Rosa's DopplerBench

Percentage Accurate: 72.6% → 97.8%

Time: 9.0s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-commutative 67.9%

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

   2. times-frac 98.3%

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

   3. neg-mul-1 98.3%

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

   4. associate-/l* 98.3%

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

   5. associate-*r/ 98.3%

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

   6. associate-/l* 98.3%

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

   7. associate-/l/ 98.3%

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

   8. neg-mul-1 98.3%

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

   9. *-lft-identity 98.3%

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

   10. metadata-eval 98.3%

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

   11. times-frac 98.3%

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

   12. neg-mul-1 98.3%

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

   13. remove-double-neg 98.3%

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

   14. neg-mul-1 98.3%

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

   15. sub0-neg 98.3%

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

   16. associate--r+ 98.3%

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

   17. neg-sub0 98.3%

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

   18. div-sub 98.4%

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

   19. distribute-frac-neg 98.4%

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

   20. *-inverses 98.4%

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

   21. metadata-eval 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4 \cdot 10^{-62}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 2.35 \cdot 10^{-35} \lor \neg \left(t1 \leq 5.3 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -9.8e-107)
   (/ (- v) (+ t1 u))
   (if (<= t1 4e-62)
     (* (/ (- t1) u) (/ v u))
     (if (or (<= t1 2.35e-35) (not (<= t1 5.3e+39)))
       (/ (- v) t1)
       (/ (- t1) (* u (/ u v)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.8e-107) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 4e-62) {
		tmp = (-t1 / u) * (v / u);
	} else if ((t1 <= 2.35e-35) || !(t1 <= 5.3e+39)) {
		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 (t1 <= (-9.8d-107)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 4d-62) then
        tmp = (-t1 / u) * (v / u)
    else if ((t1 <= 2.35d-35) .or. (.not. (t1 <= 5.3d+39))) 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 (t1 <= -9.8e-107) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 4e-62) {
		tmp = (-t1 / u) * (v / u);
	} else if ((t1 <= 2.35e-35) || !(t1 <= 5.3e+39)) {
		tmp = -v / t1;
	} else {
		tmp = -t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -9.8e-107:
		tmp = -v / (t1 + u)
	elif t1 <= 4e-62:
		tmp = (-t1 / u) * (v / u)
	elif (t1 <= 2.35e-35) or not (t1 <= 5.3e+39):
		tmp = -v / t1
	else:
		tmp = -t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -9.8e-107)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 4e-62)
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	elseif ((t1 <= 2.35e-35) || !(t1 <= 5.3e+39))
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-t1) / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -9.8e-107)
		tmp = -v / (t1 + u);
	elseif (t1 <= 4e-62)
		tmp = (-t1 / u) * (v / u);
	elseif ((t1 <= 2.35e-35) || ~((t1 <= 5.3e+39)))
		tmp = -v / t1;
	else
		tmp = -t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -9.8e-107], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4e-62], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 2.35e-35], N[Not[LessEqual[t1, 5.3e+39]], $MachinePrecision]], N[((-v) / t1), $MachinePrecision], N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -9.79999999999999959e-107

   1. Initial program 66.5%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 76.1%

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

   if -9.79999999999999959e-107 < t1 < 4.0000000000000002e-62

   1. Initial program 82.8%

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

      1. times-frac 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in t1 around 0 87.6%

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

   5. Taylor expanded in t1 around 0 89.0%

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

      1. associate-*r/ 89.0%

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

      2. neg-mul-1 89.0%

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

   7. Simplified 89.0%

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

   if 4.0000000000000002e-62 < t1 < 2.35e-35 or 5.29999999999999979e39 < t1

   1. Initial program 49.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 83.0%

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

      1. associate-*r/ 83.0%

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

      2. neg-mul-1 83.0%

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

   6. Simplified 83.0%

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

   if 2.35e-35 < t1 < 5.29999999999999979e39

   1. Initial program 86.2%

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

      1. associate-/l* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in t1 around 0 86.1%

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

      1. unpow2 86.1%

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

   6. Simplified 86.1%

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

   7. Taylor expanded in u around 0 86.1%

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

      1. unpow2 86.1%

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

      2. associate-*r/ 92.6%

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

   9. Simplified 92.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4 \cdot 10^{-62}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 2.35 \cdot 10^{-35} \lor \neg \left(t1 \leq 5.3 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 3: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.7 \cdot 10^{-107}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 3.5 \cdot 10^{-34} \lor \neg \left(t1 \leq 10^{+42}\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -9.7e-107)
   (/ (- v) (+ t1 u))
   (if (<= t1 2e-62)
     (/ (- v) (* u (/ u t1)))
     (if (or (<= t1 3.5e-34) (not (<= t1 1e+42)))
       (/ (- v) t1)
       (/ (- t1) (* u (/ u v)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.7e-107) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 2e-62) {
		tmp = -v / (u * (u / t1));
	} else if ((t1 <= 3.5e-34) || !(t1 <= 1e+42)) {
		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 (t1 <= (-9.7d-107)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 2d-62) then
        tmp = -v / (u * (u / t1))
    else if ((t1 <= 3.5d-34) .or. (.not. (t1 <= 1d+42))) 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 (t1 <= -9.7e-107) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 2e-62) {
		tmp = -v / (u * (u / t1));
	} else if ((t1 <= 3.5e-34) || !(t1 <= 1e+42)) {
		tmp = -v / t1;
	} else {
		tmp = -t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -9.7e-107:
		tmp = -v / (t1 + u)
	elif t1 <= 2e-62:
		tmp = -v / (u * (u / t1))
	elif (t1 <= 3.5e-34) or not (t1 <= 1e+42):
		tmp = -v / t1
	else:
		tmp = -t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -9.7e-107)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 2e-62)
		tmp = Float64(Float64(-v) / Float64(u * Float64(u / t1)));
	elseif ((t1 <= 3.5e-34) || !(t1 <= 1e+42))
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-t1) / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -9.7e-107)
		tmp = -v / (t1 + u);
	elseif (t1 <= 2e-62)
		tmp = -v / (u * (u / t1));
	elseif ((t1 <= 3.5e-34) || ~((t1 <= 1e+42)))
		tmp = -v / t1;
	else
		tmp = -t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -9.7e-107], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2e-62], N[((-v) / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 3.5e-34], N[Not[LessEqual[t1, 1e+42]], $MachinePrecision]], N[((-v) / t1), $MachinePrecision], N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -9.6999999999999997e-107

