Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.1%

Time: 9.9s

Alternatives: 19

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: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq -1.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -6.2 \cdot 10^{-99} \lor \neg \left(u \leq 4 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.2e+15)
   (/ (/ t1 (/ u v)) (- u))
   (if (<= u -1.6e-71)
     (/ (- v) (+ t1 u))
     (if (or (<= u -6.2e-99) (not (<= u 4e-79)))
       (* (/ v (+ t1 u)) (/ (- t1) u))
       (/ (- v) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.2e+15) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= -1.6e-71) {
		tmp = -v / (t1 + u);
	} else if ((u <= -6.2e-99) || !(u <= 4e-79)) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-7.2d+15)) then
        tmp = (t1 / (u / v)) / -u
    else if (u <= (-1.6d-71)) then
        tmp = -v / (t1 + u)
    else if ((u <= (-6.2d-99)) .or. (.not. (u <= 4d-79))) then
        tmp = (v / (t1 + u)) * (-t1 / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.2e+15) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= -1.6e-71) {
		tmp = -v / (t1 + u);
	} else if ((u <= -6.2e-99) || !(u <= 4e-79)) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.2e+15:
		tmp = (t1 / (u / v)) / -u
	elif u <= -1.6e-71:
		tmp = -v / (t1 + u)
	elif (u <= -6.2e-99) or not (u <= 4e-79):
		tmp = (v / (t1 + u)) * (-t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.2e+15)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(-u));
	elseif (u <= -1.6e-71)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif ((u <= -6.2e-99) || !(u <= 4e-79))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.2e+15)
		tmp = (t1 / (u / v)) / -u;
	elseif (u <= -1.6e-71)
		tmp = -v / (t1 + u);
	elseif ((u <= -6.2e-99) || ~((u <= 4e-79)))
		tmp = (v / (t1 + u)) * (-t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.2e+15], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, -1.6e-71], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -6.2e-99], N[Not[LessEqual[u, 4e-79]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if u < -7.2e15

