Rosa's DopplerBench

Percentage Accurate: 72.5% → 98.2%

Time: 16.7s

Alternatives: 17

Speedup: 1.0×

• Rival profile vector overflowed, profile may not be complete

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. associate-*l/ 77.5%

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

   2. *-commutative 77.5%

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

3. Simplified 77.5%

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

   1. associate-*r/ 74.6%

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

   2. *-commutative 74.6%

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

   3. times-frac 99.1%

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

   4. frac-2neg 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-neg-in 99.1%

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

   7. sub-neg 99.1%

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

   8. associate-*r/ 99.1%

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

   9. add-sqr-sqrt 48.2%

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

   10. sqrt-unprod 49.7%

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

   11. sqr-neg 49.7%

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

   12. sqrt-unprod 22.3%

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

   13. add-sqr-sqrt 41.8%

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

   14. sub-neg 41.8%

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

   15. +-commutative 41.8%

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

   16. add-sqr-sqrt 19.5%

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

   17. sqrt-unprod 50.1%

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

   18. sqr-neg 50.1%

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

   19. sqrt-unprod 37.4%

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

   20. add-sqr-sqrt 19.0%

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

   21. sqrt-unprod 41.3%

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

   22. sqr-neg 41.3%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+134}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{+120}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.6e+134)
   (- (/ v t1))
   (if (<= t1 8.5e+120)
     (* v (/ t1 (* (+ t1 u) (- (- u) t1))))
     (/ -1.0 (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.6e+134) {
		tmp = -(v / t1);
	} else if (t1 <= 8.5e+120) {
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.6d+134)) then
        tmp = -(v / t1)
    else if (t1 <= 8.5d+120) then
        tmp = v * (t1 / ((t1 + u) * (-u - t1)))
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.6e+134) {
		tmp = -(v / t1);
	} else if (t1 <= 8.5e+120) {
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.6e+134:
		tmp = -(v / t1)
	elif t1 <= 8.5e+120:
		tmp = v * (t1 / ((t1 + u) * (-u - t1)))
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.6e+134)
		tmp = Float64(-Float64(v / t1));
	elseif (t1 <= 8.5e+120)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(Float64(-u) - t1))));
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.6e+134)
		tmp = -(v / t1);
	elseif (t1 <= 8.5e+120)
		tmp = v * (t1 / ((t1 + u) * (-u - t1)));
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.6e+134], (-N[(v / t1), $MachinePrecision]), If[LessEqual[t1, 8.5e+120], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.6e134

   1. Initial program 40.7%

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

      1. associate-*l/ 44.6%

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

      2. *-commutative 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in t1 around inf 83.1%

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

      1. associate-*r/ 83.1%

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

      2. neg-mul-1 83.1%

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

   7. Simplified 83.1%

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

   if -1.6e134 < t1 < 8.50000000000000026e120

   1. Initial program 84.2%

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

      1. associate-*l/ 86.8%

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

      2. *-commutative 86.8%

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

   3. Simplified 86.8%

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

   if 8.50000000000000026e120 < t1

   1. Initial program 56.2%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 93.3%

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

      1. clear-num 93.5%

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

      2. un-div-inv 93.5%

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

   7. Applied egg-rr 93.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+134}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{+120}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{u}{-v}}\\ \mathbf{elif}\;u \leq 4.6 \cdot 10^{-51}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.6e-75)
   (/ (/ t1 (+ t1 u)) (/ u (- v)))
   (if (<= u 4.6e-51) (- (/ v t1)) (* t1 (/ (/ v u) (- t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e-75) {
		tmp = (t1 / (t1 + u)) / (u / -v);
	} else if (u <= 4.6e-51) {
		tmp = -(v / t1);
	} else {
		tmp = t1 * ((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 (u <= (-3.6d-75)) then
        tmp = (t1 / (t1 + u)) / (u / -v)
    else if (u <= 4.6d-51) then
        tmp = -(v / t1)
    else
        tmp = t1 * ((v / u) / (t1 - u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e-75) {
		tmp = (t1 / (t1 + u)) / (u / -v);
	} else if (u <= 4.6e-51) {
		tmp = -(v / t1);
	} else {
		tmp = t1 * ((v / u) / (t1 - u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.6e-75:
		tmp = (t1 / (t1 + u)) / (u / -v)
	elif u <= 4.6e-51:
		tmp = -(v / t1)
	else:
		tmp = t1 * ((v / u) / (t1 - u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.6e-75)
		tmp = Float64(Float64(t1 / Float64(t1 + u)) / Float64(u / Float64(-v)));
	elseif (u <= 4.6e-51)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(t1 - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.6e-75)
		tmp = (t1 / (t1 + u)) / (u / -v);
	elseif (u <= 4.6e-51)
		tmp = -(v / t1);
	else
		tmp = t1 * ((v / u) / (t1 - u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.6e-75], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(u / (-v)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.6e-51], (-N[(v / t1), $MachinePrecision]), N[(t1 * N[(N[(v / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.6e-75