   1. Initial program 66.5%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 76.1%

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

   if -9.6999999999999997e-107 < t1 < 2.0000000000000001e-62

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. times-frac 94.6%

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

      3. neg-mul-1 94.6%

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

      4. associate-/l* 94.6%

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

      5. associate-*r/ 94.7%

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

      6. associate-/l* 94.7%

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

      7. associate-/l/ 94.7%

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

      8. neg-mul-1 94.7%

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

      9. *-lft-identity 94.7%

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

      10. metadata-eval 94.7%

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

      11. times-frac 94.7%

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

      12. neg-mul-1 94.7%

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

      13. remove-double-neg 94.7%

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

      14. neg-mul-1 94.7%

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

      15. sub0-neg 94.7%

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

      16. associate--r+ 94.7%

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

      17. neg-sub0 94.7%

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

      18. div-sub 94.7%

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

      19. distribute-frac-neg 94.7%

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

      20. *-inverses 94.7%

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

      21. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in v around 0 97.1%

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

      1. associate-*r/ 97.1%

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

      2. neg-mul-1 97.1%

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

      3. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in t1 around 0 83.8%

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

      1. unpow2 83.8%

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

      2. associate-*r/ 91.5%

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

   9. Simplified 91.5%

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

   if 2.0000000000000001e-62 < t1 < 3.5e-34 or 1.00000000000000004e42 < t1

   1. Initial program 49.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 83.0%

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

      1. associate-*r/ 83.0%

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

      2. neg-mul-1 83.0%

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

   6. Simplified 83.0%

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

   if 3.5e-34 < t1 < 1.00000000000000004e42

   1. Initial program 86.2%

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

      1. associate-/l* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in t1 around 0 86.1%

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

      1. unpow2 86.1%

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

   6. Simplified 86.1%

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

   7. Taylor expanded in u around 0 86.1%

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

      1. unpow2 86.1%

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

      2. associate-*r/ 92.6%

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

   9. Simplified 92.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.7 \cdot 10^{-107}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 3.5 \cdot 10^{-34} \lor \neg \left(t1 \leq 10^{+42}\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 4: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -5.3 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 4.85 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-43} \lor \neg \left(t1 \leq 4.1 \cdot 10^{+46}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))))
   (if (<= t1 -5.3e-108)
     t_1
     (if (<= t1 4.85e-67)
       (/ (- v) (* u (/ u t1)))
       (if (or (<= t1 4.8e-43) (not (<= t1 4.1e+46)))
         t_1
         (/ (- t1) (* u (/ u v))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -5.3e-108) {
		tmp = t_1;
	} else if (t1 <= 4.85e-67) {
		tmp = -v / (u * (u / t1));
	} else if ((t1 <= 4.8e-43) || !(t1 <= 4.1e+46)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    if (t1 <= (-5.3d-108)) then
        tmp = t_1
    else if (t1 <= 4.85d-67) then
        tmp = -v / (u * (u / t1))
    else if ((t1 <= 4.8d-43) .or. (.not. (t1 <= 4.1d+46))) then
        tmp = t_1
    else
        tmp = -t1 / (u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -5.3e-108) {
		tmp = t_1;
	} else if (t1 <= 4.85e-67) {
		tmp = -v / (u * (u / t1));
	} else if ((t1 <= 4.8e-43) || !(t1 <= 4.1e+46)) {
		tmp = t_1;
	} else {
		tmp = -t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	tmp = 0
	if t1 <= -5.3e-108:
		tmp = t_1
	elif t1 <= 4.85e-67:
		tmp = -v / (u * (u / t1))
	elif (t1 <= 4.8e-43) or not (t1 <= 4.1e+46):
		tmp = t_1
	else:
		tmp = -t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -5.3e-108)
		tmp = t_1;
	elseif (t1 <= 4.85e-67)
		tmp = Float64(Float64(-v) / Float64(u * Float64(u / t1)));
	elseif ((t1 <= 4.8e-43) || !(t1 <= 4.1e+46))
		tmp = t_1;
	else
		tmp = Float64(Float64(-t1) / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	tmp = 0.0;
	if (t1 <= -5.3e-108)
		tmp = t_1;
	elseif (t1 <= 4.85e-67)
		tmp = -v / (u * (u / t1));
	elseif ((t1 <= 4.8e-43) || ~((t1 <= 4.1e+46)))
		tmp = t_1;
	else
		tmp = -t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.3e-108], t$95$1, If[LessEqual[t1, 4.85e-67], N[((-v) / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 4.8e-43], N[Not[LessEqual[t1, 4.1e+46]], $MachinePrecision]], t$95$1, N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.29999999999999989e-108 or 4.8500000000000003e-67 < t1 < 4.8000000000000004e-43 or 4.1e46 < t1

