Rosa's DopplerBench

Percentage Accurate: 72.5% → 98.1%

Time: 12.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.5% 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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-/r* 85.5%

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

   2. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.3%

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

   2. clear-num 99.2%

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

   3. add-sqr-sqrt 50.9%

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

   4. sqrt-unprod 45.8%

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

   5. sqr-neg 45.8%

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

   6. sqrt-unprod 15.6%

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

   7. add-sqr-sqrt 35.6%

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

   8. distribute-lft-neg-in 35.6%

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

   9. distribute-rgt-neg-in 35.6%

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

   10. add-sqr-sqrt 15.6%

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

   11. sqrt-unprod 45.8%

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

   12. sqr-neg 45.8%

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

   13. sqrt-unprod 50.9%

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

   14. add-sqr-sqrt 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

Alternative 2: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.06 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -1.26 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -1.06e+154)
     (/ (- v) t1)
     (if (<= t1 -1.26e-185)
       t_1
       (if (<= t1 2.8e-146)
         (/ (/ (- t1) (/ u v)) u)
         (if (<= t1 3.2e+136) t_1 (/ v (- u t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.06e+154) {
		tmp = -v / t1;
	} else if (t1 <= -1.26e-185) {
		tmp = t_1;
	} else if (t1 <= 2.8e-146) {
		tmp = (-t1 / (u / v)) / u;
	} else if (t1 <= 3.2e+136) {
		tmp = t_1;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-1.06d+154)) then
        tmp = -v / t1
    else if (t1 <= (-1.26d-185)) then
        tmp = t_1
    else if (t1 <= 2.8d-146) then
        tmp = (-t1 / (u / v)) / u
    else if (t1 <= 3.2d+136) then
        tmp = t_1
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.06e+154) {
		tmp = -v / t1;
	} else if (t1 <= -1.26e-185) {
		tmp = t_1;
	} else if (t1 <= 2.8e-146) {
		tmp = (-t1 / (u / v)) / u;
	} else if (t1 <= 3.2e+136) {
		tmp = t_1;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -1.06e+154:
		tmp = -v / t1
	elif t1 <= -1.26e-185:
		tmp = t_1
	elif t1 <= 2.8e-146:
		tmp = (-t1 / (u / v)) / u
	elif t1 <= 3.2e+136:
		tmp = t_1
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -1.06e+154)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= -1.26e-185)
		tmp = t_1;
	elseif (t1 <= 2.8e-146)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / u);
	elseif (t1 <= 3.2e+136)
		tmp = t_1;
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -1.06e+154)
		tmp = -v / t1;
	elseif (t1 <= -1.26e-185)
		tmp = t_1;
	elseif (t1 <= 2.8e-146)
		tmp = (-t1 / (u / v)) / u;
	elseif (t1 <= 3.2e+136)
		tmp = t_1;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.06e+154], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, -1.26e-185], t$95$1, If[LessEqual[t1, 2.8e-146], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[t1, 3.2e+136], t$95$1, N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.06e154

   1. Initial program 32.6%

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

      1. associate-*l/ 34.2%

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

      2. *-commutative 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in t1 around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.06e154 < t1 < -1.2599999999999999e-185 or 2.80000000000000003e-146 < t1 < 3.19999999999999988e136

   1. Initial program 87.4%

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

      1. associate-*l/ 93.1%

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

      2. *-commutative 93.1%

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

   3. Simplified 93.1%

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

   if -1.2599999999999999e-185 < t1 < 2.80000000000000003e-146

   1. Initial program 71.3%

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

      1. associate-/r* 86.7%

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

      2. associate-/l* 98.2%

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

   3. Simplified 98.2%

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

      1. div-inv 98.1%

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

      2. clear-num 98.3%

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

      3. add-sqr-sqrt 56.4%

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

      4. sqrt-unprod 47.3%

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

      5. sqr-neg 47.3%

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

      6. sqrt-unprod 13.6%

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

      7. add-sqr-sqrt 41.8%

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

      8. distribute-lft-neg-in 41.8%

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

      9. distribute-rgt-neg-in 41.8%

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

      10. add-sqr-sqrt 13.6%

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

      11. sqrt-unprod 47.3%

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

      12. sqr-neg 47.3%

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

      13. sqrt-unprod 56.4%

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

      14. add-sqr-sqrt 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in t1 around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. *-commutative 71.3%