   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 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around 0 79.0%

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

      1. *-commutative 79.0%

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

      2. clear-num 79.3%

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

      3. frac-2neg 79.3%

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

      4. frac-times 76.8%

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

      5. *-un-lft-identity 76.8%

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

      6. neg-sub0 76.8%

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

      7. add-sqr-sqrt 76.7%

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

      8. sqrt-unprod 69.6%

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

      9. sqr-neg 69.6%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 51.3%

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

      12. associate-+l- 51.3%

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

      13. neg-sub0 51.3%

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

      14. add-sqr-sqrt 51.3%

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

      15. sqrt-unprod 51.4%

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

      16. sqr-neg 51.4%

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

      17. sqrt-unprod 0.0%

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

      18. add-sqr-sqrt 76.8%

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

      19. +-commutative 76.8%

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

   7. Applied egg-rr 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-/r* 79.4%

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

   9. Simplified 79.4%

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

   if -3.6e-75 < u < 4.60000000000000004e-51

   1. Initial program 62.5%

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

      1. associate-*l/ 69.4%

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

      2. *-commutative 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in t1 around inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   7. Simplified 84.2%

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

   if 4.60000000000000004e-51 < u

   1. Initial program 83.8%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 81.7%

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

      1. frac-times 75.0%

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

      2. frac-2neg 75.0%

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

      3. distribute-rgt-neg-out 75.0%

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

      4. add-sqr-sqrt 33.6%

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

      5. sqrt-unprod 65.5%

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

      6. sqr-neg 65.5%

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

      7. sqrt-unprod 34.0%

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

      8. add-sqr-sqrt 56.6%

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

      9. *-commutative 56.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 75.1%

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

      12. sqr-neg 75.1%

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

      13. sqrt-unprod 75.0%

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

      14. add-sqr-sqrt 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. distribute-frac-neg2 75.1%

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

      2. associate-/r* 78.0%

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

      3. associate-*r/ 82.8%

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

      4. associate-*r/ 87.1%

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

   9. Simplified 87.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{u}{-v}}\\ \mathbf{elif}\;u \leq 4.6 \cdot 10^{-51}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.6e-75)
   (/ (/ t1 (/ u v)) (- u))
   (if (<= u 3.4e-51) (- (/ v t1)) (* t1 (/ (/ v u) (- t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e-75) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= 3.4e-51) {
		tmp = -(v / t1);
	} else {
		tmp = t1 * ((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 (u <= (-3.6d-75)) then
        tmp = (t1 / (u / v)) / -u
    else if (u <= 3.4d-51) then
        tmp = -(v / t1)
    else
        tmp = t1 * ((v / u) / (t1 - u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e-75) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= 3.4e-51) {
		tmp = -(v / t1);
	} else {
		tmp = t1 * ((v / u) / (t1 - u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.6e-75:
		tmp = (t1 / (u / v)) / -u
	elif u <= 3.4e-51:
		tmp = -(v / t1)
	else:
		tmp = t1 * ((v / u) / (t1 - u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.6e-75)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(-u));
	elseif (u <= 3.4e-51)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / Float64(t1 - u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.6e-75)
		tmp = (t1 / (u / v)) / -u;
	elseif (u <= 3.4e-51)
		tmp = -(v / t1);
	else
		tmp = t1 * ((v / u) / (t1 - u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.6e-75], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, 3.4e-51], (-N[(v / t1), $MachinePrecision]), N[(t1 * N[(N[(v / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.6e-75