   1. Initial program 77.1%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t1 around 0 80.7%

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

      1. mul-1-neg 80.7%

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

   6. Simplified 80.7%

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

      1. distribute-neg-frac 80.7%

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

      2. associate-*l/ 84.1%

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

      3. add-sqr-sqrt 37.6%

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

      4. sqrt-unprod 55.8%

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

      5. sqr-neg 55.8%

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

      6. sqrt-unprod 32.0%

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

      7. add-sqr-sqrt 58.0%

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

      8. associate-*l/ 58.1%

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

      9. frac-2neg 58.1%

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

      10. associate-*l/ 58.0%

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

      11. add-sqr-sqrt 26.0%

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

      12. sqrt-unprod 58.6%

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

      13. sqr-neg 58.6%

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

      14. sqrt-unprod 46.3%

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

      15. add-sqr-sqrt 84.1%

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

      16. +-commutative 84.1%

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

   8. Applied egg-rr 84.1%

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

      1. associate-*r/ 78.6%

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

      2. associate-/l* 85.4%

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

      3. +-commutative 85.4%

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

   10. Applied egg-rr 85.4%

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

   11. Taylor expanded in t1 around 0 82.9%

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

   if -7.2e15 < u < -1.5999999999999999e-71

   1. Initial program 88.9%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 75.7%

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

   if -1.5999999999999999e-71 < u < -6.1999999999999997e-99 or 4e-79 < u

   1. Initial program 75.6%

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

      1. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around 0 81.1%

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

      1. mul-1-neg 81.1%

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

   6. Simplified 81.1%

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

   if -6.1999999999999997e-99 < u < 4e-79

   1. Initial program 65.1%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   6. Simplified 84.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq -1.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -6.2 \cdot 10^{-99} \lor \neg \left(u \leq 4 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;u \leq -8.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq -6.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -6.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{t1}{\frac{-u}{t_1}}\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))))
   (if (<= u -8.1e+15)
     (/ (/ t1 (/ u v)) (- u))
     (if (<= u -6.5e-72)
       (/ (- v) (+ t1 u))
       (if (<= u -6.2e-99)
         (/ t1 (/ (- u) t_1))
         (if (<= u 4.3e-79) (/ (- v) t1) (* t_1 (/ (- t1) u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (u <= -8.1e+15) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= -6.5e-72) {
		tmp = -v / (t1 + u);
	} else if (u <= -6.2e-99) {
		tmp = t1 / (-u / t_1);
	} else if (u <= 4.3e-79) {
		tmp = -v / t1;
	} else {
		tmp = t_1 * (-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) :: t_1
    real(8) :: tmp
    t_1 = v / (t1 + u)
    if (u <= (-8.1d+15)) then
        tmp = (t1 / (u / v)) / -u
    else if (u <= (-6.5d-72)) then
        tmp = -v / (t1 + u)
    else if (u <= (-6.2d-99)) then
        tmp = t1 / (-u / t_1)
    else if (u <= 4.3d-79) then
        tmp = -v / t1
    else
        tmp = t_1 * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (u <= -8.1e+15) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= -6.5e-72) {
		tmp = -v / (t1 + u);
	} else if (u <= -6.2e-99) {
		tmp = t1 / (-u / t_1);
	} else if (u <= 4.3e-79) {
		tmp = -v / t1;
	} else {
		tmp = t_1 * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	tmp = 0
	if u <= -8.1e+15:
		tmp = (t1 / (u / v)) / -u
	elif u <= -6.5e-72:
		tmp = -v / (t1 + u)
	elif u <= -6.2e-99:
		tmp = t1 / (-u / t_1)
	elif u <= 4.3e-79:
		tmp = -v / t1
	else:
		tmp = t_1 * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if (u <= -8.1e+15)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(-u));
	elseif (u <= -6.5e-72)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (u <= -6.2e-99)
		tmp = Float64(t1 / Float64(Float64(-u) / t_1));
	elseif (u <= 4.3e-79)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t_1 * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	tmp = 0.0;
	if (u <= -8.1e+15)
		tmp = (t1 / (u / v)) / -u;
	elseif (u <= -6.5e-72)
		tmp = -v / (t1 + u);
	elseif (u <= -6.2e-99)
		tmp = t1 / (-u / t_1);
	elseif (u <= 4.3e-79)
		tmp = -v / t1;
	else
		tmp = t_1 * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -8.1e+15], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, -6.5e-72], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -6.2e-99], N[(t1 / N[((-u) / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.3e-79], N[((-v) / t1), $MachinePrecision], N[(t$95$1 * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -8.1e15

   1. Initial program 77.1%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t1 around 0 80.7%

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

      1. mul-1-neg 80.7%

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

   6. Simplified 80.7%

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

      1. distribute-neg-frac 80.7%

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

      2. associate-*l/ 84.1%

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

      3. add-sqr-sqrt 37.6%

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

      4. sqrt-unprod 55.8%

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

      5. sqr-neg 55.8%

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

      6. sqrt-unprod 32.0%

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

      7. add-sqr-sqrt 58.0%

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

      8. associate-*l/ 58.1%

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

      9. frac-2neg 58.1%

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

      10. associate-*l/ 58.0%

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

      11. add-sqr-sqrt 26.0%

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

      12. sqrt-unprod 58.6%

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

      13. sqr-neg 58.6%

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

      14. sqrt-unprod 46.3%

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

      15. add-sqr-sqrt 84.1%

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

      16. +-commutative 84.1%

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

   8. Applied egg-rr 84.1%

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

      1. associate-*r/ 78.6%

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

      2. associate-/l* 85.4%

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

      3. +-commutative 85.4%

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

   10. Applied egg-rr 85.4%

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

   11. Taylor expanded in t1 around 0 82.9%

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

   if -8.1e15 < u < -6.4999999999999997e-72

   1. Initial program 88.9%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 75.7%

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

   if -6.4999999999999997e-72 < u < -6.1999999999999997e-99

   1. Initial program 84.2%

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

      1. times-frac 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t1 around 0 99.2%