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 95.3%

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

      1. associate-*r/ 95.3%

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

      2. neg-mul-1 95.3%

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

      3. +-commutative 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in t1 around inf 79.8%

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

      1. *-commutative 79.8%

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

   9. Simplified 79.8%

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

   if -5.29999999999999989e-108 < t1 < 4.8500000000000003e-67

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. times-frac 94.6%

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

      3. neg-mul-1 94.6%

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

      4. associate-/l* 94.6%

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

      5. associate-*r/ 94.7%

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

      6. associate-/l* 94.7%

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

      7. associate-/l/ 94.7%

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

      8. neg-mul-1 94.7%

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

      9. *-lft-identity 94.7%

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

      10. metadata-eval 94.7%

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

      11. times-frac 94.7%

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

      12. neg-mul-1 94.7%

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

      13. remove-double-neg 94.7%

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

      14. neg-mul-1 94.7%

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

      15. sub0-neg 94.7%

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

      16. associate--r+ 94.7%

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

      17. neg-sub0 94.7%

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

      18. div-sub 94.7%

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

      19. distribute-frac-neg 94.7%

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

      20. *-inverses 94.7%

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

      21. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in v around 0 97.1%

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

      1. associate-*r/ 97.1%

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

      2. neg-mul-1 97.1%

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

      3. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in t1 around 0 83.8%

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

      1. unpow2 83.8%

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

      2. associate-*r/ 91.5%

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

   9. Simplified 91.5%

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

   if 4.8000000000000004e-43 < t1 < 4.1e46

   1. Initial program 86.2%

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

      1. associate-/l* 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in t1 around 0 86.1%

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

      1. unpow2 86.1%

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

   6. Simplified 86.1%

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

   7. Taylor expanded in u around 0 86.1%

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

      1. unpow2 86.1%

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

      2. associate-*r/ 92.6%

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

   9. Simplified 92.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq 4.85 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-43} \lor \neg \left(t1 \leq 4.1 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 5: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -1.85 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{-48} \lor \neg \left(t1 \leq 2.05 \cdot 10^{+40}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 (* u 2.0)))))
   (if (<= t1 -1.85e-109)
     t_1
     (if (<= t1 9.5e-65)
       (/ (- v) (* u (/ u t1)))
       (if (or (<= t1 5.6e-48) (not (<= t1 2.05e+40)))
         t_1
         (/ (* v (/ t1 u)) (- u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -1.85e-109) {
		tmp = t_1;
	} else if (t1 <= 9.5e-65) {
		tmp = -v / (u * (u / t1));
	} else if ((t1 <= 5.6e-48) || !(t1 <= 2.05e+40)) {
		tmp = t_1;
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + (u * 2.0d0))
    if (t1 <= (-1.85d-109)) then
        tmp = t_1
    else if (t1 <= 9.5d-65) then
        tmp = -v / (u * (u / t1))
    else if ((t1 <= 5.6d-48) .or. (.not. (t1 <= 2.05d+40))) then
        tmp = t_1
    else
        tmp = (v * (t1 / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + (u * 2.0));
	double tmp;
	if (t1 <= -1.85e-109) {
		tmp = t_1;
	} else if (t1 <= 9.5e-65) {
		tmp = -v / (u * (u / t1));
	} else if ((t1 <= 5.6e-48) || !(t1 <= 2.05e+40)) {
		tmp = t_1;
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + (u * 2.0))
	tmp = 0
	if t1 <= -1.85e-109:
		tmp = t_1
	elif t1 <= 9.5e-65:
		tmp = -v / (u * (u / t1))
	elif (t1 <= 5.6e-48) or not (t1 <= 2.05e+40):
		tmp = t_1
	else:
		tmp = (v * (t1 / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)))
	tmp = 0.0
	if (t1 <= -1.85e-109)
		tmp = t_1;
	elseif (t1 <= 9.5e-65)
		tmp = Float64(Float64(-v) / Float64(u * Float64(u / t1)));
	elseif ((t1 <= 5.6e-48) || !(t1 <= 2.05e+40))
		tmp = t_1;
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + (u * 2.0));
	tmp = 0.0;
	if (t1 <= -1.85e-109)
		tmp = t_1;
	elseif (t1 <= 9.5e-65)
		tmp = -v / (u * (u / t1));
	elseif ((t1 <= 5.6e-48) || ~((t1 <= 2.05e+40)))
		tmp = t_1;
	else
		tmp = (v * (t1 / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.85e-109], t$95$1, If[LessEqual[t1, 9.5e-65], N[((-v) / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 5.6e-48], N[Not[LessEqual[t1, 2.05e+40]], $MachinePrecision]], t$95$1, N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.8499999999999999e-109 or 9.5000000000000004e-65 < t1 < 5.6000000000000001e-48 or 2.0500000000000001e40 < t1

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 95.3%

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

      1. associate-*r/ 95.3%

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

      2. neg-mul-1 95.3%

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

      3. +-commutative 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in t1 around inf 80.1%