   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 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around 0 79.0%

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

   6. Taylor expanded in t1 around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. mul-1-neg 78.3%

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

   8. Simplified 78.3%

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

      1. *-commutative 78.3%

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

      2. frac-2neg 78.3%

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

      3. associate-*l/ 78.3%

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

      4. add-sqr-sqrt 39.4%

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

      5. sqrt-unprod 51.0%

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

      6. sqr-neg 51.0%

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

      7. sqrt-unprod 20.5%

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

      8. add-sqr-sqrt 44.5%

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

      9. *-commutative 44.5%

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

      10. associate-*l/ 44.4%

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

      11. associate-*r/ 45.5%

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

      12. clear-num 46.5%

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

      13. un-div-inv 46.5%

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

      14. add-sqr-sqrt 22.6%

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

      15. sqrt-unprod 54.2%

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

      16. sqr-neg 54.2%

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

      17. sqrt-unprod 41.4%

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

      18. add-sqr-sqrt 79.3%

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

   10. Applied egg-rr 79.3%

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

   if -3.6e-75 < u < 3.40000000000000003e-51

   1. Initial program 62.5%

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

      1. associate-*l/ 69.4%

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

      2. *-commutative 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in t1 around inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   7. Simplified 84.2%

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

   if 3.40000000000000003e-51 < u

   1. Initial program 83.8%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 81.7%

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

      1. frac-times 75.0%

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

      2. frac-2neg 75.0%

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

      3. distribute-rgt-neg-out 75.0%

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

      4. add-sqr-sqrt 33.6%

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

      5. sqrt-unprod 65.5%

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

      6. sqr-neg 65.5%

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

      7. sqrt-unprod 34.0%

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

      8. add-sqr-sqrt 56.6%

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

      9. *-commutative 56.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 75.1%

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

      12. sqr-neg 75.1%

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

      13. sqrt-unprod 75.0%

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

      14. add-sqr-sqrt 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. distribute-frac-neg2 75.1%

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

      2. associate-/r* 78.0%

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

      3. associate-*r/ 82.8%

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

      4. associate-*r/ 87.1%

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

   9. Simplified 87.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{-75} \lor \neg \left(u \leq 9.5 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.2e-75) (not (<= u 9.5e-78)))
   (/ (/ t1 (/ u v)) (- u))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e-75) || !(u <= 9.5e-78)) {
		tmp = (t1 / (u / 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 <= (-3.2d-75)) .or. (.not. (u <= 9.5d-78))) then
        tmp = (t1 / (u / 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 <= -3.2e-75) || !(u <= 9.5e-78)) {
		tmp = (t1 / (u / v)) / -u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.2e-75) or not (u <= 9.5e-78):
		tmp = (t1 / (u / v)) / -u
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.2e-75) || !(u <= 9.5e-78))
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(-u));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.2e-75) || ~((u <= 9.5e-78)))
		tmp = (t1 / (u / v)) / -u;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.2e-75], N[Not[LessEqual[u, 9.5e-78]], $MachinePrecision]], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -3.19999999999999977e-75 or 9.4999999999999997e-78 < u