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

      1. mul-1-neg 99.2%

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

   6. Simplified 99.2%

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

      1. distribute-neg-frac 99.2%

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

      2. associate-*l/ 92.3%

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

      3. add-sqr-sqrt 42.1%

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

      4. sqrt-unprod 34.5%

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

      5. sqr-neg 34.5%

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

      6. sqrt-unprod 17.2%

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

      7. add-sqr-sqrt 18.6%

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

      8. associate-*l/ 18.6%

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

      9. frac-2neg 18.6%

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

      10. associate-*l/ 18.6%

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

      11. add-sqr-sqrt 1.3%

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

      12. sqrt-unprod 34.9%

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

      13. sqr-neg 34.9%

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

      14. sqrt-unprod 49.7%

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

      15. add-sqr-sqrt 92.3%

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

      16. +-commutative 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

   if -6.1999999999999997e-99 < u < 4.29999999999999982e-79

   1. Initial program 65.1%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 84.5%

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

      1. associate-*r/ 84.5%

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

      2. neg-mul-1 84.5%

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

   6. Simplified 84.5%

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

   if 4.29999999999999982e-79 < u

   1. Initial program 74.8%

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

      1. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around 0 79.4%

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

      1. mul-1-neg 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq -6.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -6.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{t1 + u}}}\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 4: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ t_2 := \frac{t1 \cdot t_1}{-u}\\ \mathbf{if}\;u \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -1.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -1.22 \cdot 10^{-114}:\\ \;\;\;\;\frac{t1}{\frac{-u}{t_1}}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))) (t_2 (/ (* t1 t_1) (- u))))
   (if (<= u -3.2e+15)
     t_2
     (if (<= u -1.5e-68)
       (/ (- v) (+ t1 u))
       (if (<= u -1.22e-114)
         (/ t1 (/ (- u) t_1))
         (if (<= u 3.6e-79) (/ (- v) t1) t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double t_2 = (t1 * t_1) / -u;
	double tmp;
	if (u <= -3.2e+15) {
		tmp = t_2;
	} else if (u <= -1.5e-68) {
		tmp = -v / (t1 + u);
	} else if (u <= -1.22e-114) {
		tmp = t1 / (-u / t_1);
	} else if (u <= 3.6e-79) {
		tmp = -v / t1;
	} else {
		tmp = t_2;
	}
	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 * t_1) / -u
    if (u <= (-3.2d+15)) then
        tmp = t_2
    else if (u <= (-1.5d-68)) then
        tmp = -v / (t1 + u)
    else if (u <= (-1.22d-114)) then
        tmp = t1 / (-u / t_1)
    else if (u <= 3.6d-79) then
        tmp = -v / t1
    else
        tmp = t_2
    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 * t_1) / -u;
	double tmp;
	if (u <= -3.2e+15) {
		tmp = t_2;
	} else if (u <= -1.5e-68) {
		tmp = -v / (t1 + u);
	} else if (u <= -1.22e-114) {
		tmp = t1 / (-u / t_1);
	} else if (u <= 3.6e-79) {
		tmp = -v / t1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	t_2 = (t1 * t_1) / -u
	tmp = 0
	if u <= -3.2e+15:
		tmp = t_2
	elif u <= -1.5e-68:
		tmp = -v / (t1 + u)
	elif u <= -1.22e-114:
		tmp = t1 / (-u / t_1)
	elif u <= 3.6e-79:
		tmp = -v / t1
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	t_2 = Float64(Float64(t1 * t_1) / Float64(-u))
	tmp = 0.0
	if (u <= -3.2e+15)
		tmp = t_2;
	elseif (u <= -1.5e-68)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (u <= -1.22e-114)
		tmp = Float64(t1 / Float64(Float64(-u) / t_1));
	elseif (u <= 3.6e-79)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	t_2 = (t1 * t_1) / -u;
	tmp = 0.0;
	if (u <= -3.2e+15)
		tmp = t_2;
	elseif (u <= -1.5e-68)
		tmp = -v / (t1 + u);
	elseif (u <= -1.22e-114)
		tmp = t1 / (-u / t_1);
	elseif (u <= 3.6e-79)
		tmp = -v / t1;
	else
		tmp = t_2;
	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 * t$95$1), $MachinePrecision] / (-u)), $MachinePrecision]}, If[LessEqual[u, -3.2e+15], t$95$2, If[LessEqual[u, -1.5e-68], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.22e-114], N[(t1 / N[((-u) / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.6e-79], N[((-v) / t1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -3.2e15 or 3.6000000000000002e-79 < u

