Rosa's DopplerBench

Percentage Accurate: 72.6% → 98.0%

Time: 10.5s

Alternatives: 14

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: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. times-frac 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{v}{t1 - u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+120}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{\frac{t1 - u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1e-11)
   (/ (* t1 (/ v u)) (- t1 u))
   (if (<= u 3.6e-23)
     (/ (- v) t1)
     (if (<= u 5e+43)
       (* (/ v (- t1 u)) (/ t1 u))
       (if (<= u 9e+120)
         (/ v (- (* u -2.0) t1))
         (/ (/ v (+ t1 u)) (/ (- t1 u) t1)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e-11) {
		tmp = (t1 * (v / u)) / (t1 - u);
	} else if (u <= 3.6e-23) {
		tmp = -v / t1;
	} else if (u <= 5e+43) {
		tmp = (v / (t1 - u)) * (t1 / u);
	} else if (u <= 9e+120) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v / (t1 + u)) / ((t1 - 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 <= (-1d-11)) then
        tmp = (t1 * (v / u)) / (t1 - u)
    else if (u <= 3.6d-23) then
        tmp = -v / t1
    else if (u <= 5d+43) then
        tmp = (v / (t1 - u)) * (t1 / u)
    else if (u <= 9d+120) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (v / (t1 + u)) / ((t1 - u) / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1e-11) {
		tmp = (t1 * (v / u)) / (t1 - u);
	} else if (u <= 3.6e-23) {
		tmp = -v / t1;
	} else if (u <= 5e+43) {
		tmp = (v / (t1 - u)) * (t1 / u);
	} else if (u <= 9e+120) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (v / (t1 + u)) / ((t1 - u) / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1e-11:
		tmp = (t1 * (v / u)) / (t1 - u)
	elif u <= 3.6e-23:
		tmp = -v / t1
	elif u <= 5e+43:
		tmp = (v / (t1 - u)) * (t1 / u)
	elif u <= 9e+120:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (v / (t1 + u)) / ((t1 - u) / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1e-11)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	elseif (u <= 3.6e-23)
		tmp = Float64(Float64(-v) / t1);
	elseif (u <= 5e+43)
		tmp = Float64(Float64(v / Float64(t1 - u)) * Float64(t1 / u));
	elseif (u <= 9e+120)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(t1 - u) / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1e-11)
		tmp = (t1 * (v / u)) / (t1 - u);
	elseif (u <= 3.6e-23)
		tmp = -v / t1;
	elseif (u <= 5e+43)
		tmp = (v / (t1 - u)) * (t1 / u);
	elseif (u <= 9e+120)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (v / (t1 + u)) / ((t1 - u) / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1e-11], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.6e-23], N[((-v) / t1), $MachinePrecision], If[LessEqual[u, 5e+43], N[(N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 9e+120], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -9.99999999999999939e-12

   1. Initial program 81.0%

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

      1. times-frac 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in t1 around 0 81.6%

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

      1. frac-2neg 81.6%

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

      2. remove-double-neg 81.6%

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

      3. associate-*l/ 85.7%

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

      4. distribute-neg-in 85.7%

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

      5. add-sqr-sqrt 55.4%

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

      6. sqrt-unprod 81.8%

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

      7. sqr-neg 81.8%

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

      8. sqrt-unprod 30.5%

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

      9. add-sqr-sqrt 85.6%

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

      10. sub-neg 85.6%

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

   7. Applied egg-rr 85.6%

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

   if -9.99999999999999939e-12 < u < 3.5999999999999998e-23

   1. Initial program 59.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   7. Simplified 86.5%

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

   if 3.5999999999999998e-23 < u < 5.0000000000000004e43

   1. Initial program 78.5%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 62.1%

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

      1. frac-2neg 62.1%

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

      2. clear-num 62.1%

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

      3. frac-times 66.0%

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

      4. remove-double-neg 66.0%

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

      5. *-commutative 66.0%

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

      6. *-un-lft-identity 66.0%

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

      7. distribute-neg-in 66.0%

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

      8. add-sqr-sqrt 26.5%

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

      9. sqrt-unprod 66.1%

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

      10. sqr-neg 66.1%

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

      11. sqrt-unprod 39.0%

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

      12. add-sqr-sqrt 67.0%

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

      13. sub-neg 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 66.9%

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

      2. associate-/l* 62.8%

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

      3. *-commutative 62.8%

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

      4. times-frac 67.0%

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

   9. Simplified 67.0%

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

   if 5.0000000000000004e43 < u < 8.99999999999999953e120

   1. Initial program 60.5%

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

      1. associate-/r* 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 100.0%

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

      9. sub-neg 100.0%

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

      10. *-inverses 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. mul-1-neg 71.6%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 71.6%