   1. Initial program 80.2%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around 0 80.1%

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

   6. Taylor expanded in t1 around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. mul-1-neg 79.6%

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

   8. Simplified 79.6%

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

      1. *-commutative 79.6%

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

      2. frac-2neg 79.6%

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

      3. associate-*l/ 79.7%

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

      4. add-sqr-sqrt 38.0%

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

      5. sqrt-unprod 56.7%

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

      6. sqr-neg 56.7%

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

      7. sqrt-unprod 26.3%

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

      8. add-sqr-sqrt 48.8%

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

      9. *-commutative 48.8%

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

      10. associate-*l/ 48.8%

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

      11. associate-*r/ 49.3%

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

      12. clear-num 49.9%

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

      13. un-div-inv 49.9%

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

      14. add-sqr-sqrt 25.2%

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

      15. sqrt-unprod 55.2%

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

      16. sqr-neg 55.2%

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

      17. sqrt-unprod 40.2%

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

      18. add-sqr-sqrt 80.7%

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

   10. Applied egg-rr 80.7%

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

   if -3.19999999999999977e-75 < u < 9.4999999999999997e-78

   1. Initial program 62.3%

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

      1. associate-*l/ 69.5%

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

      2. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in t1 around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. neg-mul-1 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{-75} \lor \neg \left(u \leq 9.5 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.12 \cdot 10^{-75} \lor \neg \left(u \leq 1.05 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.12e-75) (not (<= u 1.05e-77)))
   (* (/ v u) (/ t1 (- u)))
   (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.12e-75) || !(u <= 1.05e-77)) {
		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 <= (-2.12d-75)) .or. (.not. (u <= 1.05d-77))) 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 <= -2.12e-75) || !(u <= 1.05e-77)) {
		tmp = (v / u) * (t1 / -u);
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.12e-75) or not (u <= 1.05e-77):
		tmp = (v / u) * (t1 / -u)
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.12e-75) || !(u <= 1.05e-77))
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	else
		tmp = Float64(-Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.12e-75) || ~((u <= 1.05e-77)))
		tmp = (v / u) * (t1 / -u);
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.12e-75], N[Not[LessEqual[u, 1.05e-77]], $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 < -2.12e-75 or 1.05000000000000008e-77 < u

   1. Initial program 80.2%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around 0 80.1%

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

   6. Taylor expanded in t1 around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. mul-1-neg 79.6%

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

   8. Simplified 79.6%

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

   if -2.12e-75 < u < 1.05000000000000008e-77

   1. Initial program 62.3%

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

      1. associate-*l/ 69.5%

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

      2. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in t1 around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. neg-mul-1 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.12 \cdot 10^{-75} \lor \neg \left(u \leq 1.05 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{-51}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.4e-75)
   (* (/ v u) (/ t1 (- u)))
   (if (<= u 6.5e-51) (- (/ v t1)) (/ t1 (* u (/ u (- v)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e-75) {
		tmp = (v / u) * (t1 / -u);
	} else if (u <= 6.5e-51) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * (u / -v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.4d-75)) then
        tmp = (v / u) * (t1 / -u)
    else if (u <= 6.5d-51) then
        tmp = -(v / t1)
    else
        tmp = t1 / (u * (u / -v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e-75) {
		tmp = (v / u) * (t1 / -u);
	} else if (u <= 6.5e-51) {
		tmp = -(v / t1);
	} else {
		tmp = t1 / (u * (u / -v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.4e-75:
		tmp = (v / u) * (t1 / -u)
	elif u <= 6.5e-51:
		tmp = -(v / t1)
	else:
		tmp = t1 / (u * (u / -v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.4e-75)
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	elseif (u <= 6.5e-51)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(u / Float64(-v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.4e-75)
		tmp = (v / u) * (t1 / -u);
	elseif (u <= 6.5e-51)
		tmp = -(v / t1);
	else
		tmp = t1 / (u * (u / -v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.4e-75], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 6.5e-51], (-N[(v / t1), $MachinePrecision]), N[(t1 / N[(u * N[(u / (-v)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.40000000000000015e-75

   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 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around 0 79.0%

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

   6. Taylor expanded in t1 around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. mul-1-neg 78.3%

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

   8. Simplified 78.3%

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

   if -3.40000000000000015e-75 < u < 6.5000000000000003e-51

   1. Initial program 62.5%

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

      1. associate-*l/ 69.4%

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

      2. *-commutative 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in t1 around inf 84.2%