   1. Initial program 76.0%

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

      1. times-frac 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 0 80.1%

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

      1. mul-1-neg 80.1%

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

   6. Simplified 80.1%

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

      1. distribute-neg-frac 80.1%

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

      2. associate-*l/ 82.1%

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

      3. add-sqr-sqrt 44.4%

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

      4. sqrt-unprod 56.4%

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

      5. sqr-neg 56.4%

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

      6. sqrt-unprod 25.1%

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

      7. add-sqr-sqrt 55.5%

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

      8. associate-*l/ 55.5%

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

      9. frac-2neg 55.5%

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

      10. associate-*l/ 55.5%

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

      11. add-sqr-sqrt 30.4%

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

      12. sqrt-unprod 55.7%

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

      13. sqr-neg 55.7%

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

      14. sqrt-unprod 37.6%

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

      15. add-sqr-sqrt 82.1%

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

      16. +-commutative 82.1%

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

   8. Applied egg-rr 82.1%

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

   if -3.2e15 < u < -1.5e-68

   1. Initial program 88.9%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 75.7%

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

   if -1.5e-68 < u < -1.22e-114

   1. Initial program 76.2%

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

      1. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t1 around 0 86.9%

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

      1. mul-1-neg 86.9%

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

   6. Simplified 86.9%

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

      1. distribute-neg-frac 86.9%

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

      2. associate-*l/ 94.0%

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

      3. add-sqr-sqrt 44.1%

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

      4. sqrt-unprod 26.5%

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

      5. sqr-neg 26.5%

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

      6. sqrt-unprod 13.4%

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

      7. add-sqr-sqrt 26.9%

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

      8. associate-*l/ 14.4%

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

      9. frac-2neg 14.4%

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

      10. associate-*l/ 26.9%

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

      11. add-sqr-sqrt 13.5%

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

      12. sqrt-unprod 26.8%

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

      13. sqr-neg 26.8%

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

      14. sqrt-unprod 49.6%

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

      15. add-sqr-sqrt 94.0%

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

      16. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-/l* 99.8%

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

   10. Simplified 99.8%

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

   if -1.22e-114 < u < 3.6000000000000002e-79

   1. Initial program 65.4%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. neg-mul-1 85.2%

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

   6. Simplified 85.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq -1.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -1.22 \cdot 10^{-114}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{t1 + u}}}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \end{array} \]

Alternative 5: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;u \leq -3.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{t1 \cdot t_1}{-u}\\ \mathbf{elif}\;u \leq -4.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -1.25 \cdot 10^{-114}:\\ \;\;\;\;\frac{t1}{\frac{-u}{t_1}}\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{-u}}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))))
   (if (<= u -3.85e+15)
     (/ (* t1 t_1) (- u))
     (if (<= u -4.3e-72)
       (/ (- v) (+ t1 u))
       (if (<= u -1.25e-114)
         (/ t1 (/ (- u) t_1))
         (if (<= u 4.3e-79) (/ (- v) t1) (/ (/ t1 (- u)) (/ (+ t1 u) v))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (u <= -3.85e+15) {
		tmp = (t1 * t_1) / -u;
	} else if (u <= -4.3e-72) {
		tmp = -v / (t1 + u);
	} else if (u <= -1.25e-114) {
		tmp = t1 / (-u / t_1);
	} else if (u <= 4.3e-79) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / -u) / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (t1 + u)
    if (u <= (-3.85d+15)) then
        tmp = (t1 * t_1) / -u
    else if (u <= (-4.3d-72)) then
        tmp = -v / (t1 + u)
    else if (u <= (-1.25d-114)) then
        tmp = t1 / (-u / t_1)
    else if (u <= 4.3d-79) then
        tmp = -v / t1
    else
        tmp = (t1 / -u) / ((t1 + 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);
	double tmp;
	if (u <= -3.85e+15) {
		tmp = (t1 * t_1) / -u;
	} else if (u <= -4.3e-72) {
		tmp = -v / (t1 + u);
	} else if (u <= -1.25e-114) {
		tmp = t1 / (-u / t_1);
	} else if (u <= 4.3e-79) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / -u) / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	tmp = 0
	if u <= -3.85e+15:
		tmp = (t1 * t_1) / -u
	elif u <= -4.3e-72:
		tmp = -v / (t1 + u)
	elif u <= -1.25e-114:
		tmp = t1 / (-u / t_1)
	elif u <= 4.3e-79:
		tmp = -v / t1
	else:
		tmp = (t1 / -u) / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if (u <= -3.85e+15)
		tmp = Float64(Float64(t1 * t_1) / Float64(-u));
	elseif (u <= -4.3e-72)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (u <= -1.25e-114)
		tmp = Float64(t1 / Float64(Float64(-u) / t_1));
	elseif (u <= 4.3e-79)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 / Float64(-u)) / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	tmp = 0.0;
	if (u <= -3.85e+15)
		tmp = (t1 * t_1) / -u;
	elseif (u <= -4.3e-72)
		tmp = -v / (t1 + u);
	elseif (u <= -1.25e-114)
		tmp = t1 / (-u / t_1);
	elseif (u <= 4.3e-79)
		tmp = -v / t1;
	else
		tmp = (t1 / -u) / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.85e+15], N[(N[(t1 * t$95$1), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, -4.3e-72], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -1.25e-114], N[(t1 / N[((-u) / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.3e-79], N[((-v) / t1), $MachinePrecision], N[(N[(t1 / (-u)), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if u < -3.85e15