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

      3. unpow2 71.3%

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

      4. associate-/r* 77.2%

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

      5. associate-*r/ 80.3%

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

   8. Simplified 80.3%

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

      1. associate-*r/ 77.2%

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

      2. associate-*l/ 81.1%

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

      3. clear-num 81.1%

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

      4. associate-*l/ 81.2%

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

      5. *-un-lft-identity 81.2%

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

   10. Applied egg-rr 81.2%

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

   if 3.19999999999999988e136 < t1

   1. Initial program 55.7%

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

      1. associate-*l/ 56.7%

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

      2. *-commutative 56.7%

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

   3. Simplified 56.7%

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

      1. associate-/r* 99.3%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/r/ 96.4%

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

      5. div-inv 96.0%

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

      6. clear-num 95.9%

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

      7. associate-/r/ 95.9%

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

      8. clear-num 99.6%

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

      9. *-commutative 99.6%

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

      10. clear-num 95.9%

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

      11. div-inv 96.0%

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

      12. frac-2neg 96.0%

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

      13. remove-double-neg 96.0%

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

      14. associate-*l/ 96.2%

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

   5. Applied egg-rr 51.0%

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

   6. Taylor expanded in t1 around inf 45.7%

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

      1. frac-2neg 45.7%

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

      2. associate-/r/ 45.7%

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

      3. sub-neg 45.7%

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

      4. distribute-neg-in 45.7%

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

      5. add-sqr-sqrt 26.4%

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

      6. sqrt-unprod 47.9%

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

      7. sqr-neg 47.9%

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

      8. sqrt-unprod 19.2%

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

      9. add-sqr-sqrt 45.8%

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

      10. distribute-neg-in 45.8%

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

      11. +-commutative 45.8%

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

      12. distribute-neg-in 45.8%

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

      13. add-sqr-sqrt 26.6%

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

      14. sqrt-unprod 47.9%

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

      15. sqr-neg 47.9%

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

      16. sqrt-unprod 19.2%

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

      17. add-sqr-sqrt 45.7%

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

      18. add-sqr-sqrt 21.0%

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

      19. sqrt-unprod 63.0%

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

      20. sqr-neg 63.0%

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

      21. sqrt-unprod 49.9%

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

      22. add-sqr-sqrt 94.5%

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

   8. Applied egg-rr 94.5%

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

      1. associate-*l/ 94.8%

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

      2. *-lft-identity 94.8%

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

      3. sub-neg 94.8%

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

   10. Simplified 94.8%

       \[\leadsto \color{blue}{\frac{v}{u - t1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.06 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -1.26 \cdot 10^{-185}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+136}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 3: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 7.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e-67)
   (/ (- v) (+ t1 u))
   (if (<= t1 7.3e-41) (/ (* v (/ (- t1) u)) u) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e-67) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 7.3e-41) {
		tmp = (v * (-t1 / u)) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.2d-67)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 7.3d-41) then
        tmp = (v * (-t1 / u)) / u
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e-67) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 7.3e-41) {
		tmp = (v * (-t1 / u)) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e-67:
		tmp = -v / (t1 + u)
	elif t1 <= 7.3e-41:
		tmp = (v * (-t1 / u)) / u
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e-67)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 7.3e-41)
		tmp = Float64(Float64(v * Float64(Float64(-t1) / u)) / u);
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e-67)
		tmp = -v / (t1 + u);
	elseif (t1 <= 7.3e-41)
		tmp = (v * (-t1 / u)) / u;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e-67], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 7.3e-41], N[(N[(v * N[((-t1) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.2e-67

   1. Initial program 72.7%

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

      1. associate-/r* 82.6%

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

      2. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 80.7%

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

      1. neg-mul-1 80.7%

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

   6. Simplified 80.7%

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

   if -1.2e-67 < t1 < 7.30000000000000026e-41

   1. Initial program 75.8%

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

      1. associate-/r* 89.0%

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

      2. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

      1. div-inv 98.0%

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

      2. clear-num 98.1%

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

      3. add-sqr-sqrt 51.6%

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

      4. sqrt-unprod 54.6%

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

      5. sqr-neg 54.6%

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

      6. sqrt-unprod 15.5%

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

      7. add-sqr-sqrt 37.2%

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

      8. distribute-lft-neg-in 37.2%

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

      9. distribute-rgt-neg-in 37.2%

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

      10. add-sqr-sqrt 15.5%

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

      11. sqrt-unprod 54.6%

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

      12. sqr-neg 54.6%

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

      13. sqrt-unprod 51.6%

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

      14. add-sqr-sqrt 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in t1 around 0 69.0%

