Rosa's DopplerBench

Percentage Accurate: 73.0% → 98.3%

Time: 12.9s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% 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: 98.3% 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 73.1%

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

   1. times-frac 98.5%

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

3. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+73}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -3.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -8.5e+73)
   (/ (* t1 (/ (- v) u)) u)
   (if (<= u -3.6e+31)
     (/ v (- (* u -2.0) t1))
     (if (<= u -2.3e-32)
       (/ t1 (* (- t1 u) (/ u v)))
       (if (<= u 4.8e-55) (/ (- v) t1) (* (/ v (+ t1 u)) (/ (- t1) u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.5e+73) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= -3.6e+31) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -2.3e-32) {
		tmp = t1 / ((t1 - u) * (u / v));
	} else if (u <= 4.8e-55) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-8.5d+73)) then
        tmp = (t1 * (-v / u)) / u
    else if (u <= (-3.6d+31)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (u <= (-2.3d-32)) then
        tmp = t1 / ((t1 - u) * (u / v))
    else if (u <= 4.8d-55) then
        tmp = -v / t1
    else
        tmp = (v / (t1 + u)) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.5e+73) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= -3.6e+31) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -2.3e-32) {
		tmp = t1 / ((t1 - u) * (u / v));
	} else if (u <= 4.8e-55) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -8.5e+73:
		tmp = (t1 * (-v / u)) / u
	elif u <= -3.6e+31:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -2.3e-32:
		tmp = t1 / ((t1 - u) * (u / v))
	elif u <= 4.8e-55:
		tmp = -v / t1
	else:
		tmp = (v / (t1 + u)) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -8.5e+73)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	elseif (u <= -3.6e+31)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (u <= -2.3e-32)
		tmp = Float64(t1 / Float64(Float64(t1 - u) * Float64(u / v)));
	elseif (u <= 4.8e-55)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -8.5e+73)
		tmp = (t1 * (-v / u)) / u;
	elseif (u <= -3.6e+31)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -2.3e-32)
		tmp = t1 / ((t1 - u) * (u / v));
	elseif (u <= 4.8e-55)
		tmp = -v / t1;
	else
		tmp = (v / (t1 + u)) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -8.5e+73], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, -3.6e+31], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -2.3e-32], N[(t1 / N[(N[(t1 - u), $MachinePrecision] * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.8e-55], N[((-v) / t1), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -8.4999999999999998e73

   1. Initial program 78.7%

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

      1. times-frac 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in t1 around 0 84.2%

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

      1. associate-*r/ 84.2%

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

      2. mul-1-neg 84.2%

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

   7. Simplified 84.2%

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

   9. Applied egg-rr 83.0%

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

       1. distribute-neg-frac 83.0%

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

       2. distribute-neg-frac 83.0%

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

   11. Simplified 83.0%

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

   12. Taylor expanded in u around inf 87.1%

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

       1. associate-*r/ 89.2%

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

       2. neg-mul-1 89.2%

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

       3. distribute-rgt-neg-in 89.2%

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

       4. distribute-neg-frac 89.2%

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

   14. Simplified 89.2%

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

   if -8.4999999999999998e73 < u < -3.59999999999999996e31

   1. Initial program 62.9%

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

      1. associate-/r* 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-/l* 100.0%

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

      4. associate-/l/ 100.0%

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

      5. +-commutative 100.0%

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

      6. remove-double-neg 100.0%

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

      7. unsub-neg 100.0%

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

      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 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. *-commutative 88.3%

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

   7. Simplified 88.3%

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

   if -3.59999999999999996e31 < u < -2.3000000000000001e-32

   1. Initial program 82.2%

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

      1. times-frac 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. clear-num 99.6%

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

      3. frac-2neg 99.6%

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

      4. frac-times 87.9%

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

      5. *-un-lft-identity 87.9%

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

      6. remove-double-neg 87.9%

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

      7. distribute-neg-in 87.9%

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

      8. add-sqr-sqrt 12.5%

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

      9. sqrt-unprod 86.0%

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

      10. sqr-neg 86.0%

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

      11. sqrt-unprod 73.5%

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

      12. add-sqr-sqrt 86.0%

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

      13. sub-neg 86.0%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in t1 around 0 85.8%