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

      1. *-commutative 80.1%

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

   9. Simplified 80.1%

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

   if -1.8499999999999999e-109 < t1 < 9.5000000000000004e-65

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. times-frac 94.6%

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

      3. neg-mul-1 94.6%

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

      4. associate-/l* 94.6%

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

      5. associate-*r/ 94.7%

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

      6. associate-/l* 94.7%

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

      7. associate-/l/ 94.7%

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

      8. neg-mul-1 94.7%

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

      9. *-lft-identity 94.7%

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

      10. metadata-eval 94.7%

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

      11. times-frac 94.7%

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

      12. neg-mul-1 94.7%

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

      13. remove-double-neg 94.7%

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

      14. neg-mul-1 94.7%

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

      15. sub0-neg 94.7%

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

      16. associate--r+ 94.7%

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

      17. neg-sub0 94.7%

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

      18. div-sub 94.7%

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

      19. distribute-frac-neg 94.7%

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

      20. *-inverses 94.7%

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

      21. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in v around 0 97.1%

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

      1. associate-*r/ 97.1%

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

      2. neg-mul-1 97.1%

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

      3. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in t1 around 0 83.8%

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

      1. unpow2 83.8%

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

      2. associate-*r/ 91.5%

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

   9. Simplified 91.5%

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

   if 5.6000000000000001e-48 < t1 < 2.0500000000000001e40

   1. Initial program 87.8%

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

      1. associate-/l* 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in t1 around 0 81.6%

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

      1. unpow2 81.6%

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

   6. Simplified 81.6%

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

      1. associate-/r/ 81.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 26.7%

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

      4. sqr-neg 26.7%

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

      5. sqrt-unprod 26.7%

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

      6. add-sqr-sqrt 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. associate-*l/ 26.7%

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

      2. *-commutative 26.7%

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

      3. times-frac 26.6%

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

      4. frac-2neg 26.6%

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

      5. associate-*l/ 26.6%

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

      6. add-sqr-sqrt 25.9%

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

      7. sqrt-unprod 38.9%

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

      8. sqr-neg 38.9%

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

      9. sqrt-unprod 25.2%

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

      10. add-sqr-sqrt 87.6%

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

   10. Applied egg-rr 87.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{-109}:\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 5.6 \cdot 10^{-48} \lor \neg \left(t1 \leq 2.05 \cdot 10^{+40}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 6: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 4.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{-56} \lor \neg \left(t1 \leq 2.6 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.2e-109)
   (/ (/ v t1) (- -1.0 (/ u t1)))
   (if (<= t1 4.9e-67)
     (/ (- v) (* u (/ u t1)))
     (if (or (<= t1 2.3e-56) (not (<= t1 2.6e+41)))
       (/ (- v) (+ t1 (* u 2.0)))
       (/ (* v (/ t1 u)) (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.2e-109) {
		tmp = (v / t1) / (-1.0 - (u / t1));
	} else if (t1 <= 4.9e-67) {
		tmp = -v / (u * (u / t1));
	} else if ((t1 <= 2.3e-56) || !(t1 <= 2.6e+41)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.2d-109)) then
        tmp = (v / t1) / ((-1.0d0) - (u / t1))
    else if (t1 <= 4.9d-67) then
        tmp = -v / (u * (u / t1))
    else if ((t1 <= 2.3d-56) .or. (.not. (t1 <= 2.6d+41))) then
        tmp = -v / (t1 + (u * 2.0d0))
    else
        tmp = (v * (t1 / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.2e-109) {
		tmp = (v / t1) / (-1.0 - (u / t1));
	} else if (t1 <= 4.9e-67) {
		tmp = -v / (u * (u / t1));
	} else if ((t1 <= 2.3e-56) || !(t1 <= 2.6e+41)) {
		tmp = -v / (t1 + (u * 2.0));
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.2e-109:
		tmp = (v / t1) / (-1.0 - (u / t1))
	elif t1 <= 4.9e-67:
		tmp = -v / (u * (u / t1))
	elif (t1 <= 2.3e-56) or not (t1 <= 2.6e+41):
		tmp = -v / (t1 + (u * 2.0))
	else:
		tmp = (v * (t1 / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.2e-109)
		tmp = Float64(Float64(v / t1) / Float64(-1.0 - Float64(u / t1)));
	elseif (t1 <= 4.9e-67)
		tmp = Float64(Float64(-v) / Float64(u * Float64(u / t1)));
	elseif ((t1 <= 2.3e-56) || !(t1 <= 2.6e+41))
		tmp = Float64(Float64(-v) / Float64(t1 + Float64(u * 2.0)));
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.2e-109)
		tmp = (v / t1) / (-1.0 - (u / t1));
	elseif (t1 <= 4.9e-67)
		tmp = -v / (u * (u / t1));
	elseif ((t1 <= 2.3e-56) || ~((t1 <= 2.6e+41)))
		tmp = -v / (t1 + (u * 2.0));
	else
		tmp = (v * (t1 / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.2e-109], N[(N[(v / t1), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.9e-67], N[((-v) / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 2.3e-56], N[Not[LessEqual[t1, 2.6e+41]], $MachinePrecision]], N[((-v) / N[(t1 + N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.1999999999999999e-109

   1. Initial program 66.5%

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

      1. *-commutative 66.5%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 77.0%