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

      3. *-commutative 71.6%

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

   7. Simplified 71.6%

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

   if 8.99999999999999953e120 < u

   1. Initial program 66.0%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 83.5%

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

      3. *-un-lft-identity 83.5%

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

      4. frac-2neg 83.5%

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

      5. distribute-neg-in 83.5%

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

      6. add-sqr-sqrt 47.7%

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

      7. sqrt-unprod 71.7%

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

      8. sqr-neg 71.7%

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

      9. sqrt-unprod 30.9%

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

      10. add-sqr-sqrt 73.9%

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

      11. sub-neg 73.9%

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

      12. remove-double-neg 73.9%

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

   6. Applied egg-rr 73.9%

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

      1. associate-/l/ 90.2%

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

   8. Simplified 90.2%

      \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 - u}{t1}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{v}{t1 - u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+120}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{\frac{t1 - u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{if}\;u \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+44} \lor \neg \left(u \leq 9 \cdot 10^{+120}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v u) (/ t1 (- t1 u)))))
   (if (<= u -8.2e-13)
     t_1
     (if (<= u 2.8e-25)
       (/ (- v) t1)
       (if (or (<= u 1.05e+44) (not (<= u 9e+120)))
         t_1
         (/ v (- (* u -2.0) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / u) * (t1 / (t1 - u));
	double tmp;
	if (u <= -8.2e-13) {
		tmp = t_1;
	} else if (u <= 2.8e-25) {
		tmp = -v / t1;
	} else if ((u <= 1.05e+44) || !(u <= 9e+120)) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / u) * (t1 / (t1 - u))
    if (u <= (-8.2d-13)) then
        tmp = t_1
    else if (u <= 2.8d-25) then
        tmp = -v / t1
    else if ((u <= 1.05d+44) .or. (.not. (u <= 9d+120))) then
        tmp = t_1
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / u) * (t1 / (t1 - u));
	double tmp;
	if (u <= -8.2e-13) {
		tmp = t_1;
	} else if (u <= 2.8e-25) {
		tmp = -v / t1;
	} else if ((u <= 1.05e+44) || !(u <= 9e+120)) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / u) * (t1 / (t1 - u))
	tmp = 0
	if u <= -8.2e-13:
		tmp = t_1
	elif u <= 2.8e-25:
		tmp = -v / t1
	elif (u <= 1.05e+44) or not (u <= 9e+120):
		tmp = t_1
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / u) * Float64(t1 / Float64(t1 - u)))
	tmp = 0.0
	if (u <= -8.2e-13)
		tmp = t_1;
	elseif (u <= 2.8e-25)
		tmp = Float64(Float64(-v) / t1);
	elseif ((u <= 1.05e+44) || !(u <= 9e+120))
		tmp = t_1;
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / u) * (t1 / (t1 - u));
	tmp = 0.0;
	if (u <= -8.2e-13)
		tmp = t_1;
	elseif (u <= 2.8e-25)
		tmp = -v / t1;
	elseif ((u <= 1.05e+44) || ~((u <= 9e+120)))
		tmp = t_1;
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / u), $MachinePrecision] * N[(t1 / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -8.2e-13], t$95$1, If[LessEqual[u, 2.8e-25], N[((-v) / t1), $MachinePrecision], If[Or[LessEqual[u, 1.05e+44], N[Not[LessEqual[u, 9e+120]], $MachinePrecision]], t$95$1, N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.2000000000000004e-13 or 2.79999999999999988e-25 < u < 1.04999999999999993e44 or 8.99999999999999953e120 < u