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

   if -2.3000000000000001e-32 < u < 4.79999999999999983e-55

   1. Initial program 68.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. neg-mul-1 84.3%

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

   7. Simplified 84.3%

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

   if 4.79999999999999983e-55 < u

   1. Initial program 74.7%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around 0 80.2%

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

      1. associate-*r/ 80.2%

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

      2. mul-1-neg 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+73}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -3.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -3.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -1.15 \cdot 10^{-41}:\\ \;\;\;\;\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.25e+74)
   (/ (* t1 (/ (- v) u)) u)
   (if (<= u -3.5e+31)
     (/ v (- (* u -2.0) t1))
     (if (<= u -1.15e-41)
       (/ t1 (* (/ (+ t1 u) v) (- t1 u)))
       (if (<= u 4.8e-55) (/ (- v) t1) (* (/ v (+ t1 u)) (/ (- t1) u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.25e+74) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= -3.5e+31) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -1.15e-41) {
		tmp = t1 / (((t1 + u) / v) * (t1 - u));
	} else if (u <= 4.8e-55) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.25d+74)) then
        tmp = (t1 * (-v / u)) / u
    else if (u <= (-3.5d+31)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (u <= (-1.15d-41)) then
        tmp = t1 / (((t1 + u) / v) * (t1 - u))
    else if (u <= 4.8d-55) then
        tmp = -v / t1
    else
        tmp = (v / (t1 + u)) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.25e+74) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= -3.5e+31) {
		tmp = v / ((u * -2.0) - t1);
	} else if (u <= -1.15e-41) {
		tmp = t1 / (((t1 + u) / v) * (t1 - u));
	} else if (u <= 4.8e-55) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.25e+74:
		tmp = (t1 * (-v / u)) / u
	elif u <= -3.5e+31:
		tmp = v / ((u * -2.0) - t1)
	elif u <= -1.15e-41:
		tmp = t1 / (((t1 + u) / v) * (t1 - u))
	elif u <= 4.8e-55:
		tmp = -v / t1
	else:
		tmp = (v / (t1 + u)) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.25e+74)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	elseif (u <= -3.5e+31)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (u <= -1.15e-41)
		tmp = Float64(t1 / Float64(Float64(Float64(t1 + u) / v) * Float64(t1 - u)));
	elseif (u <= 4.8e-55)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.25e+74)
		tmp = (t1 * (-v / u)) / u;
	elseif (u <= -3.5e+31)
		tmp = v / ((u * -2.0) - t1);
	elseif (u <= -1.15e-41)
		tmp = t1 / (((t1 + u) / v) * (t1 - u));
	elseif (u <= 4.8e-55)
		tmp = -v / t1;
	else
		tmp = (v / (t1 + u)) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.25e+74], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, -3.5e+31], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.15e-41], N[(t1 / N[(N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision] * N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.8e-55], N[((-v) / t1), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -1.24999999999999991e74

   1. Initial program 78.7%

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

      1. times-frac 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in t1 around 0 84.2%

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

      1. associate-*r/ 84.2%

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

      2. mul-1-neg 84.2%

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

   7. Simplified 84.2%

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

   9. Applied egg-rr 83.0%

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

       1. distribute-neg-frac 83.0%

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

       2. distribute-neg-frac 83.0%

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

   11. Simplified 83.0%

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

   12. Taylor expanded in u around inf 87.1%

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

       1. associate-*r/ 89.2%

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

       2. neg-mul-1 89.2%

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

       3. distribute-rgt-neg-in 89.2%

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

       4. distribute-neg-frac 89.2%

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

   14. Simplified 89.2%

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

   if -1.24999999999999991e74 < u < -3.5e31

   1. Initial program 62.9%

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

      1. associate-/r* 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-/l* 100.0%

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

      4. associate-/l/ 100.0%

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

      5. +-commutative 100.0%

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

      6. remove-double-neg 100.0%

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

      7. unsub-neg 100.0%

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

      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 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. *-commutative 88.3%

