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in t1 around inf 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   6. Simplified 64.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+112} \lor \neg \left(u \leq 2.5 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10: 60.4% 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 69.8%

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

   1. times-frac 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in t1 around inf 61.0%

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

5. Final simplification 61.0%

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

Alternative 11: 53.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

   1. associate-/l* 73.3%

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

   2. neg-mul-1 73.3%

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

   3. *-commutative 73.3%

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

   4. associate-*r/ 73.3%

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

   5. associate-/l* 73.5%

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

   6. neg-mul-1 73.5%

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

   7. associate-/r* 85.4%

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

3. Simplified 85.4%

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

4. Taylor expanded in t1 around inf 52.7%

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

   1. associate-*r/ 52.7%

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

   2. neg-mul-1 52.7%

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

6. Simplified 52.7%

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

7. Final simplification 52.7%

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

Alternative 12: 14.0% 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 69.8%

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

   1. associate-/l* 73.3%

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

   2. neg-mul-1 73.3%

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

   3. *-commutative 73.3%

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

   4. associate-*r/ 73.3%

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

   5. associate-/l* 73.5%

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

   6. neg-mul-1 73.5%

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

   7. associate-/r* 85.4%

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

3. Simplified 85.4%

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

   1. associate-*r/ 98.1%

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

   2. add-sqr-sqrt 49.5%

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

   3. sqrt-unprod 55.8%

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

   4. sqr-neg 55.8%

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

   5. sqrt-unprod 19.6%

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

   6. add-sqr-sqrt 40.3%

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

   7. frac-2neg 40.3%

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

   8. add-sqr-sqrt 20.6%

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

   9. sqrt-unprod 56.2%

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

   10. sqr-neg 56.2%

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

   11. sqrt-unprod 48.3%

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

   12. add-sqr-sqrt 98.1%

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

   13. distribute-neg-in 98.1%

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

   14. add-sqr-sqrt 51.9%

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

   15. sqrt-unprod 74.5%

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

   16. sqr-neg 74.5%

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

   17. sqrt-unprod 30.5%

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

   18. add-sqr-sqrt 62.7%

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

   19. sub-neg 62.7%

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

5. Applied egg-rr 62.7%

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

6. Taylor expanded in t1 around inf 17.5%

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

7. Final simplification 17.5%

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

Reproduce

herbie shell --seed 2023258 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))