   1. Initial program 75.9%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t1 around 0 80.4%

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

      1. *-commutative 80.4%

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

      2. clear-num 81.0%

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

      3. frac-2neg 81.0%

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

      4. frac-times 79.2%

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

      5. *-un-lft-identity 79.2%

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

      6. remove-double-neg 79.2%

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

      7. distribute-neg-in 79.2%

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

      8. add-sqr-sqrt 43.7%

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

      9. sqrt-unprod 78.5%

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

      10. sqr-neg 78.5%

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

      11. sqrt-unprod 35.4%

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

      12. add-sqr-sqrt 79.2%

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

      13. sub-neg 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. times-frac 81.1%

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

      3. clear-num 80.5%

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

   9. Applied egg-rr 80.5%

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

   if -8.2000000000000004e-13 < u < 2.79999999999999988e-25

   1. Initial program 59.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   7. Simplified 86.5%

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

   if 1.04999999999999993e44 < u < 8.99999999999999953e120

   1. Initial program 60.5%

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

      1. associate-/r* 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 100.0%

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

      9. sub-neg 100.0%

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

      10. *-inverses 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. mul-1-neg 71.6%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 71.6%

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

      3. *-commutative 71.6%

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

   7. Simplified 71.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+44} \lor \neg \left(u \leq 9 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 - u} \cdot \frac{t1}{u}\\ \mathbf{if}\;u \leq -2.9 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+43} \lor \neg \left(u \leq 1.55 \cdot 10^{+121}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ v (- t1 u)) (/ t1 u))))
   (if (<= u -2.9e-12)
     t_1
     (if (<= u 1.4e-22)
       (/ (- v) t1)
       (if (or (<= u 2.5e+43) (not (<= u 1.55e+121)))
         t_1
         (/ v (- (* u -2.0) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = (v / (t1 - u)) * (t1 / u);
	double tmp;
	if (u <= -2.9e-12) {
		tmp = t_1;
	} else if (u <= 1.4e-22) {
		tmp = -v / t1;
	} else if ((u <= 2.5e+43) || !(u <= 1.55e+121)) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (v / (t1 - u)) * (t1 / u)
    if (u <= (-2.9d-12)) then
        tmp = t_1
    else if (u <= 1.4d-22) then
        tmp = -v / t1
    else if ((u <= 2.5d+43) .or. (.not. (u <= 1.55d+121))) then
        tmp = t_1
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (v / (t1 - u)) * (t1 / u);
	double tmp;
	if (u <= -2.9e-12) {
		tmp = t_1;
	} else if (u <= 1.4e-22) {
		tmp = -v / t1;
	} else if ((u <= 2.5e+43) || !(u <= 1.55e+121)) {
		tmp = t_1;
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (v / (t1 - u)) * (t1 / u)
	tmp = 0
	if u <= -2.9e-12:
		tmp = t_1
	elif u <= 1.4e-22:
		tmp = -v / t1
	elif (u <= 2.5e+43) or not (u <= 1.55e+121):
		tmp = t_1
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(v / Float64(t1 - u)) * Float64(t1 / u))
	tmp = 0.0
	if (u <= -2.9e-12)
		tmp = t_1;
	elseif (u <= 1.4e-22)
		tmp = Float64(Float64(-v) / t1);
	elseif ((u <= 2.5e+43) || !(u <= 1.55e+121))
		tmp = t_1;
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (v / (t1 - u)) * (t1 / u);
	tmp = 0.0;
	if (u <= -2.9e-12)
		tmp = t_1;
	elseif (u <= 1.4e-22)
		tmp = -v / t1;
	elseif ((u <= 2.5e+43) || ~((u <= 1.55e+121)))
		tmp = t_1;
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.9e-12], t$95$1, If[LessEqual[u, 1.4e-22], N[((-v) / t1), $MachinePrecision], If[Or[LessEqual[u, 2.5e+43], N[Not[LessEqual[u, 1.55e+121]], $MachinePrecision]], t$95$1, N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.9000000000000002e-12 or 1.39999999999999997e-22 < u < 2.5000000000000002e43 or 1.55000000000000004e121 < u