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

   7. Simplified 88.3%

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

   if -3.5e31 < u < -1.15000000000000005e-41

   1. Initial program 80.1%

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

      1. times-frac 99.3%

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

   3. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. clear-num 99.6%

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

      3. frac-2neg 99.6%

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

      4. frac-times 84.9%

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

      5. *-un-lft-identity 84.9%

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

      6. remove-double-neg 84.9%

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

      7. distribute-neg-in 84.9%

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

      8. add-sqr-sqrt 10.9%

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

      9. sqrt-unprod 83.3%

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

      10. sqr-neg 83.3%

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

      11. sqrt-unprod 72.4%

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

      12. add-sqr-sqrt 83.3%

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

      13. sub-neg 83.3%

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

   6. Applied egg-rr 83.3%

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

   if -1.15000000000000005e-41 < u < 4.79999999999999983e-55

   1. Initial program 68.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. neg-mul-1 84.8%

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

   7. Simplified 84.8%

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

   if 4.79999999999999983e-55 < u

   1. Initial program 74.7%

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

      1. times-frac 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around 0 80.2%

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

      1. associate-*r/ 80.2%

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

      2. mul-1-neg 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq -3.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -1.15 \cdot 10^{-41}:\\ \;\;\;\;\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.38 \cdot 10^{+66} \lor \neg \left(t1 \leq -1.55 \cdot 10^{+27} \lor \neg \left(t1 \leq -3.2 \cdot 10^{-40}\right) \land t1 \leq 116000000000\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.38e+66)
         (not
          (or (<= t1 -1.55e+27)
              (and (not (<= t1 -3.2e-40)) (<= t1 116000000000.0)))))
   (/ v (- (* u -2.0) t1))
   (* (/ (- v) u) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.38e+66) || !((t1 <= -1.55e+27) || (!(t1 <= -3.2e-40) && (t1 <= 116000000000.0)))) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-v / u) * (t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.38d+66)) .or. (.not. (t1 <= (-1.55d+27)) .or. (.not. (t1 <= (-3.2d-40))) .and. (t1 <= 116000000000.0d0))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (-v / u) * (t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.38e+66) || !((t1 <= -1.55e+27) || (!(t1 <= -3.2e-40) && (t1 <= 116000000000.0)))) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-v / u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.38e+66) or not ((t1 <= -1.55e+27) or (not (t1 <= -3.2e-40) and (t1 <= 116000000000.0))):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-v / u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.38e+66) || !((t1 <= -1.55e+27) || (!(t1 <= -3.2e-40) && (t1 <= 116000000000.0))))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.38e+66) || ~(((t1 <= -1.55e+27) || (~((t1 <= -3.2e-40)) && (t1 <= 116000000000.0)))))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-v / u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.38e+66], N[Not[Or[LessEqual[t1, -1.55e+27], And[N[Not[LessEqual[t1, -3.2e-40]], $MachinePrecision], LessEqual[t1, 116000000000.0]]]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.38e66 or -1.54999999999999998e27 < t1 < -3.20000000000000002e-40 or 1.16e11 < t1

   1. Initial program 65.5%

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

      1. associate-/r* 79.9%

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

      2. *-commutative 79.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 95.3%

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

      5. +-commutative 95.3%

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

      6. remove-double-neg 95.3%

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

      7. unsub-neg 95.3%

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

      8. div-sub 95.3%

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

      9. sub-neg 95.3%

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

      10. *-inverses 95.3%

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

      11. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in t1 around inf 84.2%

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

      1. mul-1-neg 84.2%

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

      2. unsub-neg 84.2%

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

      3. *-commutative 84.2%

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

   7. Simplified 84.2%

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

   if -1.38e66 < t1 < -1.54999999999999998e27 or -3.20000000000000002e-40 < t1 < 1.16e11