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

   if -2.1999999999999999e-109 < t1 < 4.89999999999999993e-67

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. times-frac 94.6%

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

      3. neg-mul-1 94.6%

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

      4. associate-/l* 94.6%

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

      5. associate-*r/ 94.7%

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

      6. associate-/l* 94.7%

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

      7. associate-/l/ 94.7%

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

      8. neg-mul-1 94.7%

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

      9. *-lft-identity 94.7%

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

      10. metadata-eval 94.7%

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

      11. times-frac 94.7%

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

      12. neg-mul-1 94.7%

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

      13. remove-double-neg 94.7%

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

      14. neg-mul-1 94.7%

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

      15. sub0-neg 94.7%

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

      16. associate--r+ 94.7%

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

      17. neg-sub0 94.7%

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

      18. div-sub 94.7%

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

      19. distribute-frac-neg 94.7%

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

      20. *-inverses 94.7%

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

      21. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in v around 0 97.1%

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

      1. associate-*r/ 97.1%

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

      2. neg-mul-1 97.1%

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

      3. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in t1 around 0 83.8%

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

      1. unpow2 83.8%

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

      2. associate-*r/ 91.5%

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

   9. Simplified 91.5%

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

   if 4.89999999999999993e-67 < t1 < 2.30000000000000002e-56 or 2.6000000000000001e41 < t1

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 96.9%

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

      1. associate-*r/ 96.9%

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

      2. neg-mul-1 96.9%

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

      3. +-commutative 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in t1 around inf 85.0%

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

      1. *-commutative 85.0%

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

   9. Simplified 85.0%

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

   if 2.30000000000000002e-56 < t1 < 2.6000000000000001e41

   1. Initial program 87.8%

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

      1. associate-/l* 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in t1 around 0 81.6%

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

      1. unpow2 81.6%

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

   6. Simplified 81.6%

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

      1. associate-/r/ 81.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 26.7%

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

      4. sqr-neg 26.7%

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

      5. sqrt-unprod 26.7%

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

      6. add-sqr-sqrt 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. associate-*l/ 26.7%

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

      2. *-commutative 26.7%

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

      3. times-frac 26.6%

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

      4. frac-2neg 26.6%

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

      5. associate-*l/ 26.6%

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

      6. add-sqr-sqrt 25.9%

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

      7. sqrt-unprod 38.9%

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

      8. sqr-neg 38.9%

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

      9. sqrt-unprod 25.2%

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

      10. add-sqr-sqrt 87.6%

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

   10. Applied egg-rr 87.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 4.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{-56} \lor \neg \left(t1 \leq 2.6 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 7: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{if}\;t1 \leq -9 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.18 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.02 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (* (- v) (/ t1 (* u u)))))
   (if (<= t1 -9e-107)
     t_1
     (if (<= t1 1.18e-64)
       t_2
       (if (<= t1 4.4e-51) t_1 (if (<= t1 1.02e+42) t_2 (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = -v * (t1 / (u * u));
	double tmp;
	if (t1 <= -9e-107) {
		tmp = t_1;
	} else if (t1 <= 1.18e-64) {
		tmp = t_2;
	} else if (t1 <= 4.4e-51) {
		tmp = t_1;
	} else if (t1 <= 1.02e+42) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    t_2 = -v * (t1 / (u * u))
    if (t1 <= (-9d-107)) then
        tmp = t_1
    else if (t1 <= 1.18d-64) then
        tmp = t_2
    else if (t1 <= 4.4d-51) then
        tmp = t_1
    else if (t1 <= 1.02d+42) then
        tmp = t_2
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = -v * (t1 / (u * u));
	double tmp;
	if (t1 <= -9e-107) {
		tmp = t_1;
	} else if (t1 <= 1.18e-64) {
		tmp = t_2;
	} else if (t1 <= 4.4e-51) {
		tmp = t_1;
	} else if (t1 <= 1.02e+42) {
		tmp = t_2;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	t_2 = -v * (t1 / (u * u))
	tmp = 0
	if t1 <= -9e-107:
		tmp = t_1
	elif t1 <= 1.18e-64:
		tmp = t_2
	elif t1 <= 4.4e-51:
		tmp = t_1
	elif t1 <= 1.02e+42:
		tmp = t_2
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	t_2 = Float64(Float64(-v) * Float64(t1 / Float64(u * u)))
	tmp = 0.0
	if (t1 <= -9e-107)
		tmp = t_1;
	elseif (t1 <= 1.18e-64)
		tmp = t_2;
	elseif (t1 <= 4.4e-51)
		tmp = t_1;
	elseif (t1 <= 1.02e+42)
		tmp = t_2;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	t_2 = -v * (t1 / (u * u));
	tmp = 0.0;
	if (t1 <= -9e-107)
		tmp = t_1;
	elseif (t1 <= 1.18e-64)
		tmp = t_2;
	elseif (t1 <= 4.4e-51)
		tmp = t_1;
	elseif (t1 <= 1.02e+42)
		tmp = t_2;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-v) * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -9e-107], t$95$1, If[LessEqual[t1, 1.18e-64], t$95$2, If[LessEqual[t1, 4.4e-51], t$95$1, If[LessEqual[t1, 1.02e+42], t$95$2, N[((-v) / t1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.00000000000000032e-107 or 1.17999999999999996e-64 < t1 < 4.4e-51

   1. Initial program 67.2%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 76.8%

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

   if -9.00000000000000032e-107 < t1 < 1.17999999999999996e-64 or 4.4e-51 < t1 < 1.01999999999999996e42