   1. Initial program 75.9%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in t1 around 0 80.4%

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

      1. frac-2neg 80.4%

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

      2. clear-num 81.0%

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

      3. frac-times 79.2%

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

      4. remove-double-neg 79.2%

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

      5. *-commutative 79.2%

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

      6. *-un-lft-identity 79.2%

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

      7. distribute-neg-in 79.2%

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

      8. add-sqr-sqrt 43.7%

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

      9. sqrt-unprod 78.5%

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

      10. sqr-neg 78.5%

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

      11. sqrt-unprod 35.4%

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

      12. add-sqr-sqrt 79.2%

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

      13. sub-neg 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. associate-*r/ 71.9%

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

      2. associate-/l* 71.3%

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

      3. *-commutative 71.3%

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

      4. times-frac 82.6%

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

   9. Simplified 82.6%

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

   if -2.9000000000000002e-12 < u < 1.39999999999999997e-22

   1. Initial program 59.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   7. Simplified 86.5%

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

   if 2.5000000000000002e43 < u < 1.55000000000000004e121

   1. Initial program 60.5%

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

      1. associate-/r* 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 100.0%

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

      9. sub-neg 100.0%

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

      10. *-inverses 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. mul-1-neg 71.6%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 71.6%

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

      3. *-commutative 71.6%

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

   7. Simplified 71.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{v}{t1 - u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+43} \lor \neg \left(u \leq 1.55 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{v}{t1 - u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 8.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+43} \lor \neg \left(u \leq 9 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{v}{t1 - u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.4e-12)
   (/ (* t1 (/ v u)) (- t1 u))
   (if (<= u 8.8e-27)
     (/ (- v) t1)
     (if (or (<= u 3.8e+43) (not (<= u 9e+120)))
       (* (/ v (- t1 u)) (/ t1 u))
       (/ v (- (* u -2.0) t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.4e-12) {
		tmp = (t1 * (v / u)) / (t1 - u);
	} else if (u <= 8.8e-27) {
		tmp = -v / t1;
	} else if ((u <= 3.8e+43) || !(u <= 9e+120)) {
		tmp = (v / (t1 - u)) * (t1 / u);
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.4d-12)) then
        tmp = (t1 * (v / u)) / (t1 - u)
    else if (u <= 8.8d-27) then
        tmp = -v / t1
    else if ((u <= 3.8d+43) .or. (.not. (u <= 9d+120))) then
        tmp = (v / (t1 - u)) * (t1 / u)
    else
        tmp = v / ((u * (-2.0d0)) - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.4e-12) {
		tmp = (t1 * (v / u)) / (t1 - u);
	} else if (u <= 8.8e-27) {
		tmp = -v / t1;
	} else if ((u <= 3.8e+43) || !(u <= 9e+120)) {
		tmp = (v / (t1 - u)) * (t1 / u);
	} else {
		tmp = v / ((u * -2.0) - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.4e-12:
		tmp = (t1 * (v / u)) / (t1 - u)
	elif u <= 8.8e-27:
		tmp = -v / t1
	elif (u <= 3.8e+43) or not (u <= 9e+120):
		tmp = (v / (t1 - u)) * (t1 / u)
	else:
		tmp = v / ((u * -2.0) - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.4e-12)
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(t1 - u));
	elseif (u <= 8.8e-27)
		tmp = Float64(Float64(-v) / t1);
	elseif ((u <= 3.8e+43) || !(u <= 9e+120))
		tmp = Float64(Float64(v / Float64(t1 - u)) * Float64(t1 / u));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.4e-12)
		tmp = (t1 * (v / u)) / (t1 - u);
	elseif (u <= 8.8e-27)
		tmp = -v / t1;
	elseif ((u <= 3.8e+43) || ~((u <= 9e+120)))
		tmp = (v / (t1 - u)) * (t1 / u);
	else
		tmp = v / ((u * -2.0) - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.4e-12], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8.8e-27], N[((-v) / t1), $MachinePrecision], If[Or[LessEqual[u, 3.8e+43], N[Not[LessEqual[u, 9e+120]], $MachinePrecision]], N[(N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if u < -4.39999999999999983e-12

   1. Initial program 81.0%

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

      1. times-frac 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in t1 around 0 81.6%