   1. Initial program 79.9%

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

      1. times-frac 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in t1 around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. mul-1-neg 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t1 around 0 80.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.38 \cdot 10^{+66} \lor \neg \left(t1 \leq -1.55 \cdot 10^{+27} \lor \neg \left(t1 \leq -3.2 \cdot 10^{-40}\right) \land t1 \leq 116000000000\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.38 \cdot 10^{+66}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;t\_1 \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq -2.95 \cdot 10^{-40} \lor \neg \left(t1 \leq 3500000000000\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot t\_1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) u)) (t_2 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.38e+66)
     t_2
     (if (<= t1 -1.1e+26)
       (* t_1 (/ t1 u))
       (if (or (<= t1 -2.95e-40) (not (<= t1 3500000000000.0)))
         t_2
         (/ (* t1 t_1) u))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / u;
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.38e+66) {
		tmp = t_2;
	} else if (t1 <= -1.1e+26) {
		tmp = t_1 * (t1 / u);
	} else if ((t1 <= -2.95e-40) || !(t1 <= 3500000000000.0)) {
		tmp = t_2;
	} else {
		tmp = (t1 * t_1) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / u
    t_2 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-1.38d+66)) then
        tmp = t_2
    else if (t1 <= (-1.1d+26)) then
        tmp = t_1 * (t1 / u)
    else if ((t1 <= (-2.95d-40)) .or. (.not. (t1 <= 3500000000000.0d0))) then
        tmp = t_2
    else
        tmp = (t1 * t_1) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / u;
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.38e+66) {
		tmp = t_2;
	} else if (t1 <= -1.1e+26) {
		tmp = t_1 * (t1 / u);
	} else if ((t1 <= -2.95e-40) || !(t1 <= 3500000000000.0)) {
		tmp = t_2;
	} else {
		tmp = (t1 * t_1) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / u
	t_2 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.38e+66:
		tmp = t_2
	elif t1 <= -1.1e+26:
		tmp = t_1 * (t1 / u)
	elif (t1 <= -2.95e-40) or not (t1 <= 3500000000000.0):
		tmp = t_2
	else:
		tmp = (t1 * t_1) / u
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / u)
	t_2 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.38e+66)
		tmp = t_2;
	elseif (t1 <= -1.1e+26)
		tmp = Float64(t_1 * Float64(t1 / u));
	elseif ((t1 <= -2.95e-40) || !(t1 <= 3500000000000.0))
		tmp = t_2;
	else
		tmp = Float64(Float64(t1 * t_1) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / u;
	t_2 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.38e+66)
		tmp = t_2;
	elseif (t1 <= -1.1e+26)
		tmp = t_1 * (t1 / u);
	elseif ((t1 <= -2.95e-40) || ~((t1 <= 3500000000000.0)))
		tmp = t_2;
	else
		tmp = (t1 * t_1) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / u), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.38e+66], t$95$2, If[LessEqual[t1, -1.1e+26], N[(t$95$1 * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -2.95e-40], N[Not[LessEqual[t1, 3500000000000.0]], $MachinePrecision]], t$95$2, N[(N[(t1 * t$95$1), $MachinePrecision] / u), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.38e66 or -1.10000000000000004e26 < t1 < -2.94999999999999983e-40 or 3.5e12 < t1

   1. Initial program 65.5%

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

      1. associate-/r* 79.9%

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

      2. *-commutative 79.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 95.3%

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

      5. +-commutative 95.3%

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

      6. remove-double-neg 95.3%

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

      7. unsub-neg 95.3%

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

      8. div-sub 95.3%

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

      9. sub-neg 95.3%

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

      10. *-inverses 95.3%

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

      11. metadata-eval 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in t1 around inf 84.2%

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

      1. mul-1-neg 84.2%

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

      2. unsub-neg 84.2%

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

      3. *-commutative 84.2%

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

   7. Simplified 84.2%

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

   if -1.38e66 < t1 < -1.10000000000000004e26

   1. Initial program 60.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. Taylor expanded in t1 around 0 77.6%

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

      1. associate-*r/ 77.6%

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

      2. mul-1-neg 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in t1 around 0 77.8%

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

   if -2.94999999999999983e-40 < t1 < 3.5e12

   1. Initial program 82.0%

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

      1. times-frac 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t1 around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. mul-1-neg 77.0%