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

      1. mul-1-neg 69.0%

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

      2. *-commutative 69.0%

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

      3. unpow2 69.0%

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

      4. associate-/r* 74.8%

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

      5. associate-*r/ 76.7%

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

   8. Simplified 76.7%

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

   if 7.30000000000000026e-41 < t1

   1. Initial program 72.2%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

   3. Simplified 76.5%

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

      1. associate-/r* 97.1%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r/ 98.1%

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

      5. div-inv 97.9%

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

      6. clear-num 97.8%

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

      7. associate-/r/ 97.8%

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

      8. clear-num 99.6%

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

      9. *-commutative 99.6%

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

      10. clear-num 97.8%

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

      11. div-inv 97.9%

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

      12. frac-2neg 97.9%

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

      13. remove-double-neg 97.9%

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

      14. associate-*l/ 98.0%

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

   5. Applied egg-rr 45.0%

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

   6. Taylor expanded in t1 around inf 34.9%

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

      1. frac-2neg 34.9%

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

      2. associate-/r/ 33.7%

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

      3. sub-neg 33.7%

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

      4. distribute-neg-in 33.7%

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

      5. add-sqr-sqrt 19.9%

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

      6. sqrt-unprod 37.1%

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

      7. sqr-neg 37.1%

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

      8. sqrt-unprod 13.7%

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

      9. add-sqr-sqrt 33.9%

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

      10. distribute-neg-in 33.9%

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

      11. +-commutative 33.9%

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

      12. distribute-neg-in 33.9%

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

      13. add-sqr-sqrt 20.2%

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

      14. sqrt-unprod 37.3%

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

      15. sqr-neg 37.3%

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

      16. sqrt-unprod 13.8%

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

      17. add-sqr-sqrt 33.7%

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

      18. add-sqr-sqrt 14.2%

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

      19. sqrt-unprod 53.0%

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

      20. sqr-neg 53.0%

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

      21. sqrt-unprod 52.5%

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

      22. add-sqr-sqrt 87.7%

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

   8. Applied egg-rr 87.7%

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

      1. associate-*l/ 87.9%

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

      2. *-lft-identity 87.9%

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

      3. sub-neg 87.9%

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

   10. Simplified 87.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 7.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.25e-67)
   (/ (- v) (+ t1 u))
   (if (<= t1 3.3e-41) (/ (/ (- t1) (/ u v)) u) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.25e-67) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 3.3e-41) {
		tmp = (-t1 / (u / v)) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.25d-67)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 3.3d-41) then
        tmp = (-t1 / (u / v)) / u
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.25e-67) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 3.3e-41) {
		tmp = (-t1 / (u / v)) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.25e-67:
		tmp = -v / (t1 + u)
	elif t1 <= 3.3e-41:
		tmp = (-t1 / (u / v)) / u
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.25e-67)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 3.3e-41)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / u);
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.25e-67)
		tmp = -v / (t1 + u);
	elseif (t1 <= 3.3e-41)
		tmp = (-t1 / (u / v)) / u;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.25e-67], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.3e-41], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.25e-67

   1. Initial program 72.7%

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

      1. associate-/r* 82.6%

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

      2. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 80.7%

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

      1. neg-mul-1 80.7%

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

   6. Simplified 80.7%

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

   if -1.25e-67 < t1 < 3.30000000000000024e-41

   1. Initial program 75.8%

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

      1. associate-/r* 89.0%

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

      2. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

      1. div-inv 98.0%

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

      2. clear-num 98.1%

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

      3. add-sqr-sqrt 51.6%

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

      4. sqrt-unprod 54.6%

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

      5. sqr-neg 54.6%

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

      6. sqrt-unprod 15.5%

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

      7. add-sqr-sqrt 37.2%

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

      8. distribute-lft-neg-in 37.2%

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

      9. distribute-rgt-neg-in 37.2%

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

      10. add-sqr-sqrt 15.5%

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

      11. sqrt-unprod 54.6%

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

      12. sqr-neg 54.6%

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

      13. sqrt-unprod 51.6%

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

      14. add-sqr-sqrt 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in t1 around 0 69.0%