   1. Initial program 77.1%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t1 around 0 80.7%

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

      1. mul-1-neg 80.7%

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

   6. Simplified 80.7%

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

      1. distribute-neg-frac 80.7%

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

      2. associate-*l/ 84.1%

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

      3. add-sqr-sqrt 37.6%

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

      4. sqrt-unprod 55.8%

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

      5. sqr-neg 55.8%

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

      6. sqrt-unprod 32.0%

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

      7. add-sqr-sqrt 58.0%

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

      8. associate-*l/ 58.1%

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

      9. frac-2neg 58.1%

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

      10. associate-*l/ 58.0%

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

      11. add-sqr-sqrt 26.0%

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

      12. sqrt-unprod 58.6%

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

      13. sqr-neg 58.6%

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

      14. sqrt-unprod 46.3%

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

      15. add-sqr-sqrt 84.1%

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

      16. +-commutative 84.1%

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

   8. Applied egg-rr 84.1%

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

   if -3.85e15 < u < -4.2999999999999999e-72

   1. Initial program 88.9%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around inf 75.7%

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

   if -4.2999999999999999e-72 < u < -1.24999999999999997e-114

   1. Initial program 76.2%

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

      1. times-frac 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t1 around 0 86.9%

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

      1. mul-1-neg 86.9%

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

   6. Simplified 86.9%

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

      1. distribute-neg-frac 86.9%

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

      2. associate-*l/ 94.0%

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

      3. add-sqr-sqrt 44.1%

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

      4. sqrt-unprod 26.5%

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

      5. sqr-neg 26.5%

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

      6. sqrt-unprod 13.4%

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

      7. add-sqr-sqrt 26.9%

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

      8. associate-*l/ 14.4%

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

      9. frac-2neg 14.4%

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

      10. associate-*l/ 26.9%

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

      11. add-sqr-sqrt 13.5%

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

      12. sqrt-unprod 26.8%

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

      13. sqr-neg 26.8%

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

      14. sqrt-unprod 49.6%

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

      15. add-sqr-sqrt 94.0%

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

      16. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-/l* 99.8%

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

   10. Simplified 99.8%

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

   if -1.24999999999999997e-114 < u < 4.29999999999999982e-79

   1. Initial program 65.4%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. neg-mul-1 85.2%

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

   6. Simplified 85.2%

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

   if 4.29999999999999982e-79 < u

   1. Initial program 74.8%

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

      1. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around 0 79.4%

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

      1. mul-1-neg 79.4%

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

   6. Simplified 79.4%

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

      1. distribute-neg-frac 79.4%

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

      2. associate-*l/ 79.9%

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

      3. add-sqr-sqrt 51.8%

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

      4. sqrt-unprod 57.0%

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

      5. sqr-neg 57.0%

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

      6. sqrt-unprod 17.3%

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

      7. add-sqr-sqrt 52.7%

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

      8. associate-*l/ 52.7%

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

      9. frac-2neg 52.7%

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

      10. clear-num 53.4%

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

      11. frac-times 53.5%

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

      12. *-commutative 53.5%

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

      13. *-un-lft-identity 53.5%

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

      14. add-sqr-sqrt 36.1%

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

      15. sqrt-unprod 50.8%

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

      16. sqr-neg 50.8%

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

      17. sqrt-unprod 26.5%

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

      18. add-sqr-sqrt 79.2%

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

      19. +-commutative 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. associate-/r* 80.2%

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

   10. Simplified 80.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \mathbf{elif}\;u \leq -4.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -1.25 \cdot 10^{-114}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{t1 + u}}}\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{-u}}{\frac{t1 + u}{v}}\\ \end{array} \]

Alternative 6: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8.6 \cdot 10^{-76} \lor \neg \left(t1 \leq 1.05 \cdot 10^{-131}\right) \land \left(t1 \leq 2 \cdot 10^{-40} \lor \neg \left(t1 \leq 1.06 \cdot 10^{+18}\right)\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -8.6e-76)
         (and (not (<= t1 1.05e-131))
              (or (<= t1 2e-40) (not (<= t1 1.06e+18)))))
   (/ (- v) (+ t1 u))
   (* t1 (/ (- v) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8.6e-76) || (!(t1 <= 1.05e-131) && ((t1 <= 2e-40) || !(t1 <= 1.06e+18)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 * (-v / (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 <= (-8.6d-76)) .or. (.not. (t1 <= 1.05d-131)) .and. (t1 <= 2d-40) .or. (.not. (t1 <= 1.06d+18))) then
        tmp = -v / (t1 + u)
    else
        tmp = t1 * (-v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -8.6e-76) || (!(t1 <= 1.05e-131) && ((t1 <= 2e-40) || !(t1 <= 1.06e+18)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = t1 * (-v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -8.6e-76) or (not (t1 <= 1.05e-131) and ((t1 <= 2e-40) or not (t1 <= 1.06e+18))):
		tmp = -v / (t1 + u)
	else:
		tmp = t1 * (-v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -8.6e-76) || (!(t1 <= 1.05e-131) && ((t1 <= 2e-40) || !(t1 <= 1.06e+18))))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -8.6e-76) || (~((t1 <= 1.05e-131)) && ((t1 <= 2e-40) || ~((t1 <= 1.06e+18)))))
		tmp = -v / (t1 + u);
	else
		tmp = t1 * (-v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -8.6e-76], And[N[Not[LessEqual[t1, 1.05e-131]], $MachinePrecision], Or[LessEqual[t1, 2e-40], N[Not[LessEqual[t1, 1.06e+18]], $MachinePrecision]]]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.5999999999999998e-76 or 1.04999999999999999e-131 < t1 < 1.9999999999999999e-40 or 1.06e18 < t1