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

   7. Simplified 77.0%

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

   9. Applied egg-rr 76.0%

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

       1. distribute-neg-frac 76.0%

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

       2. distribute-neg-frac 76.0%

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

   11. Simplified 76.0%

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

   12. Taylor expanded in u around inf 78.4%

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

       1. associate-*r/ 82.5%

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

       2. neg-mul-1 82.5%

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

       3. distribute-rgt-neg-in 82.5%

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

       4. distribute-neg-frac 82.5%

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

   14. Simplified 82.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.38 \cdot 10^{+66}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq -2.95 \cdot 10^{-40} \lor \neg \left(t1 \leq 3500000000000\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+104}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+161}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.15e+104)
   (/ (* t1 (/ (- v) u)) u)
   (if (<= u 1.25e+161)
     (/ v (* (+ t1 u) (- -1.0 (/ u t1))))
     (* (/ v (+ t1 u)) (/ (- t1) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+104) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= 1.25e+161) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.15d+104)) then
        tmp = (t1 * (-v / u)) / u
    else if (u <= 1.25d+161) then
        tmp = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
    else
        tmp = (v / (t1 + u)) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+104) {
		tmp = (t1 * (-v / u)) / u;
	} else if (u <= 1.25e+161) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.15e+104:
		tmp = (t1 * (-v / u)) / u
	elif u <= 1.25e+161:
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)))
	else:
		tmp = (v / (t1 + u)) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.15e+104)
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	elseif (u <= 1.25e+161)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 - Float64(u / t1))));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.15e+104)
		tmp = (t1 * (-v / u)) / u;
	elseif (u <= 1.25e+161)
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	else
		tmp = (v / (t1 + u)) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.15e+104], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 1.25e+161], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.14999999999999992e104

   1. Initial program 77.4%

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

      1. times-frac 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in t1 around 0 87.5%

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

      1. associate-*r/ 87.5%

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

      2. mul-1-neg 87.5%

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

   7. Simplified 87.5%

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

   9. Applied egg-rr 86.0%

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

       1. distribute-neg-frac 86.0%

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

       2. distribute-neg-frac 86.0%

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

   11. Simplified 86.0%

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

   12. Taylor expanded in u around inf 90.6%

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

       1. associate-*r/ 92.7%

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

       2. neg-mul-1 92.7%

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

       3. distribute-rgt-neg-in 92.7%

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

       4. distribute-neg-frac 92.7%

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

   14. Simplified 92.7%

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

   if -1.14999999999999992e104 < u < 1.2499999999999999e161

   1. Initial program 74.5%

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

      1. associate-/r* 83.4%

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

      2. *-commutative 83.4%

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

      3. associate-/l* 98.9%

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

      4. associate-/l/ 97.3%

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

      5. +-commutative 97.3%

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

      6. remove-double-neg 97.3%

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

      7. unsub-neg 97.3%

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

      8. div-sub 97.3%

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

      9. sub-neg 97.3%

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

      10. *-inverses 97.3%

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

      11. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in u around 0 97.3%

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

      1. sub-neg 97.3%

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

      2. mul-1-neg 97.3%

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

      3. distribute-neg-in 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-neg-in 97.3%

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

      6. metadata-eval 97.3%

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

      7. sub-neg 97.3%

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

   7. Simplified 97.3%

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

   if 1.2499999999999999e161 < u

   1. Initial program 59.6%

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

      1. times-frac 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t1 around 0 91.1%

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

      1. associate-*r/ 91.1%

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

      2. mul-1-neg 91.1%

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

   7. Simplified 91.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+104}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{+161}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+88} \lor \neg \left(u \leq 3.8 \cdot 10^{+63}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.2e+88) (not (<= u 3.8e+63)))
   (* (/ v u) (/ t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.2e+88) || !(u <= 3.8e+63)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-7.2d+88)) .or. (.not. (u <= 3.8d+63))) then
        tmp = (v / u) * (t1 / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.2e+88) || !(u <= 3.8e+63)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.2e+88) or not (u <= 3.8e+63):
		tmp = (v / u) * (t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.2e+88) || !(u <= 3.8e+63))
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.2e+88) || ~((u <= 3.8e+63)))
		tmp = (v / u) * (t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.2e+88], N[Not[LessEqual[u, 3.8e+63]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.2000000000000004e88 or 3.8000000000000001e63 < u