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

      1. mul-1-neg 69.0%

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

      2. *-commutative 69.0%

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

      3. unpow2 69.0%

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

      4. associate-/r* 74.8%

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

      5. associate-*r/ 76.7%

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

   8. Simplified 76.7%

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

      1. associate-*r/ 74.8%

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

      2. associate-*l/ 77.9%

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

      3. clear-num 77.9%

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

      4. associate-*l/ 77.9%

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

      5. *-un-lft-identity 77.9%

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

   10. Applied egg-rr 77.9%

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

   if 3.30000000000000024e-41 < t1

   1. Initial program 72.2%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

   3. Simplified 76.5%

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

      1. associate-/r* 97.1%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r/ 98.1%

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

      5. div-inv 97.9%

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

      6. clear-num 97.8%

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

      7. associate-/r/ 97.8%

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

      8. clear-num 99.6%

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

      9. *-commutative 99.6%

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

      10. clear-num 97.8%

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

      11. div-inv 97.9%

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

      12. frac-2neg 97.9%

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

      13. remove-double-neg 97.9%

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

      14. associate-*l/ 98.0%

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

   5. Applied egg-rr 45.0%

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

   6. Taylor expanded in t1 around inf 34.9%

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

      1. frac-2neg 34.9%

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

      2. associate-/r/ 33.7%

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

      3. sub-neg 33.7%

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

      4. distribute-neg-in 33.7%

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

      5. add-sqr-sqrt 19.9%

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

      6. sqrt-unprod 37.1%

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

      7. sqr-neg 37.1%

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

      8. sqrt-unprod 13.7%

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

      9. add-sqr-sqrt 33.9%

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

      10. distribute-neg-in 33.9%

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

      11. +-commutative 33.9%

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

      12. distribute-neg-in 33.9%

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

      13. add-sqr-sqrt 20.2%

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

      14. sqrt-unprod 37.3%

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

      15. sqr-neg 37.3%

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

      16. sqrt-unprod 13.8%

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

      17. add-sqr-sqrt 33.7%

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

      18. add-sqr-sqrt 14.2%

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

      19. sqrt-unprod 53.0%

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

      20. sqr-neg 53.0%

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

      21. sqrt-unprod 52.5%

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

      22. add-sqr-sqrt 87.7%

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

   8. Applied egg-rr 87.7%

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

      1. associate-*l/ 87.9%

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

      2. *-lft-identity 87.9%

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

      3. sub-neg 87.9%

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

   10. Simplified 87.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.25 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 5: 68.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+118} \lor \neg \left(u \leq 5.5 \cdot 10^{+158}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.55e+118) (not (<= u 5.5e+158)))
   (* v (/ (/ t1 u) u))
   (/ v (- u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.55e+118) || !(u <= 5.5e+158)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.55d+118)) .or. (.not. (u <= 5.5d+158))) then
        tmp = v * ((t1 / u) / u)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.55e+118) || !(u <= 5.5e+158)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.55e+118) or not (u <= 5.5e+158):
		tmp = v * ((t1 / u) / u)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.55e+118) || !(u <= 5.5e+158))
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.55e+118) || ~((u <= 5.5e+158)))
		tmp = v * ((t1 / u) / u);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.55e+118], N[Not[LessEqual[u, 5.5e+158]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.54999999999999993e118 or 5.4999999999999998e158 < u

   1. Initial program 86.3%

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

      1. associate-*l/ 85.0%

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

      2. *-commutative 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in t1 around 0 85.0%