   1. Initial program 67.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 81.2%

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

   if -8.5999999999999998e-76 < t1 < 1.04999999999999999e-131 or 1.9999999999999999e-40 < t1 < 1.06e18

   1. Initial program 82.2%

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

      1. associate-/l* 85.2%

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

      2. neg-mul-1 85.2%

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

      3. *-commutative 85.2%

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

      4. associate-*r/ 85.2%

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

      5. associate-/l* 85.3%

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

      6. neg-mul-1 85.3%

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

      7. associate-/r* 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in t1 around 0 81.7%

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

      1. associate-*r/ 81.7%

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

      2. neg-mul-1 81.7%

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

      3. unpow2 81.7%

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

   6. Simplified 81.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.6 \cdot 10^{-76} \lor \neg \left(t1 \leq 1.05 \cdot 10^{-131}\right) \land \left(t1 \leq 2 \cdot 10^{-40} \lor \neg \left(t1 \leq 1.06 \cdot 10^{+18}\right)\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 7: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{-127}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6.7 \cdot 10^{-40} \lor \neg \left(t1 \leq 62000000000000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -1.15e-77)
     t_1
     (if (<= t1 1.25e-127)
       (* v (/ (/ (- t1) u) u))
       (if (or (<= t1 6.7e-40) (not (<= t1 62000000000000.0)))
         t_1
         (* t1 (/ (- v) (* u u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1.15e-77) {
		tmp = t_1;
	} else if (t1 <= 1.25e-127) {
		tmp = v * ((-t1 / u) / u);
	} else if ((t1 <= 6.7e-40) || !(t1 <= 62000000000000.0)) {
		tmp = t_1;
	} else {
		tmp = t1 * (-v / (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)
    if (t1 <= (-1.15d-77)) then
        tmp = t_1
    else if (t1 <= 1.25d-127) then
        tmp = v * ((-t1 / u) / u)
    else if ((t1 <= 6.7d-40) .or. (.not. (t1 <= 62000000000000.0d0))) then
        tmp = t_1
    else
        tmp = t1 * (-v / (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);
	double tmp;
	if (t1 <= -1.15e-77) {
		tmp = t_1;
	} else if (t1 <= 1.25e-127) {
		tmp = v * ((-t1 / u) / u);
	} else if ((t1 <= 6.7e-40) || !(t1 <= 62000000000000.0)) {
		tmp = t_1;
	} else {
		tmp = t1 * (-v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -1.15e-77:
		tmp = t_1
	elif t1 <= 1.25e-127:
		tmp = v * ((-t1 / u) / u)
	elif (t1 <= 6.7e-40) or not (t1 <= 62000000000000.0):
		tmp = t_1
	else:
		tmp = t1 * (-v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1.15e-77)
		tmp = t_1;
	elseif (t1 <= 1.25e-127)
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	elseif ((t1 <= 6.7e-40) || !(t1 <= 62000000000000.0))
		tmp = t_1;
	else
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -1.15e-77)
		tmp = t_1;
	elseif (t1 <= 1.25e-127)
		tmp = v * ((-t1 / u) / u);
	elseif ((t1 <= 6.7e-40) || ~((t1 <= 62000000000000.0)))
		tmp = t_1;
	else
		tmp = t1 * (-v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.15e-77], t$95$1, If[LessEqual[t1, 1.25e-127], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 6.7e-40], N[Not[LessEqual[t1, 62000000000000.0]], $MachinePrecision]], t$95$1, N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.14999999999999999e-77 or 1.2499999999999999e-127 < t1 < 6.6999999999999998e-40 or 6.2e13 < t1

   1. Initial program 67.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 81.2%

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

   if -1.14999999999999999e-77 < t1 < 1.2499999999999999e-127

   1. Initial program 80.2%

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

      1. associate-/l* 83.6%

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

      2. associate-/l* 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in t1 around 0 80.7%