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. times-frac 95.4%

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

      3. neg-mul-1 95.4%

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

      4. associate-/l* 95.5%

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

      5. associate-*r/ 95.6%

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

      6. associate-/l* 95.6%

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

      7. associate-/l/ 95.6%

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

      8. neg-mul-1 95.6%

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

      9. *-lft-identity 95.6%

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

      10. metadata-eval 95.6%

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

      11. times-frac 95.6%

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

      12. neg-mul-1 95.6%

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

      13. remove-double-neg 95.6%

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

      14. neg-mul-1 95.6%

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

      15. sub0-neg 95.6%

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

      16. associate--r+ 95.6%

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

      17. neg-sub0 95.6%

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

      18. div-sub 95.6%

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

      19. distribute-frac-neg 95.6%

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

      20. *-inverses 95.6%

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

      21. metadata-eval 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in v around 0 96.5%

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

      1. associate-*r/ 96.5%

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

      2. neg-mul-1 96.5%

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

      3. +-commutative 96.5%

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

   6. Simplified 96.5%

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

   7. Taylor expanded in t1 around 0 80.6%

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

      1. mul-1-neg 80.6%

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

      2. unpow2 80.6%

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

      3. associate-*l/ 83.5%

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

      4. *-commutative 83.5%

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

   9. Simplified 83.5%

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

   if 1.01999999999999996e42 < t1

   1. Initial program 44.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 83.3%

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

      1. associate-*r/ 83.3%

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

      2. neg-mul-1 83.3%

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

   6. Simplified 83.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-107}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.18 \cdot 10^{-64}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.02 \cdot 10^{+42}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;t1 \leq -5.3 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3.5 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (* (/ (- t1) u) (/ v u))))
   (if (<= t1 -5.3e-108)
     t_1
     (if (<= t1 3.5e-65)
       t_2
       (if (<= t1 2e-56) t_1 (if (<= t1 2.1e+44) t_2 (/ (- v) t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = (-t1 / u) * (v / u);
	double tmp;
	if (t1 <= -5.3e-108) {
		tmp = t_1;
	} else if (t1 <= 3.5e-65) {
		tmp = t_2;
	} else if (t1 <= 2e-56) {
		tmp = t_1;
	} else if (t1 <= 2.1e+44) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    t_2 = (-t1 / u) * (v / u)
    if (t1 <= (-5.3d-108)) then
        tmp = t_1
    else if (t1 <= 3.5d-65) then
        tmp = t_2
    else if (t1 <= 2d-56) then
        tmp = t_1
    else if (t1 <= 2.1d+44) then
        tmp = t_2
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = (-t1 / u) * (v / u);
	double tmp;
	if (t1 <= -5.3e-108) {
		tmp = t_1;
	} else if (t1 <= 3.5e-65) {
		tmp = t_2;
	} else if (t1 <= 2e-56) {
		tmp = t_1;
	} else if (t1 <= 2.1e+44) {
		tmp = t_2;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	t_2 = (-t1 / u) * (v / u)
	tmp = 0
	if t1 <= -5.3e-108:
		tmp = t_1
	elif t1 <= 3.5e-65:
		tmp = t_2
	elif t1 <= 2e-56:
		tmp = t_1
	elif t1 <= 2.1e+44:
		tmp = t_2
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	t_2 = Float64(Float64(Float64(-t1) / u) * Float64(v / u))
	tmp = 0.0
	if (t1 <= -5.3e-108)
		tmp = t_1;
	elseif (t1 <= 3.5e-65)
		tmp = t_2;
	elseif (t1 <= 2e-56)
		tmp = t_1;
	elseif (t1 <= 2.1e+44)
		tmp = t_2;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	t_2 = (-t1 / u) * (v / u);
	tmp = 0.0;
	if (t1 <= -5.3e-108)
		tmp = t_1;
	elseif (t1 <= 3.5e-65)
		tmp = t_2;
	elseif (t1 <= 2e-56)
		tmp = t_1;
	elseif (t1 <= 2.1e+44)
		tmp = t_2;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.3e-108], t$95$1, If[LessEqual[t1, 3.5e-65], t$95$2, If[LessEqual[t1, 2e-56], t$95$1, If[LessEqual[t1, 2.1e+44], t$95$2, N[((-v) / t1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.29999999999999989e-108 or 3.50000000000000005e-65 < t1 < 2.0000000000000001e-56

   1. Initial program 67.2%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 76.8%

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

   if -5.29999999999999989e-108 < t1 < 3.50000000000000005e-65 or 2.0000000000000001e-56 < t1 < 2.09999999999999987e44

   1. Initial program 83.7%

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

      1. times-frac 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in t1 around 0 87.6%

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

   5. Taylor expanded in t1 around 0 88.7%

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

      1. associate-*r/ 88.7%

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

      2. neg-mul-1 88.7%

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

   7. Simplified 88.7%

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

   if 2.09999999999999987e44 < t1

   1. Initial program 44.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 83.3%

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

      1. associate-*r/ 83.3%

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

      2. neg-mul-1 83.3%

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

   6. Simplified 83.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.1 \cdot 10^{+44}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9: 68.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+163} \lor \neg \left(u \leq 3.6 \cdot 10^{+162}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.6e+163) (not (<= u 3.6e+162)))
   (* t1 (/ v (* u u)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.6e+163) || !(u <= 3.6e+162)) {
		tmp = t1 * (v / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.6d+163)) .or. (.not. (u <= 3.6d+162))) then
        tmp = t1 * (v / (u * u))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.6e+163) || !(u <= 3.6e+162)) {
		tmp = t1 * (v / (u * u));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.6e+163) or not (u <= 3.6e+162):
		tmp = t1 * (v / (u * u))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.6e+163) || !(u <= 3.6e+162))
		tmp = Float64(t1 * Float64(v / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.6e+163) || ~((u <= 3.6e+162)))
		tmp = t1 * (v / (u * u));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.6e+163], N[Not[LessEqual[u, 3.6e+162]], $MachinePrecision]], N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.60000000000000003e163 or 3.59999999999999994e162 < u