   1. Initial program 73.1%

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

      1. times-frac 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in t1 around 0 84.1%

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

      1. associate-*r/ 84.1%

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

      2. mul-1-neg 84.1%

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

   7. Simplified 84.1%

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

   9. Applied egg-rr 83.5%

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

       1. distribute-neg-frac 83.5%

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

       2. distribute-neg-frac 83.5%

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

   11. Simplified 83.5%

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

   12. Taylor expanded in u around inf 81.8%

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

       1. div-inv 81.8%

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

       2. associate-/r/ 87.4%

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

       3. associate-*l* 84.3%

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

       4. add-sqr-sqrt 43.8%

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

       5. sqrt-unprod 58.8%

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

       6. sqr-neg 58.8%

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

       7. sqrt-unprod 25.9%

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

       8. add-sqr-sqrt 58.1%

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

       9. div-inv 58.1%

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

   14. Applied egg-rr 58.1%

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

   if -7.2000000000000004e88 < u < 3.8000000000000001e63

   1. Initial program 73.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 72.2%

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

      1. associate-*r/ 72.2%

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

      2. neg-mul-1 72.2%

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

   7. Simplified 72.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+88} \lor \neg \left(u \leq 3.8 \cdot 10^{+63}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 22.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.2 \cdot 10^{+148} \lor \neg \left(t1 \leq 2 \cdot 10^{+150}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.2e+148) (not (<= t1 2e+150))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.2e+148) || !(t1 <= 2e+150)) {
		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 <= (-3.2d+148)) .or. (.not. (t1 <= 2d+150))) 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 <= -3.2e+148) || !(t1 <= 2e+150)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.2e+148) or not (t1 <= 2e+150):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.2e+148) || !(t1 <= 2e+150))
		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 <= -3.2e+148) || ~((t1 <= 2e+150)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.2e+148], N[Not[LessEqual[t1, 2e+150]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.1999999999999999e148 or 1.99999999999999996e150 < t1

   1. Initial program 43.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. clear-num 99.7%

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

      3. frac-2neg 99.7%

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

      4. frac-times 58.7%

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

      5. *-un-lft-identity 58.7%

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

      6. remove-double-neg 58.7%

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

      7. distribute-neg-in 58.7%

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

      8. add-sqr-sqrt 22.0%

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

      9. sqrt-unprod 43.5%

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

      10. sqr-neg 43.5%

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

      11. sqrt-unprod 24.7%

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

      12. add-sqr-sqrt 43.2%

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

      13. sub-neg 43.2%

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

   6. Applied egg-rr 43.2%

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

   7. Taylor expanded in t1 around inf 40.8%

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

   if -3.1999999999999999e148 < t1 < 1.99999999999999996e150

   1. Initial program 81.5%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in t1 around 0 63.9%

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

      1. associate-*r/ 63.9%

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

      2. mul-1-neg 63.9%

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

   7. Simplified 63.9%

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

      1. associate-*r/ 63.5%

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

      2. clear-num 63.1%

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

      3. associate-*l/ 63.2%

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

      4. *-un-lft-identity 63.2%

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

      5. add-sqr-sqrt 30.3%

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

      6. sqrt-unprod 42.8%

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

      7. sqr-neg 42.8%

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

      8. sqrt-unprod 17.7%

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

      9. add-sqr-sqrt 32.7%

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

   9. Applied egg-rr 32.7%

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

   10. Taylor expanded in u around 0 15.4%

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

4. Final simplification 21.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.2 \cdot 10^{+148} \lor \neg \left(t1 \leq 2 \cdot 10^{+150}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. associate-/r* 85.1%