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

      1. associate-*r/ 84.2%

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

      2. neg-mul-1 84.2%

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

   7. Simplified 84.2%

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

   if 6.5000000000000003e-51 < u

   1. Initial program 83.8%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 81.7%

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

   6. Taylor expanded in t1 around 0 81.7%

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

      1. associate-*r/ 81.7%

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

      2. mul-1-neg 81.7%

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

   8. Simplified 81.7%

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

      1. *-commutative 81.7%

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

      2. clear-num 81.5%

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

      3. frac-2neg 81.5%

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

      4. frac-times 82.8%

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

      5. *-un-lft-identity 82.8%

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

      6. remove-double-neg 82.8%

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

   10. Applied egg-rr 82.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{-51}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{+38} \lor \neg \left(u \leq 1.55 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.5e+38) (not (<= u 1.55e+136)))
   (/ t1 (* u (/ u v)))
   (/ v (- (- u) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.5e+38) || !(u <= 1.55e+136)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.5d+38)) .or. (.not. (u <= 1.55d+136))) then
        tmp = t1 / (u * (u / v))
    else
        tmp = v / (-u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.5e+38) || !(u <= 1.55e+136)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / (-u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.5e+38) or not (u <= 1.55e+136):
		tmp = t1 / (u * (u / v))
	else:
		tmp = v / (-u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.5e+38) || !(u <= 1.55e+136))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(v / Float64(Float64(-u) - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.5e+38) || ~((u <= 1.55e+136)))
		tmp = t1 / (u * (u / v));
	else
		tmp = v / (-u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.5e+38], N[Not[LessEqual[u, 1.55e+136]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.5000000000000001e38 or 1.54999999999999992e136 < u

   1. Initial program 78.8%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 91.6%

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

   6. Taylor expanded in t1 around 0 91.5%

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

      1. associate-*r/ 91.5%

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

      2. mul-1-neg 91.5%

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

   8. Simplified 91.5%

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

      1. *-commutative 91.5%

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

      2. clear-num 91.5%

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

      3. frac-times 89.1%

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

      4. *-un-lft-identity 89.1%

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

      5. add-sqr-sqrt 43.3%

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

      6. sqrt-unprod 69.5%

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

      7. sqr-neg 69.5%

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

      8. sqrt-unprod 37.4%

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

      9. add-sqr-sqrt 71.7%

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

   10. Applied egg-rr 71.7%

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

   if -1.5000000000000001e38 < u < 1.54999999999999992e136

   1. Initial program 71.4%

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

      1. associate-*l/ 76.2%

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

      2. *-commutative 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*r/ 71.4%

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

      2. *-commutative 71.4%

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

      3. times-frac 98.5%

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

      4. frac-2neg 98.5%

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

      5. +-commutative 98.5%

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

      6. distribute-neg-in 98.5%

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

      7. sub-neg 98.5%

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

      8. associate-*r/ 98.6%

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

      9. add-sqr-sqrt 48.7%

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

      10. sqrt-unprod 34.4%

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

      11. sqr-neg 34.4%

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

      12. sqrt-unprod 10.5%

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

      13. add-sqr-sqrt 19.0%

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

      14. sub-neg 19.0%

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

      15. +-commutative 19.0%

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

      16. add-sqr-sqrt 8.4%

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

      17. sqrt-unprod 33.3%

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

      18. sqr-neg 33.3%

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

      19. sqrt-unprod 34.8%

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

      20. add-sqr-sqrt 15.0%

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

      21. sqrt-unprod 42.8%

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

      22. sqr-neg 42.8%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in t1 around inf 65.6%

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

      1. mul-1-neg 65.6%

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

   9. Simplified 65.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.5 \cdot 10^{+38} \lor \neg \left(u \leq 1.55 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.8e+36)
   (/ t1 (* u (/ u v)))
   (if (<= u 2.05e+136) (/ v (- (- u) t1)) (/ v (* u (/ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+36) {
		tmp = t1 / (u * (u / v));
	} else if (u <= 2.05e+136) {
		tmp = v / (-u - t1);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.8d+36)) then
        tmp = t1 / (u * (u / v))
    else if (u <= 2.05d+136) then
        tmp = v / (-u - t1)
    else
        tmp = v / (u * (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e+36) {
		tmp = t1 / (u * (u / v));
	} else if (u <= 2.05e+136) {
		tmp = v / (-u - t1);
	} else {
		tmp = v / (u * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.8e+36:
		tmp = t1 / (u * (u / v))
	elif u <= 2.05e+136:
		tmp = v / (-u - t1)
	else:
		tmp = v / (u * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.8e+36)
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	elseif (u <= 2.05e+136)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.8e+36)
		tmp = t1 / (u * (u / v));
	elseif (u <= 2.05e+136)
		tmp = v / (-u - t1);
	else
		tmp = v / (u * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.8e+36], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.05e+136], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.80000000000000025e36