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

      1. frac-2neg 81.6%

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

      2. remove-double-neg 81.6%

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

      3. associate-*l/ 85.7%

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

      4. distribute-neg-in 85.7%

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

      5. add-sqr-sqrt 55.4%

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

      6. sqrt-unprod 81.8%

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

      7. sqr-neg 81.8%

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

      8. sqrt-unprod 30.5%

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

      9. add-sqr-sqrt 85.6%

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

      10. sub-neg 85.6%

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

   7. Applied egg-rr 85.6%

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

   if -4.39999999999999983e-12 < u < 8.79999999999999948e-27

   1. Initial program 59.8%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t1 around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

   7. Simplified 86.5%

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

   if 8.79999999999999948e-27 < u < 3.80000000000000008e43 or 8.99999999999999953e120 < u

   1. Initial program 70.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 79.2%

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

      1. frac-2neg 79.2%

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

      2. clear-num 79.9%

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

      3. frac-times 78.3%

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

      4. remove-double-neg 78.3%

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

      5. *-commutative 78.3%

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

      6. *-un-lft-identity 78.3%

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

      7. distribute-neg-in 78.3%

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

      8. add-sqr-sqrt 39.7%

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

      9. sqrt-unprod 78.3%

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

      10. sqr-neg 78.3%

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

      11. sqrt-unprod 38.4%

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

      12. add-sqr-sqrt 78.6%

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

      13. sub-neg 78.6%

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

   7. Applied egg-rr 78.6%

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

      1. associate-*r/ 66.5%

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

      2. associate-/l* 64.8%

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

      3. *-commutative 64.8%

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

      4. times-frac 81.8%

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

   9. Simplified 81.8%

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

   if 3.80000000000000008e43 < u < 8.99999999999999953e120

   1. Initial program 60.5%

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

      1. associate-/r* 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. unsub-neg 99.9%

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

      8. div-sub 100.0%

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

      9. sub-neg 100.0%

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

      10. *-inverses 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 71.6%

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

      1. mul-1-neg 71.6%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 71.6%

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

      3. *-commutative 71.6%

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

   7. Simplified 71.6%

      \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq 8.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+43} \lor \neg \left(u \leq 9 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{v}{t1 - u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+142}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 1.72 \cdot 10^{+168}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.3e+142)
   (* (/ v u) -0.5)
   (if (<= u 1.72e+168) (/ (- v) t1) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e+142) {
		tmp = (v / u) * -0.5;
	} else if (u <= 1.72e+168) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.3d+142)) then
        tmp = (v / u) * (-0.5d0)
    else if (u <= 1.72d+168) then
        tmp = -v / t1
    else
        tmp = v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e+142) {
		tmp = (v / u) * -0.5;
	} else if (u <= 1.72e+168) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.3e+142:
		tmp = (v / u) * -0.5
	elif u <= 1.72e+168:
		tmp = -v / t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.3e+142)
		tmp = Float64(Float64(v / u) * -0.5);
	elseif (u <= 1.72e+168)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.3e+142)
		tmp = (v / u) * -0.5;
	elseif (u <= 1.72e+168)
		tmp = -v / t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.3e+142], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[u, 1.72e+168], N[((-v) / t1), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.30000000000000011e142

   1. Initial program 76.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. *-commutative 89.0%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 85.5%

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

      5. +-commutative 85.5%

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

      6. remove-double-neg 85.5%

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

      7. unsub-neg 85.5%

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

      8. div-sub 85.5%

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

      9. sub-neg 85.5%

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

      10. *-inverses 85.5%

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

      11. metadata-eval 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in t1 around inf 52.1%