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

   2. *-commutative 85.1%

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

   3. associate-/l* 98.1%

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

   4. associate-/l/ 91.6%

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

   5. +-commutative 91.6%

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

   6. remove-double-neg 91.6%

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

   7. unsub-neg 91.6%

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

   8. div-sub 91.6%

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

   9. sub-neg 91.6%

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

   10. *-inverses 91.6%

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

   11. metadata-eval 91.6%

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

3. Simplified 91.6%

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

5. Taylor expanded in v around 0 91.6%

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

   1. sub-neg 91.6%

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

   2. mul-1-neg 91.6%

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

   3. distribute-neg-in 91.6%

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

   4. associate-/r* 98.3%

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

   5. +-commutative 98.3%

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

   6. distribute-neg-in 98.3%

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

   7. metadata-eval 98.3%

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

   8. sub-neg 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 10: 56.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4e+217) (* (/ v u) -0.5) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+217) {
		tmp = (v / u) * -0.5;
	} 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 <= (-4d+217)) then
        tmp = (v / u) * (-0.5d0)
    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 <= -4e+217) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4e+217:
		tmp = (v / u) * -0.5
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4e+217)
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4e+217)
		tmp = (v / u) * -0.5;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4e+217], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.99999999999999984e217

   1. Initial program 81.7%

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

      1. associate-/r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 81.7%

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

      5. +-commutative 81.7%

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

      6. remove-double-neg 81.7%

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

      7. unsub-neg 81.7%

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

      8. div-sub 81.7%

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

      9. sub-neg 81.7%

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

      10. *-inverses 81.7%

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

      11. metadata-eval 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in t1 around inf 43.1%

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

      1. mul-1-neg 43.1%

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

      2. unsub-neg 43.1%

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

      3. *-commutative 43.1%

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

   7. Simplified 43.1%

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

   8. Taylor expanded in u around inf 43.1%

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

   if -3.99999999999999984e217 < u

   1. Initial program 72.4%

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

      1. times-frac 98.4%

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

   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 55.4%

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

      1. associate-*r/ 55.4%

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

      2. neg-mul-1 55.4%

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

   7. Simplified 55.4%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.2e+217) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.2e+217) {
		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 <= (-5.2d+217)) 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 <= -5.2e+217) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.2e+217:
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.2e+217)
		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 <= -5.2e+217)
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.2e+217], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.20000000000000023e217

   1. Initial program 81.7%

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

      1. times-frac 99.8%

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

   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%

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

      1. associate-*r/ 99.8%

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

      2. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. add-sqr-sqrt 59.8%

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

      6. sqrt-unprod 85.2%

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

      7. sqr-neg 85.2%

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

      8. sqrt-unprod 40.0%

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

      9. add-sqr-sqrt 81.3%

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

   9. Applied egg-rr 81.3%

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

   10. Taylor expanded in u around 0 42.9%

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

   if -5.20000000000000023e217 < u

   1. Initial program 72.4%

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

      1. times-frac 98.4%

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

   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 55.4%

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

      1. associate-*r/ 55.4%

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

      2. neg-mul-1 55.4%

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

   7. Simplified 55.4%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+217}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+217}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.1e+217) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.1e+217) {
		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 <= (-4.1d+217)) 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 <= -4.1e+217) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.1e+217:
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.1e+217)
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.1e+217)
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.1e+217], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.1000000000000002e217

   1. Initial program 81.7%

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

      1. times-frac 99.8%

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

   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%

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

      1. associate-*r/ 99.8%

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

      2. mul-1-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in t1 around inf 43.1%

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

      1. associate-*r/ 43.1%

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

      2. neg-mul-1 43.1%

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

   10. Simplified 43.1%

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

   if -4.1000000000000002e217 < u

   1. Initial program 72.4%

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

      1. times-frac 98.4%

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

   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 55.4%

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

      1. associate-*r/ 55.4%

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

      2. neg-mul-1 55.4%

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

   7. Simplified 55.4%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+217}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.7% 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 73.1%