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

      1. unpow2 80.7%

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

   6. Simplified 80.7%

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

      1. add-sqr-sqrt 50.2%

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

      2. sqrt-unprod 51.7%

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

      3. sqr-neg 51.7%

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

      4. sqrt-unprod 15.3%

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

      5. add-sqr-sqrt 42.4%

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

      6. associate-/r/ 42.5%

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

   8. Applied egg-rr 42.5%

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

      1. clear-num 42.5%

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

      2. associate-*l/ 42.5%

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

      3. *-un-lft-identity 42.5%

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

      4. associate-/l* 42.5%

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

   10. Applied egg-rr 42.5%

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

       1. frac-2neg 42.5%

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

       2. div-inv 42.5%

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

       3. distribute-neg-frac 42.5%

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

       4. add-sqr-sqrt 22.6%

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

       5. sqrt-unprod 59.4%

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

       6. sqr-neg 59.4%

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

       7. sqrt-unprod 37.8%

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

       8. add-sqr-sqrt 81.2%

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

       9. clear-num 81.2%

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

   12. Applied egg-rr 81.2%

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

   if 6.6999999999999998e-40 < t1 < 6.2e13

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

      2. neg-mul-1 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. associate-/l* 99.8%

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

      6. neg-mul-1 99.8%

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

      7. associate-/r* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 90.5%

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

      1. associate-*r/ 90.5%

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

      2. neg-mul-1 90.5%

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

      3. unpow2 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-77}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{-127}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 6.7 \cdot 10^{-40} \lor \neg \left(t1 \leq 62000000000000\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 8: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{-115} \lor \neg \left(u \leq 3.9 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.6e-115) (not (<= u 3.9e-79)))
   (/ (/ t1 (/ (+ t1 u) v)) (- u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.6e-115) || !(u <= 3.9e-79)) {
		tmp = (t1 / ((t1 + u) / v)) / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-7.6d-115)) .or. (.not. (u <= 3.9d-79))) then
        tmp = (t1 / ((t1 + u) / v)) / -u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.6e-115) || !(u <= 3.9e-79)) {
		tmp = (t1 / ((t1 + u) / v)) / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.6e-115) or not (u <= 3.9e-79):
		tmp = (t1 / ((t1 + u) / v)) / -u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.6e-115) || !(u <= 3.9e-79))
		tmp = Float64(Float64(t1 / Float64(Float64(t1 + u) / v)) / Float64(-u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.6e-115) || ~((u <= 3.9e-79)))
		tmp = (t1 / ((t1 + u) / v)) / -u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.6e-115], N[Not[LessEqual[u, 3.9e-79]], $MachinePrecision]], N[(N[(t1 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.59999999999999984e-115 or 3.90000000000000006e-79 < u

   1. Initial program 77.4%

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

      1. times-frac 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around 0 76.8%

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

      1. mul-1-neg 76.8%

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

   6. Simplified 76.8%

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

      1. distribute-neg-frac 76.8%

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

      2. associate-*l/ 79.4%

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

      3. add-sqr-sqrt 44.0%

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

      4. sqrt-unprod 51.5%

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

      5. sqr-neg 51.5%

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

      6. sqrt-unprod 23.0%

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

      7. add-sqr-sqrt 51.8%

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

      8. associate-*l/ 50.6%

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

      9. frac-2neg 50.6%

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

      10. associate-*l/ 51.8%

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

      11. add-sqr-sqrt 28.8%

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

      12. sqrt-unprod 50.1%

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

      13. sqr-neg 50.1%

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

      14. sqrt-unprod 35.3%

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

      15. add-sqr-sqrt 79.4%

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

      16. +-commutative 79.4%

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

   8. Applied egg-rr 79.4%

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

      1. associate-*r/ 69.3%

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

      2. associate-/l* 80.7%

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

      3. +-commutative 80.7%

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

   10. Applied egg-rr 80.7%

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

   if -7.59999999999999984e-115 < u < 3.90000000000000006e-79

   1. Initial program 65.4%

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

      1. times-frac 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. neg-mul-1 85.2%

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

   6. Simplified 85.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.6 \cdot 10^{-115} \lor \neg \left(u \leq 3.9 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{\frac{t1}{\frac{t1 + u}{v}}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+15} \lor \neg \left(u \leq 4 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4e+15) (not (<= u 4e-79)))
   (* (/ v u) (/ (- t1) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4e+15) || !(u <= 4e-79)) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4d+15)) .or. (.not. (u <= 4d-79))) then
        tmp = (v / u) * (-t1 / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4e+15) || !(u <= 4e-79)) {
		tmp = (v / u) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4e+15) or not (u <= 4e-79):
		tmp = (v / u) * (-t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4e+15) || !(u <= 4e-79))
		tmp = Float64(Float64(v / u) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4e+15) || ~((u <= 4e-79)))
		tmp = (v / u) * (-t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4e+15], N[Not[LessEqual[u, 4e-79]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4e15 or 4e-79 < u