   1. Initial program 72.4%

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

      1. associate-/l* 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in t1 around 0 72.9%

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

      1. unpow2 72.9%

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

   6. Simplified 72.9%

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

      1. div-inv 72.9%

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

      2. clear-num 72.9%

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

      3. add-sqr-sqrt 40.3%

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

      4. sqrt-unprod 56.6%

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

      5. sqr-neg 56.6%

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

      6. sqrt-unprod 32.6%

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

      7. add-sqr-sqrt 72.9%

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

   8. Applied egg-rr 72.9%

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

   if -4.60000000000000003e163 < u < 3.59999999999999994e162

   1. Initial program 66.5%

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

      1. times-frac 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in t1 around inf 67.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+163} \lor \neg \left(u \leq 3.6 \cdot 10^{+162}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 10: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+165} \lor \neg \left(u \leq 3.9 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.75e+165) (not (<= u 3.9e+164)))
   (* (/ v u) -0.5)
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+165) || !(u <= 3.9e+164)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.75d+165)) .or. (.not. (u <= 3.9d+164))) then
        tmp = (v / u) * (-0.5d0)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+165) || !(u <= 3.9e+164)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.75e+165) or not (u <= 3.9e+164):
		tmp = (v / u) * -0.5
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.75e+165) || !(u <= 3.9e+164))
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.75e+165) || ~((u <= 3.9e+164)))
		tmp = (v / u) * -0.5;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.75e+165], N[Not[LessEqual[u, 3.9e+164]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.74999999999999998e165 or 3.89999999999999985e164 < u

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

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

      16. associate--r+ 100.0%

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

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 84.9%

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

      1. associate-*r/ 84.9%

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

      2. neg-mul-1 84.9%

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

      3. +-commutative 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in t1 around inf 42.2%

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

      1. *-commutative 42.2%

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

   9. Simplified 42.2%

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

   10. Taylor expanded in t1 around 0 39.4%

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

   if -1.74999999999999998e165 < u < 3.89999999999999985e164

   1. Initial program 66.5%

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

      1. times-frac 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in t1 around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   6. Simplified 66.2%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+165} \lor \neg \left(u \leq 3.9 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 59.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+165}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \mathbf{elif}\;u \leq 3.55 \cdot 10^{+162}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4e+165)
   (* v (/ -0.5 u))
   (if (<= u 3.55e+162) (/ (- v) t1) (* (/ v u) -0.5))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+165) {
		tmp = v * (-0.5 / u);
	} else if (u <= 3.55e+162) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4d+165)) then
        tmp = v * ((-0.5d0) / u)
    else if (u <= 3.55d+162) then
        tmp = -v / t1
    else
        tmp = (v / u) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+165) {
		tmp = v * (-0.5 / u);
	} else if (u <= 3.55e+162) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * -0.5;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4e+165:
		tmp = v * (-0.5 / u)
	elif u <= 3.55e+162:
		tmp = -v / t1
	else:
		tmp = (v / u) * -0.5
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4e+165)
		tmp = Float64(v * Float64(-0.5 / u));
	elseif (u <= 3.55e+162)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / u) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4e+165)
		tmp = v * (-0.5 / u);
	elseif (u <= 3.55e+162)
		tmp = -v / t1;
	else
		tmp = (v / u) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4e+165], N[(v * N[(-0.5 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.55e+162], N[((-v) / t1), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.9999999999999996e165

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

      2. times-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 82.7%

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

      1. associate-*r/ 82.7%

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

      2. neg-mul-1 82.7%

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

      3. +-commutative 82.7%

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

   6. Simplified 82.7%

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

   7. Taylor expanded in t1 around inf 41.7%

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

      1. *-commutative 41.7%

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

   9. Simplified 41.7%

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

   10. Taylor expanded in t1 around 0 39.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 39.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \frac{v}{u}\right)\right)} \]

       2. expm1-udef 74.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{v}{u}\right)} - 1} \]

   12. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \frac{v}{u}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 39.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \frac{v}{u}\right)\right)} \]

       2. expm1-log1p 39.1%

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

       3. associate-*r/ 39.1%

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

       4. associate-*l/ 39.1%

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

       5. *-commutative 39.1%

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

   14. Simplified 39.1%

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

   if -3.9999999999999996e165 < u < 3.5499999999999999e162

   1. Initial program 66.5%

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

      1. times-frac 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in t1 around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   6. Simplified 66.2%

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

   if 3.5499999999999999e162 < u

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

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

      16. associate--r+ 100.0%

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

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

      3. +-commutative 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in t1 around inf 42.7%

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

      1. *-commutative 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in t1 around 0 39.7%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+165}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \mathbf{elif}\;u \leq 3.55 \cdot 10^{+162}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \]