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

      1. associate-*r/ 85.0%

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

      2. neg-mul-1 85.0%

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

      3. unpow2 85.0%

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

   6. Simplified 85.0%

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

      1. add-sqr-sqrt 32.8%

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

      2. sqrt-unprod 70.6%

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

      3. sqr-neg 70.6%

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

      4. sqrt-unprod 50.6%

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

      5. times-frac 49.1%

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

   8. Applied egg-rr 49.1%

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

      1. times-frac 50.6%

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

      2. rem-square-sqrt 83.4%

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

      3. *-rgt-identity 83.4%

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

      4. times-frac 80.4%

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

      5. associate-*r/ 80.4%

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

      6. *-rgt-identity 80.4%

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

   10. Simplified 80.4%

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

   if -1.54999999999999993e118 < u < 5.4999999999999998e158

   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/ 75.0%

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

      2. *-commutative 75.0%

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

   3. Simplified 75.0%

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

      1. associate-/r* 98.3%

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

      2. associate-*r/ 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-/r/ 98.0%

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

      5. div-inv 97.7%

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

      6. clear-num 97.6%

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

      7. associate-/r/ 97.5%

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

      8. clear-num 98.6%

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

      9. *-commutative 98.6%

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

      10. clear-num 97.5%

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

      11. div-inv 97.7%

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

      12. frac-2neg 97.7%

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

      13. remove-double-neg 97.7%

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

      14. associate-*l/ 97.9%

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

   5. Applied egg-rr 42.1%

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

   6. Taylor expanded in t1 around inf 16.6%

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

      1. frac-2neg 16.6%

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

      2. associate-/r/ 16.6%

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

      3. sub-neg 16.6%

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

      4. distribute-neg-in 16.6%

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

      5. add-sqr-sqrt 7.8%

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

      6. sqrt-unprod 16.6%

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

      7. sqr-neg 16.6%

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

      8. sqrt-unprod 9.0%

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

      9. add-sqr-sqrt 17.2%

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

      10. distribute-neg-in 17.2%

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

      11. +-commutative 17.2%

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

      12. distribute-neg-in 17.2%

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

      13. add-sqr-sqrt 8.3%

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

      14. sqrt-unprod 17.0%

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

      15. sqr-neg 17.0%

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

      16. sqrt-unprod 8.7%

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

      17. add-sqr-sqrt 16.6%

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

      18. add-sqr-sqrt 10.0%

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

      19. sqrt-unprod 37.3%

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

      20. sqr-neg 37.3%

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

      21. sqrt-unprod 32.5%

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

      22. add-sqr-sqrt 70.6%

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

   8. Applied egg-rr 70.6%

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

      1. associate-*l/ 70.9%

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

      2. *-lft-identity 70.9%

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

      3. sub-neg 70.9%

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

   10. Simplified 70.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+118} \lor \neg \left(u \leq 5.5 \cdot 10^{+158}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]

Alternative 6: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+119}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.6e+119)
   (/ v (/ (* u u) t1))
   (if (<= u 3.8e+158) (/ v (- u t1)) (* v (/ (/ t1 u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e+119) {
		tmp = v / ((u * u) / t1);
	} else if (u <= 3.8e+158) {
		tmp = v / (u - t1);
	} else {
		tmp = v * ((t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.6d+119)) then
        tmp = v / ((u * u) / t1)
    else if (u <= 3.8d+158) then
        tmp = v / (u - t1)
    else
        tmp = v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.6e+119) {
		tmp = v / ((u * u) / t1);
	} else if (u <= 3.8e+158) {
		tmp = v / (u - t1);
	} else {
		tmp = v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.6e+119:
		tmp = v / ((u * u) / t1)
	elif u <= 3.8e+158:
		tmp = v / (u - t1)
	else:
		tmp = v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.6e+119)
		tmp = Float64(v / Float64(Float64(u * u) / t1));
	elseif (u <= 3.8e+158)
		tmp = Float64(v / Float64(u - t1));
	else
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.6e+119)
		tmp = v / ((u * u) / t1);
	elseif (u <= 3.8e+158)
		tmp = v / (u - t1);
	else
		tmp = v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.6e+119], N[(v / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.8e+158], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.6e119

   1. Initial program 84.2%

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

      1. associate-*l/ 81.7%

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

      2. *-commutative 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in t1 around 0 81.7%

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

      1. associate-*r/ 81.7%

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

      2. neg-mul-1 81.7%

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

      3. unpow2 81.7%

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

   6. Simplified 81.7%

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

      1. clear-num 81.7%

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

      2. un-div-inv 81.7%

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

      3. add-sqr-sqrt 38.2%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-unprod 40.8%