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

      1. mul-1-neg 52.1%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 52.1%

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

      3. *-commutative 52.1%

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

   7. Simplified 52.1%

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

   8. Taylor expanded in u around inf 46.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]

   if -1.30000000000000011e142 < u < 1.7200000000000001e168

   1. Initial program 65.8%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. neg-mul-1 68.7%

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

   7. Simplified 68.7%

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

   if 1.7200000000000001e168 < u

   1. Initial program 69.8%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 52.2%

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

      1. clear-num 52.2%

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

      2. frac-times 40.3%

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

      3. *-commutative 40.3%

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

      4. *-un-lft-identity 40.3%

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

      5. add-sqr-sqrt 26.1%

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

      6. sqrt-unprod 38.9%

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

      7. sqr-neg 38.9%

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

      8. sqrt-unprod 13.9%

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

      9. add-sqr-sqrt 40.3%

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

   7. Applied egg-rr 40.3%

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

      1. *-commutative 40.3%

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

      2. associate-*l/ 49.6%

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

      3. associate-/l* 48.8%

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

      4. times-frac 34.9%

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

      5. *-inverses 34.9%

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

      6. associate-*r/ 34.9%

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

      7. *-lft-identity 34.9%

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

      8. +-commutative 34.9%

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

   9. Simplified 34.9%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+142}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 1.72 \cdot 10^{+168}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+142} \lor \neg \left(u \leq 2.2 \cdot 10^{+191}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.45e+142) (not (<= u 2.2e+191))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.45e+142) || !(u <= 2.2e+191)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.45d+142)) .or. (.not. (u <= 2.2d+191))) then
        tmp = v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.45e+142) || !(u <= 2.2e+191)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.45e+142) or not (u <= 2.2e+191):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.45e+142) || !(u <= 2.2e+191))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.45e+142) || ~((u <= 2.2e+191)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.45e+142], N[Not[LessEqual[u, 2.2e+191]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.45000000000000007e142 or 2.2e191 < u

   1. Initial program 74.5%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 96.6%

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

      1. *-commutative 96.6%

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

      2. clear-num 96.5%

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

      3. frac-2neg 96.5%

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

      4. frac-times 87.1%

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

      5. *-un-lft-identity 87.1%

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

      6. remove-double-neg 87.1%

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

      7. distribute-neg-in 87.1%

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

      8. add-sqr-sqrt 45.6%

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

      9. sqrt-unprod 85.8%

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

      10. sqr-neg 85.8%

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

      11. sqrt-unprod 41.5%

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

      12. add-sqr-sqrt 87.0%

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

      13. sub-neg 87.0%

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

   7. Applied egg-rr 87.0%

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

   8. Taylor expanded in t1 around inf 40.7%

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

   if -1.45000000000000007e142 < u < 2.2e191

   1. Initial program 65.7%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   7. Simplified 68.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+142} \lor \neg \left(u \leq 2.2 \cdot 10^{+191}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{+191}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.4e+142) (/ (- v) u) (if (<= u 2.8e+191) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+142) {
		tmp = -v / u;
	} else if (u <= 2.8e+191) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.4d+142)) then
        tmp = -v / u
    else if (u <= 2.8d+191) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+142) {
		tmp = -v / u;
	} else if (u <= 2.8e+191) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.4e+142:
		tmp = -v / u
	elif u <= 2.8e+191:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.4e+142)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 2.8e+191)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.4e+142)
		tmp = -v / u;
	elseif (u <= 2.8e+191)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.4e+142], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 2.8e+191], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.4e142

   1. Initial program 76.9%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 54.4%

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

   6. Taylor expanded in t1 around 0 46.8%

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

      1. associate-*r/ 46.8%

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

      2. neg-mul-1 46.8%

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

   8. Simplified 46.8%

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

   if -1.4e142 < u < 2.7999999999999999e191

   1. Initial program 65.7%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   7. Simplified 68.0%

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

   if 2.7999999999999999e191 < u

   1. Initial program 71.2%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.7%

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

      3. frac-2neg 99.7%

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

      4. frac-times 93.2%

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

      5. *-un-lft-identity 93.2%

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

      6. remove-double-neg 93.2%

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

      7. distribute-neg-in 93.2%

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

      8. add-sqr-sqrt 48.3%

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

      9. sqrt-unprod 93.2%

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

      10. sqr-neg 93.2%

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

      11. sqrt-unprod 44.9%

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

      12. add-sqr-sqrt 93.2%

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

      13. sub-neg 93.2%

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

   7. Applied egg-rr 93.2%

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

   8. Taylor expanded in t1 around inf 33.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+142}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{+191}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+191}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.7e+141)
   (* (/ v u) -0.5)
   (if (<= u 2.4e+191) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.7e+141) {
		tmp = (v / u) * -0.5;
	} else if (u <= 2.4e+191) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.7d+141)) then
        tmp = (v / u) * (-0.5d0)
    else if (u <= 2.4d+191) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.7e+141) {
		tmp = (v / u) * -0.5;
	} else if (u <= 2.4e+191) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.7e+141:
		tmp = (v / u) * -0.5
	elif u <= 2.4e+191:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.7e+141)
		tmp = Float64(Float64(v / u) * -0.5);
	elseif (u <= 2.4e+191)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.7e+141)
		tmp = (v / u) * -0.5;
	elseif (u <= 2.4e+191)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.7e+141], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[u, 2.4e+191], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.7000000000000001e141