Alternative 12: 59.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.1 \cdot 10^{+163}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 6.4 \cdot 10^{+163}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.1e+163)
   (/ -0.5 (/ u v))
   (if (<= u 6.4e+163) (/ (- v) t1) (* (/ v u) -0.5))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.1e+163) {
		tmp = -0.5 / (u / v);
	} else if (u <= 6.4e+163) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.1d+163)) then
        tmp = (-0.5d0) / (u / v)
    else if (u <= 6.4d+163) then
        tmp = -v / t1
    else
        tmp = (v / u) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.1e+163) {
		tmp = -0.5 / (u / v);
	} else if (u <= 6.4e+163) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * -0.5;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.1e+163:
		tmp = -0.5 / (u / v)
	elif u <= 6.4e+163:
		tmp = -v / t1
	else:
		tmp = (v / u) * -0.5
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.1e+163)
		tmp = Float64(-0.5 / Float64(u / v));
	elseif (u <= 6.4e+163)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / u) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.1e+163)
		tmp = -0.5 / (u / v);
	elseif (u <= 6.4e+163)
		tmp = -v / t1;
	else
		tmp = (v / u) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.1e+163], N[(-0.5 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 6.4e+163], N[((-v) / t1), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.1000000000000002e163

   1. Initial program 73.3%

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

      1. *-commutative 73.3%

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

      2. times-frac 100.0%

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

      3. neg-mul-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 82.7%

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

      1. associate-*r/ 82.7%

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

      2. neg-mul-1 82.7%

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

      3. +-commutative 82.7%

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

   6. Simplified 82.7%

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

   7. Taylor expanded in t1 around inf 41.7%

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

      1. *-commutative 41.7%

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

   9. Simplified 41.7%

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

   10. Taylor expanded in t1 around 0 39.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   11. Step-by-step derivation

       1. associate-*r/ 39.1%

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

       2. associate-/l* 42.9%

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

   12. Simplified 42.9%

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

   if -5.1000000000000002e163 < u < 6.3999999999999996e163

   1. Initial program 66.5%

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

      1. times-frac 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in t1 around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   6. Simplified 66.2%

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

   if 6.3999999999999996e163 < u

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

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

      16. associate--r+ 100.0%

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

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

      3. +-commutative 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in t1 around inf 42.7%

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

      1. *-commutative 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in t1 around 0 39.7%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.1 \cdot 10^{+163}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 6.4 \cdot 10^{+163}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \]

Alternative 13: 59.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+164} \lor \neg \left(u \leq 2 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.3e+164) (not (<= u 2e+166))) (/ v (- u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.3e+164) || !(u <= 2e+166)) {
		tmp = v / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.3d+164)) .or. (.not. (u <= 2d+166))) then
        tmp = v / -u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.3e+164) || !(u <= 2e+166)) {
		tmp = v / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.3e+164) or not (u <= 2e+166):
		tmp = v / -u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.3e+164) || !(u <= 2e+166))
		tmp = Float64(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 <= -1.3e+164) || ~((u <= 2e+166)))
		tmp = v / -u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.3e+164], N[Not[LessEqual[u, 2e+166]], $MachinePrecision]], N[(v / (-u)), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.3e164 or 1.99999999999999988e166 < u

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. neg-mul-1 100.0%

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

      15. sub0-neg 100.0%

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

      16. associate--r+ 100.0%

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

      17. neg-sub0 100.0%

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

      18. div-sub 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. *-inverses 100.0%

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

      21. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around inf 59.2%

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

   5. Taylor expanded in t1 around 0 39.3%

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

      1. metadata-eval 39.3%

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

      2. times-frac 39.3%

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

      3. *-lft-identity 39.3%

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

      4. neg-mul-1 39.3%

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

   7. Simplified 39.3%

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

   if -1.3e164 < u < 1.99999999999999988e166

   1. Initial program 66.5%

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

      1. times-frac 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in t1 around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   6. Simplified 66.2%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+164} \lor \neg \left(u \leq 2 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 14: 62.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. times-frac 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in t1 around inf 61.3%

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

5. Final simplification 61.3%

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

Alternative 15: 17.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-commutative 67.9%

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

   2. times-frac 98.3%

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

   3. neg-mul-1 98.3%

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

   4. associate-/l* 98.3%

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

   5. associate-*r/ 98.3%

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

   6. associate-/l* 98.3%

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

   7. associate-/l/ 98.3%

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

   8. neg-mul-1 98.3%

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

   9. *-lft-identity 98.3%

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

   10. metadata-eval 98.3%

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

   11. times-frac 98.3%

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

   12. neg-mul-1 98.3%

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

   13. remove-double-neg 98.3%

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

   14. neg-mul-1 98.3%

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

   15. sub0-neg 98.3%

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

   16. associate--r+ 98.3%

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

   17. neg-sub0 98.3%

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

   18. div-sub 98.4%

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

   19. distribute-frac-neg 98.4%

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

   20. *-inverses 98.4%

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

   21. metadata-eval 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in t1 around inf 65.8%

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

5. Taylor expanded in t1 around 0 17.0%

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

   1. metadata-eval 17.0%

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

   2. times-frac 17.0%

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

   3. *-lft-identity 17.0%

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

   4. neg-mul-1 17.0%

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

7. Simplified 17.0%

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

8. Final simplification 17.0%

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

Reproduce

herbie shell --seed 2023240 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))