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

      7. add-sqr-sqrt 78.9%

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

   8. Applied egg-rr 78.9%

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

   if -2.6e119 < u < 3.7999999999999998e158

   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/ 75.0%

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

      2. *-commutative 75.0%

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

   3. Simplified 75.0%

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

      1. associate-/r* 98.3%

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

      2. associate-*r/ 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-/r/ 98.0%

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

      5. div-inv 97.7%

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

      6. clear-num 97.6%

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

      7. associate-/r/ 97.5%

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

      8. clear-num 98.6%

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

      9. *-commutative 98.6%

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

      10. clear-num 97.5%

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

      11. div-inv 97.7%

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

      12. frac-2neg 97.7%

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

      13. remove-double-neg 97.7%

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

      14. associate-*l/ 97.9%

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

   5. Applied egg-rr 42.1%

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

   6. Taylor expanded in t1 around inf 16.6%

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

      1. frac-2neg 16.6%

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

      2. associate-/r/ 16.6%

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

      3. sub-neg 16.6%

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

      4. distribute-neg-in 16.6%

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

      5. add-sqr-sqrt 7.8%

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

      6. sqrt-unprod 16.6%

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

      7. sqr-neg 16.6%

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

      8. sqrt-unprod 9.0%

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

      9. add-sqr-sqrt 17.2%

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

      10. distribute-neg-in 17.2%

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

      11. +-commutative 17.2%

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

      12. distribute-neg-in 17.2%

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

      13. add-sqr-sqrt 8.3%

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

      14. sqrt-unprod 17.0%

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

      15. sqr-neg 17.0%

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

      16. sqrt-unprod 8.7%

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

      17. add-sqr-sqrt 16.6%

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

      18. add-sqr-sqrt 10.0%

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

      19. sqrt-unprod 37.3%

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

      20. sqr-neg 37.3%

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

      21. sqrt-unprod 32.5%

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

      22. add-sqr-sqrt 70.6%

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

   8. Applied egg-rr 70.6%

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

      1. associate-*l/ 70.9%

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

      2. *-lft-identity 70.9%

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

      3. sub-neg 70.9%

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

   10. Simplified 70.9%

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

   if 3.7999999999999998e158 < u

   1. Initial program 89.1%

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

      1. associate-*l/ 89.3%

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

      2. *-commutative 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in t1 around 0 89.3%

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

      1. associate-*r/ 89.3%

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

      2. neg-mul-1 89.3%

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

      3. unpow2 89.3%

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

   6. Simplified 89.3%

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

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 82.3%

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

      3. sqr-neg 82.3%

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

      4. sqrt-unprod 63.4%

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

      5. times-frac 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. times-frac 63.4%

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

      2. rem-square-sqrt 89.3%

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

      3. *-rgt-identity 89.3%

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

      4. times-frac 89.3%

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

      5. associate-*r/ 89.3%

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

      6. *-rgt-identity 89.3%

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

   10. Simplified 89.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+119}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 7: 58.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+118} \lor \neg \left(u \leq 2.2 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.1e+118) (not (<= u 2.2e+154))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.1e+118) || !(u <= 2.2e+154)) {
		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 <= (-2.1d+118)) .or. (.not. (u <= 2.2d+154))) 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 <= -2.1e+118) || !(u <= 2.2e+154)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.1e+118) or not (u <= 2.2e+154):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.1e+118) || !(u <= 2.2e+154))
		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 <= -2.1e+118) || ~((u <= 2.2e+154)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.1e+118], N[Not[LessEqual[u, 2.2e+154]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.1e118 or 2.2000000000000001e154 < u

   1. Initial program 85.3%

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

      1. associate-/r* 95.4%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. associate-/l* 97.0%

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

      3. distribute-neg-frac 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in t1 around inf 43.9%

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

      1. neg-mul-1 43.9%

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

      2. distribute-neg-frac 43.9%

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

   9. Simplified 43.9%

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

   if -2.1e118 < u < 2.2000000000000001e154

   1. Initial program 70.0%

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

      1. associate-*l/ 75.2%

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

      2. *-commutative 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in t1 around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. neg-mul-1 69.2%