   1. Initial program 76.0%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 87.2%

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

   6. Taylor expanded in t1 around 0 87.1%

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

      1. associate-*r/ 87.1%

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

      2. mul-1-neg 87.1%

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

   8. Simplified 87.1%

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

      1. *-commutative 87.1%

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

      2. clear-num 87.1%

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

      3. frac-times 82.9%

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

      4. *-un-lft-identity 82.9%

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

      5. add-sqr-sqrt 37.7%

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

      6. sqrt-unprod 65.0%

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

      7. sqr-neg 65.0%

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

      8. sqrt-unprod 35.4%

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

      9. add-sqr-sqrt 64.0%

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

   10. Applied egg-rr 64.0%

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

   if -3.80000000000000025e36 < u < 2.0499999999999999e136

   1. Initial program 71.4%

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

      1. associate-*l/ 76.2%

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

      2. *-commutative 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*r/ 71.4%

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

      2. *-commutative 71.4%

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

      3. times-frac 98.5%

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

      4. frac-2neg 98.5%

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

      5. +-commutative 98.5%

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

      6. distribute-neg-in 98.5%

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

      7. sub-neg 98.5%

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

      8. associate-*r/ 98.6%

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

      9. add-sqr-sqrt 48.7%

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

      10. sqrt-unprod 34.4%

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

      11. sqr-neg 34.4%

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

      12. sqrt-unprod 10.5%

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

      13. add-sqr-sqrt 19.0%

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

      14. sub-neg 19.0%

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

      15. +-commutative 19.0%

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

      16. add-sqr-sqrt 8.4%

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

      17. sqrt-unprod 33.3%

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

      18. sqr-neg 33.3%

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

      19. sqrt-unprod 34.8%

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

      20. add-sqr-sqrt 15.0%

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

      21. sqrt-unprod 42.8%

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

      22. sqr-neg 42.8%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in t1 around inf 65.6%

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

      1. mul-1-neg 65.6%

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

   9. Simplified 65.6%

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

   if 2.0499999999999999e136 < u

   1. Initial program 82.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 97.8%

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

   6. Taylor expanded in t1 around 0 97.8%

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

      1. associate-*r/ 97.8%

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

      2. mul-1-neg 97.8%

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

   8. Simplified 97.8%

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

      1. clear-num 97.8%

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

      2. frac-times 85.2%

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

      3. *-un-lft-identity 85.2%

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

      4. add-sqr-sqrt 44.8%

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

      5. sqrt-unprod 69.8%

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

      6. sqr-neg 69.8%

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

      7. sqrt-unprod 40.3%

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

      8. add-sqr-sqrt 83.0%

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

   10. Applied egg-rr 83.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 3.2 \cdot 10^{+136}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.4e+150)
   (/ v (- u))
   (if (<= u 3.2e+136) (- (/ v t1)) (/ 1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+150) {
		tmp = v / -u;
	} else if (u <= 3.2e+136) {
		tmp = -(v / t1);
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.4d+150)) then
        tmp = v / -u
    else if (u <= 3.2d+136) then
        tmp = -(v / t1)
    else
        tmp = 1.0d0 / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+150) {
		tmp = v / -u;
	} else if (u <= 3.2e+136) {
		tmp = -(v / t1);
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.4e+150:
		tmp = v / -u
	elif u <= 3.2e+136:
		tmp = -(v / t1)
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.4e+150)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 3.2e+136)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.4e+150)
		tmp = v / -u;
	elseif (u <= 3.2e+136)
		tmp = -(v / t1);
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.4e+150], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 3.2e+136], (-N[(v / t1), $MachinePrecision]), N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.40000000000000005e150

   1. Initial program 70.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 37.1%

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

   6. Taylor expanded in t1 around 0 32.5%

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

      1. associate-*r/ 32.5%

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

      2. mul-1-neg 32.5%

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

   8. Simplified 32.5%

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

   if -1.40000000000000005e150 < u < 3.19999999999999988e136

   1. Initial program 73.4%

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

      1. associate-*l/ 77.6%

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

      2. *-commutative 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in t1 around inf 60.8%