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

   1. times-frac 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r/ 98.2%

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

   2. clear-num 98.0%

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

   3. associate-*l/ 98.1%

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

   4. *-un-lft-identity 98.1%

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

   5. frac-2neg 98.1%

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

   6. distribute-neg-in 98.1%

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

   7. add-sqr-sqrt 45.1%

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

   8. sqrt-unprod 69.8%

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

   9. sqr-neg 69.8%

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

   10. sqrt-unprod 30.3%

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

   11. add-sqr-sqrt 57.5%

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

   12. sub-neg 57.5%

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

   13. remove-double-neg 57.5%

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

6. Applied egg-rr 57.5%

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

   1. associate-/r/ 58.9%

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

   2. add-sqr-sqrt 31.0%

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

   3. sqrt-unprod 44.5%

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

   4. sqr-neg 44.5%

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

   5. sqrt-unprod 32.6%

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

   6. add-sqr-sqrt 71.4%

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

   7. distribute-rgt-neg-in 71.4%

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

   8. associate-/r/ 71.4%

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

   9. neg-sub0 71.4%

      \[\leadsto \frac{\color{blue}{0 - \frac{v}{\frac{t1 - u}{t1}}}}{t1 + u} \]

   10. div-sub 71.4%

       \[\leadsto \frac{0 - \frac{v}{\color{blue}{\frac{t1}{t1} - \frac{u}{t1}}}}{t1 + u} \]

   11. *-inverses 71.4%

       \[\leadsto \frac{0 - \frac{v}{\color{blue}{1} - \frac{u}{t1}}}{t1 + u} \]

   12. sub-neg 71.4%

       \[\leadsto \frac{0 - \frac{v}{\color{blue}{1 + \left(-\frac{u}{t1}\right)}}}{t1 + u} \]

   13. distribute-neg-frac 71.4%

       \[\leadsto \frac{0 - \frac{v}{1 + \color{blue}{\frac{-u}{t1}}}}{t1 + u} \]

   14. add-sqr-sqrt 38.7%

       \[\leadsto \frac{0 - \frac{v}{1 + \frac{-u}{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}}}{t1 + u} \]

   15. sqrt-unprod 77.2%

       \[\leadsto \frac{0 - \frac{v}{1 + \frac{-u}{\color{blue}{\sqrt{t1 \cdot t1}}}}}{t1 + u} \]

   16. sqr-neg 77.2%

       \[\leadsto \frac{0 - \frac{v}{1 + \frac{-u}{\sqrt{\color{blue}{\left(-t1\right) \cdot \left(-t1\right)}}}}}{t1 + u} \]

   17. sqrt-unprod 45.3%

       \[\leadsto \frac{0 - \frac{v}{1 + \frac{-u}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}}}{t1 + u} \]

   18. add-sqr-sqrt 98.1%

       \[\leadsto \frac{0 - \frac{v}{1 + \frac{-u}{\color{blue}{-t1}}}}{t1 + u} \]

   19. frac-2neg 98.1%

       \[\leadsto \frac{0 - \frac{v}{1 + \color{blue}{\frac{u}{t1}}}}{t1 + u} \]

8. Applied egg-rr 98.1%

   \[\leadsto \frac{\color{blue}{0 - \frac{v}{1 + \frac{u}{t1}}}}{t1 + u} \]
9. Step-by-step derivation

   1. neg-sub0 98.1%

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

   2. distribute-neg-frac 98.1%

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

10. Simplified 98.1%

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

11. Taylor expanded in u around 0 56.8%

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

    1. mul-1-neg 56.8%

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

13. Simplified 56.8%

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

14. Final simplification 56.8%

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

Alternative 14: 14.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 98.5%

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

3. Simplified 98.5%

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

   1. *-commutative 98.5%

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

   2. clear-num 98.1%

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

   3. frac-2neg 98.1%

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

   4. frac-times 80.5%

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

   5. *-un-lft-identity 80.5%

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

   6. remove-double-neg 80.5%

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

   7. distribute-neg-in 80.5%

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

   8. add-sqr-sqrt 36.0%

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

   9. sqrt-unprod 64.9%

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

   10. sqr-neg 64.9%

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

   11. sqrt-unprod 30.3%

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

   12. add-sqr-sqrt 53.8%

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

   13. sub-neg 53.8%

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

6. Applied egg-rr 53.8%

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

7. Taylor expanded in t1 around inf 12.1%

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

8. Final simplification 12.1%

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

Reproduce

herbie shell --seed 2024031 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))