   1. Initial program 76.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. *-commutative 89.0%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 85.5%

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

      5. +-commutative 85.5%

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

      6. remove-double-neg 85.5%

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

      7. unsub-neg 85.5%

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

      8. div-sub 85.5%

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

      9. sub-neg 85.5%

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

      10. *-inverses 85.5%

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

      11. metadata-eval 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in t1 around inf 52.1%

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

      1. mul-1-neg 52.1%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 52.1%

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

      3. *-commutative 52.1%

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

   7. Simplified 52.1%

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

   8. Taylor expanded in u around inf 46.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]

   if -2.7000000000000001e141 < u < 2.39999999999999986e191

   1. Initial program 65.7%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in t1 around inf 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   7. Simplified 68.0%

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

   if 2.39999999999999986e191 < u

   1. Initial program 71.2%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.7%

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

      3. frac-2neg 99.7%

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

      4. frac-times 93.2%

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

      5. *-un-lft-identity 93.2%

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

      6. remove-double-neg 93.2%

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

      7. distribute-neg-in 93.2%

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

      8. add-sqr-sqrt 48.3%

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

      9. sqrt-unprod 93.2%

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

      10. sqr-neg 93.2%

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

      11. sqrt-unprod 44.9%

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

      12. add-sqr-sqrt 93.2%

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

      13. sub-neg 93.2%

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

   7. Applied egg-rr 93.2%

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

   8. Taylor expanded in t1 around inf 33.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+191}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3.4 \cdot 10^{+196}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{t1 \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 3.4e+196) (/ (- v) (+ t1 u)) (* v (/ t1 (* t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.4e+196) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * (t1 / (t1 * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 3.4d+196) then
        tmp = -v / (t1 + u)
    else
        tmp = v * (t1 / (t1 * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.4e+196) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * (t1 / (t1 * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3.4e+196:
		tmp = -v / (t1 + u)
	else:
		tmp = v * (t1 / (t1 * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3.4e+196)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(v * Float64(t1 / Float64(t1 * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 3.4e+196)
		tmp = -v / (t1 + u);
	else
		tmp = v * (t1 / (t1 * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3.4e+196], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v * N[(t1 / N[(t1 * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.4e196

   1. Initial program 67.2%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 67.7%

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

   6. Taylor expanded in v around 0 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

      3. +-commutative 66.2%

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

   8. Simplified 66.2%

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

   if 3.4e196 < u

   1. Initial program 73.5%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.8%

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

      3. frac-2neg 99.8%

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

      4. frac-times 96.3%

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

      5. *-un-lft-identity 96.3%

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

      6. remove-double-neg 96.3%

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

      7. distribute-neg-in 96.3%

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

      8. add-sqr-sqrt 50.2%

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

      9. sqrt-unprod 96.3%

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

      10. sqr-neg 96.3%

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

      11. sqrt-unprod 46.2%

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

      12. add-sqr-sqrt 96.3%

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

      13. sub-neg 96.3%

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in u around 0 52.6%

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

      1. associate-/r/ 52.6%

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

      2. *-commutative 52.6%

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

   10. Applied egg-rr 52.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 3.4 \cdot 10^{+196}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{t1 \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 3.4 \cdot 10^{+196}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{t1 \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 3.4e+196) (/ v (- (* u -2.0) t1)) (* v (/ t1 (* t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.4e+196) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v * (t1 / (t1 * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 3.4d+196) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = v * (t1 / (t1 * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 3.4e+196) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = v * (t1 / (t1 * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 3.4e+196:
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = v * (t1 / (t1 * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 3.4e+196)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(v * Float64(t1 / Float64(t1 * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 3.4e+196)
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = v * (t1 / (t1 * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 3.4e+196], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(v * N[(t1 / N[(t1 * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 3.4e196