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

   6. Simplified 69.2%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+118} \lor \neg \left(u \leq 2.2 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 58.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.6e+118) (/ v u) (if (<= u 1.55e+154) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.6e+118) {
		tmp = v / u;
	} else if (u <= 1.55e+154) {
		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 <= (-4.6d+118)) then
        tmp = v / u
    else if (u <= 1.55d+154) 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 <= -4.6e+118) {
		tmp = v / u;
	} else if (u <= 1.55e+154) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.6e+118:
		tmp = v / u
	elif u <= 1.55e+154:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.6e+118)
		tmp = Float64(v / u);
	elseif (u <= 1.55e+154)
		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 <= -4.6e+118)
		tmp = v / u;
	elseif (u <= 1.55e+154)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.6e+118], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.55e+154], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -4.60000000000000032e118 or 1.5500000000000001e154 < u

   1. Initial program 85.3%

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

      1. associate-*l/ 84.0%

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

      2. *-commutative 84.0%

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

   3. Simplified 84.0%

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

      1. associate-/r* 89.3%

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

      2. associate-*r/ 97.4%

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

      3. *-commutative 97.4%

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

      4. associate-/r/ 99.9%

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

      5. div-inv 99.9%

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

      6. clear-num 99.9%

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

      7. associate-/r/ 99.9%

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

      8. clear-num 99.9%

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

      9. *-commutative 99.9%

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

      10. clear-num 99.9%

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

      11. div-inv 99.9%

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

      12. frac-2neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. associate-*l/ 97.4%

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

   5. Applied egg-rr 94.4%

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

   6. Taylor expanded in t1 around 0 94.5%

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

      1. neg-mul-1 94.5%

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

      2. distribute-neg-frac 94.5%

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

   8. Simplified 94.5%

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

      1. clear-num 94.3%

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

      2. associate-/r/ 94.5%

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

      3. clear-num 94.5%

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

      4. add-sqr-sqrt 52.0%

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

      5. sqrt-unprod 85.5%

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

      6. sqr-neg 85.5%

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

      7. sqrt-unprod 39.3%

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

      8. add-sqr-sqrt 82.3%

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

   10. Applied egg-rr 82.3%

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

   11. Taylor expanded in u around 0 43.8%

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

   if -4.60000000000000032e118 < u < 1.5500000000000001e154

   1. Initial program 70.0%

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

      1. associate-*l/ 75.2%

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

      2. *-commutative 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in t1 around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. neg-mul-1 69.2%

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

   6. Simplified 69.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 9: 23.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+123}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.2e+52) (/ v t1) (if (<= t1 9e+123) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.2e+52) {
		tmp = v / t1;
	} else if (t1 <= 9e+123) {
		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 (t1 <= (-1.2d+52)) then
        tmp = v / t1
    else if (t1 <= 9d+123) 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 (t1 <= -1.2e+52) {
		tmp = v / t1;
	} else if (t1 <= 9e+123) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.2e+52:
		tmp = v / t1
	elif t1 <= 9e+123:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.2e+52)
		tmp = Float64(v / t1);
	elseif (t1 <= 9e+123)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.2e+52)
		tmp = v / t1;
	elseif (t1 <= 9e+123)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.2e+52], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 9e+123], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.2e52 or 8.99999999999999965e123 < t1