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

      1. associate-*r/ 60.8%

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

      2. neg-mul-1 60.8%

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

   7. Simplified 60.8%

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

   if 3.19999999999999988e136 < u

   1. Initial program 82.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 52.3%

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

      1. associate-*r/ 52.3%

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

      2. neg-mul-1 52.3%

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

      3. clear-num 53.7%

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

      4. add-sqr-sqrt 23.2%

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

      5. sqrt-unprod 53.4%

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

      6. sqr-neg 53.4%

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

      7. sqrt-unprod 30.5%

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

      8. add-sqr-sqrt 51.8%

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

   7. Applied egg-rr 51.8%

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

   8. Taylor expanded in t1 around 0 51.9%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+150}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 3.2 \cdot 10^{+136}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{+174} \lor \neg \left(u \leq 2.35 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.7e+174) (not (<= u 2.35e+136))) (/ v u) (- (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.7e+174) || !(u <= 2.35e+136)) {
		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 <= (-3.7d+174)) .or. (.not. (u <= 2.35d+136))) 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 <= -3.7e+174) || !(u <= 2.35e+136)) {
		tmp = v / u;
	} else {
		tmp = -(v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.7e+174) or not (u <= 2.35e+136):
		tmp = v / u
	else:
		tmp = -(v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.7e+174) || !(u <= 2.35e+136))
		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 <= -3.7e+174) || ~((u <= 2.35e+136)))
		tmp = v / u;
	else
		tmp = -(v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.7e+174], N[Not[LessEqual[u, 2.35e+136]], $MachinePrecision]], N[(v / u), $MachinePrecision], (-N[(v / t1), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if u < -3.7000000000000002e174 or 2.35000000000000002e136 < u

   1. Initial program 80.0%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 47.3%

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

      1. associate-*r/ 47.3%

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

      2. neg-mul-1 47.3%

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

      3. clear-num 48.0%

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

      4. add-sqr-sqrt 22.1%

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

      5. sqrt-unprod 46.3%

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

      6. sqr-neg 46.3%

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

      7. sqrt-unprod 24.7%

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

      8. add-sqr-sqrt 44.5%

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

   7. Applied egg-rr 44.5%

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

   8. Taylor expanded in t1 around 0 44.0%

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

   if -3.7000000000000002e174 < u < 2.35000000000000002e136

   1. Initial program 72.0%

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

      1. associate-*l/ 76.1%

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

      2. *-commutative 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in t1 around inf 59.2%

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

      1. associate-*r/ 59.2%

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

      2. neg-mul-1 59.2%

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

   7. Simplified 59.2%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{+174} \lor \neg \left(u \leq 2.35 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+136}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.2e+156) (/ v (- u)) (if (<= u 1.8e+136) (- (/ v t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.2e+156) {
		tmp = v / -u;
	} else if (u <= 1.8e+136) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-6.2d+156)) then
        tmp = v / -u
    else if (u <= 1.8d+136) then
        tmp = -(v / t1)
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.2e+156) {
		tmp = v / -u;
	} else if (u <= 1.8e+136) {
		tmp = -(v / t1);
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.2e+156:
		tmp = v / -u
	elif u <= 1.8e+136:
		tmp = -(v / t1)
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.2e+156)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 1.8e+136)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.2e+156)
		tmp = v / -u;
	elseif (u <= 1.8e+136)
		tmp = -(v / t1);
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.2e+156], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 1.8e+136], (-N[(v / t1), $MachinePrecision]), N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -6.2000000000000004e156

   1. Initial program 70.3%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 37.1%

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

   6. Taylor expanded in t1 around 0 32.5%

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

      1. associate-*r/ 32.5%

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

      2. mul-1-neg 32.5%

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

   8. Simplified 32.5%

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

   if -6.2000000000000004e156 < u < 1.80000000000000003e136

   1. Initial program 73.4%

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

      1. associate-*l/ 77.6%

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

      2. *-commutative 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in t1 around inf 60.8%