   1. Initial program 67.2%

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

      1. associate-/r* 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-/l* 98.6%

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

      4. associate-/l/ 95.2%

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

      5. +-commutative 95.2%

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

      6. remove-double-neg 95.2%

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

      7. unsub-neg 95.2%

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

      8. div-sub 95.2%

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

      9. sub-neg 95.2%

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

      10. *-inverses 95.2%

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

      11. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in t1 around inf 66.3%

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

      1. mul-1-neg 66.3%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 66.3%

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

      3. *-commutative 66.3%

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

   7. Simplified 66.3%

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

   if 3.4e196 < u

   1. Initial program 73.5%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.8%

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

      3. frac-2neg 99.8%

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

      4. frac-times 96.3%

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

      5. *-un-lft-identity 96.3%

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

      6. remove-double-neg 96.3%

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

      7. distribute-neg-in 96.3%

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

      8. add-sqr-sqrt 50.2%

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

      9. sqrt-unprod 96.3%

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

      10. sqr-neg 96.3%

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

      11. sqrt-unprod 46.2%

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

      12. add-sqr-sqrt 96.3%

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

      13. sub-neg 96.3%

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in u around 0 52.6%

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

      1. associate-/r/ 52.6%

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

      2. *-commutative 52.6%

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

   10. Applied egg-rr 52.6%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 3.4 \cdot 10^{+196}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{t1 \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 20.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= t1 -9.8e+131) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.8e+131) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-9.8d+131)) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.8e+131) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -9.8e+131:
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -9.8e+131)
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -9.8e+131)
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -9.8e+131], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.80000000000000064e131

   1. Initial program 32.4%

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

      1. times-frac 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. frac-times 92.5%

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

      3. *-un-lft-identity 92.5%

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

      4. frac-2neg 92.5%

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

      5. distribute-neg-in 92.5%

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

      6. add-sqr-sqrt 91.8%

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

      7. sqrt-unprod 36.5%

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

      8. sqr-neg 36.5%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 30.5%

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

      11. sub-neg 30.5%

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

      12. remove-double-neg 30.5%

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

   6. Applied egg-rr 30.5%

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

      1. associate-/l/ 37.9%

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

   8. Simplified 37.9%

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

   9. Taylor expanded in t1 around inf 21.0%

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

   if -9.80000000000000064e131 < t1

   1. Initial program 75.2%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in t1 around 0 53.8%

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

      1. *-commutative 53.8%

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

      2. clear-num 54.0%

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

      3. frac-2neg 54.0%

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

      4. frac-times 53.5%

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

      5. *-un-lft-identity 53.5%

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

      6. remove-double-neg 53.5%

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

      7. distribute-neg-in 53.5%

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

      8. add-sqr-sqrt 25.4%

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

      9. sqrt-unprod 54.9%

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

      10. sqr-neg 54.9%

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

      11. sqrt-unprod 28.2%

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

      12. add-sqr-sqrt 54.8%

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

      13. sub-neg 54.8%

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

   7. Applied egg-rr 54.8%

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

   8. Taylor expanded in t1 around inf 16.4%

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

4. Final simplification 17.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. times-frac 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in t1 around inf 65.8%

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

6. Taylor expanded in v around 0 63.0%

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

   1. associate-*r/ 63.0%

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

   2. neg-mul-1 63.0%

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

   3. +-commutative 63.0%

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

8. Simplified 63.0%

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

9. Final simplification 63.0%

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

Alternative 14: 14.3% 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 67.9%

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

   1. times-frac 98.4%

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

3. Simplified 98.4%

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

   1. clear-num 98.4%

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

   2. frac-times 93.8%

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

   3. *-un-lft-identity 93.8%

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

   4. frac-2neg 93.8%

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

   5. distribute-neg-in 93.8%

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

   6. add-sqr-sqrt 47.0%

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

   7. sqrt-unprod 59.2%

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

   8. sqr-neg 59.2%

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

   9. sqrt-unprod 22.6%

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

   10. add-sqr-sqrt 48.1%

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

   11. sub-neg 48.1%

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

   12. remove-double-neg 48.1%

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

6. Applied egg-rr 48.1%

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

   1. associate-/l/ 52.7%

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

8. Simplified 52.7%

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

9. Taylor expanded in t1 around inf 8.4%

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

10. Final simplification 8.4%

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

Reproduce

herbie shell --seed 2024041 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))