   1. Initial program 57.9%

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

      1. associate-*l/ 60.0%

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

      2. *-commutative 60.0%

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

   3. Simplified 60.0%

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

      1. associate-/r* 97.9%

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

      2. associate-*r/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/r/ 98.0%

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

      5. div-inv 97.6%

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

      6. clear-num 97.5%

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

      7. associate-/r/ 97.6%

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

      8. clear-num 99.6%

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

      9. *-commutative 99.6%

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

      10. clear-num 97.6%

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

      11. div-inv 97.6%

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

      12. frac-2neg 97.6%

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

      13. remove-double-neg 97.6%

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

      14. associate-*l/ 97.8%

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

   5. Applied egg-rr 46.9%

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

   6. Taylor expanded in t1 around inf 38.2%

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

   7. Taylor expanded in t1 around inf 35.8%

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

   if -1.2e52 < t1 < 8.99999999999999965e123

   1. Initial program 81.8%

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

      1. associate-*l/ 86.1%

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

      2. *-commutative 86.1%

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

   3. Simplified 86.1%

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

      1. associate-/r* 95.1%

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

      2. associate-*r/ 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-/r/ 98.7%

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

      5. div-inv 98.5%

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

      6. clear-num 98.5%

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

      7. associate-/r/ 98.4%

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

      8. clear-num 98.5%

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

      9. *-commutative 98.5%

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

      10. clear-num 98.4%

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

      11. div-inv 98.5%

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

      12. frac-2neg 98.5%

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

      13. remove-double-neg 98.5%

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

      14. associate-*l/ 97.8%

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

   5. Applied egg-rr 59.2%

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

   6. Taylor expanded in t1 around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. distribute-neg-frac 61.1%

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

   8. Simplified 61.1%

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

      1. clear-num 61.0%

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

      2. associate-/r/ 61.0%

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

      3. clear-num 61.0%

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

      4. add-sqr-sqrt 28.4%

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

      5. sqrt-unprod 45.3%

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

      6. sqr-neg 45.3%

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

      7. sqrt-unprod 19.0%

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

      8. add-sqr-sqrt 35.2%

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

   10. Applied egg-rr 35.2%

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

   11. Taylor expanded in u around 0 18.6%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 9 \cdot 10^{+123}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 10: 61.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-*l/ 77.4%

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

   2. *-commutative 77.4%

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

3. Simplified 77.4%

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

   1. associate-/r* 96.0%

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

   2. associate-*r/ 98.7%

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

   3. *-commutative 98.7%

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

   4. associate-/r/ 98.5%

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

   5. div-inv 98.2%

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

   6. clear-num 98.2%

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

   7. associate-/r/ 98.1%

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

   8. clear-num 98.9%

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

   9. *-commutative 98.9%

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

   10. clear-num 98.1%

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

   11. div-inv 98.2%

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

   12. frac-2neg 98.2%

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

   13. remove-double-neg 98.2%

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

   14. associate-*l/ 97.8%

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

5. Applied egg-rr 55.1%

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

6. Taylor expanded in t1 around inf 23.9%

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

   1. frac-2neg 23.9%

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

   2. associate-/r/ 23.1%

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

   3. sub-neg 23.1%

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

   4. distribute-neg-in 23.1%

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

   5. add-sqr-sqrt 12.9%

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

   6. sqrt-unprod 32.0%

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

   7. sqr-neg 32.0%

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

   8. sqrt-unprod 10.4%

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

   9. add-sqr-sqrt 23.6%

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

   10. distribute-neg-in 23.6%

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

   11. +-commutative 23.6%

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

   12. distribute-neg-in 23.6%

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

   13. add-sqr-sqrt 13.2%

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

   14. sqrt-unprod 32.3%

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

   15. sqr-neg 32.3%

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

   16. sqrt-unprod 10.3%

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

   17. add-sqr-sqrt 23.1%

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

   18. add-sqr-sqrt 11.6%

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

   19. sqrt-unprod 38.7%

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

   20. sqr-neg 38.7%

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

   21. sqrt-unprod 31.5%

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

   22. add-sqr-sqrt 64.4%

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

8. Applied egg-rr 64.4%

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

   1. associate-*l/ 64.6%

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

   2. *-lft-identity 64.6%

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

   3. sub-neg 64.6%

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

10. Simplified 64.6%

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

11. Final simplification 64.6%

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

Alternative 11: 14.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-*l/ 77.4%

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

   2. *-commutative 77.4%

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

3. Simplified 77.4%

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

   1. associate-/r* 96.0%

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

   2. associate-*r/ 98.7%

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

   3. *-commutative 98.7%

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

   4. associate-/r/ 98.5%

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

   5. div-inv 98.2%

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

   6. clear-num 98.2%

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

   7. associate-/r/ 98.1%

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

   8. clear-num 98.9%

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

   9. *-commutative 98.9%

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

   10. clear-num 98.1%

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

   11. div-inv 98.2%

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

   12. frac-2neg 98.2%

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

   13. remove-double-neg 98.2%

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

   14. associate-*l/ 97.8%

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

5. Applied egg-rr 55.1%

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

6. Taylor expanded in t1 around inf 23.9%

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

7. Taylor expanded in t1 around inf 14.0%

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

8. Final simplification 14.0%

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

Reproduce

herbie shell --seed 2023287 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))