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

      1. associate-*r/ 60.8%

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

      2. neg-mul-1 60.8%

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

   7. Simplified 60.8%

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

   if 1.80000000000000003e136 < u

   1. Initial program 82.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 52.3%

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

      1. associate-*r/ 52.3%

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

      2. neg-mul-1 52.3%

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

      3. clear-num 53.7%

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

      4. add-sqr-sqrt 23.2%

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

      5. sqrt-unprod 53.4%

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

      6. sqr-neg 53.4%

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

      7. sqrt-unprod 30.5%

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

      8. add-sqr-sqrt 51.8%

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

   7. Applied egg-rr 51.8%

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

   8. Taylor expanded in t1 around 0 50.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+156}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+136}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 23.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.5 \cdot 10^{+158} \lor \neg \left(t1 \leq 2.2 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.5e+158) (not (<= t1 2.2e+111))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.5e+158) || !(t1 <= 2.2e+111)) {
		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 <= (-3.5d+158)) .or. (.not. (t1 <= 2.2d+111))) 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 <= -3.5e+158) || !(t1 <= 2.2e+111)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.5e+158) or not (t1 <= 2.2e+111):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.5e+158) || !(t1 <= 2.2e+111))
		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 <= -3.5e+158) || ~((t1 <= 2.2e+111)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.5e+158], N[Not[LessEqual[t1, 2.2e+111]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.5000000000000001e158 or 2.19999999999999999e111 < t1

   1. Initial program 51.0%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 92.6%

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

   6. Taylor expanded in u around inf 45.5%

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

   if -3.5000000000000001e158 < t1 < 2.19999999999999999e111

   1. Initial program 82.6%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around inf 43.8%

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

      1. associate-*r/ 43.8%

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

      2. neg-mul-1 43.8%

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

      3. clear-num 43.9%

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

      4. add-sqr-sqrt 21.4%

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

      5. sqrt-unprod 28.6%

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

      6. sqr-neg 28.6%

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

      7. sqrt-unprod 8.6%

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

      8. add-sqr-sqrt 17.9%

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

   7. Applied egg-rr 17.9%

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

   8. Taylor expanded in t1 around 0 18.6%

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

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.5 \cdot 10^{+158} \lor \neg \left(t1 \leq 2.2 \cdot 10^{+111}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. times-frac 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-neg-frac2 99.1%

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

   4. +-commutative 99.1%

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

   5. distribute-neg-in 99.1%

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

   6. unsub-neg 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 15: 61.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. associate-*l/ 77.5%

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

   2. *-commutative 77.5%

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

3. Simplified 77.5%

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

   1. associate-*r/ 74.6%

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

   2. *-commutative 74.6%

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

   3. times-frac 99.1%

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

   4. frac-2neg 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-neg-in 99.1%

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

   7. sub-neg 99.1%

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

   8. associate-*r/ 99.1%

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

   9. add-sqr-sqrt 48.2%

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

   10. sqrt-unprod 49.7%

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

   11. sqr-neg 49.7%

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

   12. sqrt-unprod 22.3%

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

   13. add-sqr-sqrt 41.8%

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

   14. sub-neg 41.8%

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

   15. +-commutative 41.8%

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

   16. add-sqr-sqrt 19.5%

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

   17. sqrt-unprod 50.1%

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

   18. sqr-neg 50.1%

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

   19. sqrt-unprod 37.4%

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

   20. add-sqr-sqrt 19.0%

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

   21. sqrt-unprod 41.3%

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

   22. sqr-neg 41.3%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in t1 around inf 56.4%

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

   1. mul-1-neg 56.4%

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

9. Simplified 56.4%

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

10. Final simplification 56.4%

    \[\leadsto \frac{v}{\left(-u\right) - t1} \]
11. Add Preprocessing

Alternative 16: 61.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. times-frac 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-neg-frac2 99.1%

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

   4. +-commutative 99.1%

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

   5. distribute-neg-in 99.1%

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

   6. unsub-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in t1 around inf 56.4%

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

   1. mul-1-neg 56.4%

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

   2. distribute-neg-frac2 56.4%

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

   3. +-commutative 56.4%

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

   4. distribute-neg-in 56.4%

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

   5. sub-neg 56.4%

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

   6. add-sqr-sqrt 28.0%

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

   7. sqrt-unprod 66.6%

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

   8. sqr-neg 66.6%

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

   9. sqrt-unprod 28.3%

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

   10. add-sqr-sqrt 56.3%

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

7. Applied egg-rr 56.3%

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

Alternative 17: 13.6% 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 74.6%

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

   1. times-frac 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-neg-frac2 99.1%

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

   4. +-commutative 99.1%

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

   5. distribute-neg-in 99.1%

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

   6. unsub-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in t1 around inf 48.6%

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

6. Taylor expanded in u around inf 15.0%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))