   1. Initial program 76.0%

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

      1. times-frac 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 0 80.1%

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

      1. mul-1-neg 80.1%

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

   6. Simplified 80.1%

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

      1. distribute-neg-frac 80.1%

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

      2. associate-*l/ 82.1%

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

      3. add-sqr-sqrt 44.4%

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

      4. sqrt-unprod 56.4%

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

      5. sqr-neg 56.4%

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

      6. sqrt-unprod 25.1%

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

      7. add-sqr-sqrt 55.5%

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

      8. associate-*l/ 55.5%

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

      9. frac-2neg 55.5%

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

      10. associate-*l/ 55.5%

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

      11. add-sqr-sqrt 30.4%

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

      12. sqrt-unprod 55.7%

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

      13. sqr-neg 55.7%

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

      14. sqrt-unprod 37.6%

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

      15. add-sqr-sqrt 82.1%

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

      16. +-commutative 82.1%

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

   8. Applied egg-rr 82.1%

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

   9. Taylor expanded in t1 around 0 71.0%

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

       1. mul-1-neg 71.0%

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

       2. *-commutative 71.0%

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

       3. unpow2 71.0%

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

       4. times-frac 78.0%

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

       5. distribute-rgt-neg-in 78.0%

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

       6. distribute-neg-frac 78.0%

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

   11. Simplified 78.0%

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

   if -4e15 < u < 4e-79

   1. Initial program 69.7%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 79.9%

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

      1. associate-*r/ 79.9%

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

      2. neg-mul-1 79.9%

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

   6. Simplified 79.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+15} \lor \neg \left(u \leq 4 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+15} \lor \neg \left(u \leq 4.3 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.2e+15) (not (<= u 4.3e-79)))
   (/ (* t1 (/ v u)) (- u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.2e+15) || !(u <= 4.3e-79)) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.2d+15)) .or. (.not. (u <= 4.3d-79))) then
        tmp = (t1 * (v / u)) / -u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.2e+15) || !(u <= 4.3e-79)) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.2e+15) or not (u <= 4.3e-79):
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.2e+15) || !(u <= 4.3e-79))
		tmp = Float64(Float64(t1 * Float64(v / u)) / 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 <= -5.2e+15) || ~((u <= 4.3e-79)))
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.2e+15], N[Not[LessEqual[u, 4.3e-79]], $MachinePrecision]], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.2e15 or 4.29999999999999982e-79 < u

   1. Initial program 76.0%

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

      1. times-frac 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 0 80.1%

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

      1. mul-1-neg 80.1%

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

   6. Simplified 80.1%

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

      1. distribute-neg-frac 80.1%

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

      2. associate-*l/ 82.1%

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

      3. add-sqr-sqrt 44.4%

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

      4. sqrt-unprod 56.4%

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

      5. sqr-neg 56.4%

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

      6. sqrt-unprod 25.1%

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

      7. add-sqr-sqrt 55.5%

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

      8. associate-*l/ 55.5%

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

      9. frac-2neg 55.5%

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

      10. associate-*l/ 55.5%

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

      11. add-sqr-sqrt 30.4%

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

      12. sqrt-unprod 55.7%

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

      13. sqr-neg 55.7%

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

      14. sqrt-unprod 37.6%

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

      15. add-sqr-sqrt 82.1%

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

      16. +-commutative 82.1%

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

   8. Applied egg-rr 82.1%

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

   9. Taylor expanded in t1 around 0 75.5%

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

       1. associate-*r/ 80.0%

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

   11. Simplified 80.0%

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

   if -5.2e15 < u < 4.29999999999999982e-79

   1. Initial program 69.7%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 79.9%

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

      1. associate-*r/ 79.9%

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

      2. neg-mul-1 79.9%

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

   6. Simplified 79.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+15} \lor \neg \left(u \leq 4.3 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -7.5e+15)
   (/ (/ t1 (/ u v)) (- u))
   (if (<= u 4.3e-79) (/ (- v) t1) (/ (* t1 (/ v u)) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.5e+15) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= 4.3e-79) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-7.5d+15)) then
        tmp = (t1 / (u / v)) / -u
    else if (u <= 4.3d-79) then
        tmp = -v / t1
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -7.5e+15) {
		tmp = (t1 / (u / v)) / -u;
	} else if (u <= 4.3e-79) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -7.5e+15:
		tmp = (t1 / (u / v)) / -u
	elif u <= 4.3e-79:
		tmp = -v / t1
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -7.5e+15)
		tmp = Float64(Float64(t1 / Float64(u / v)) / Float64(-u));
	elseif (u <= 4.3e-79)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -7.5e+15)
		tmp = (t1 / (u / v)) / -u;
	elseif (u <= 4.3e-79)
		tmp = -v / t1;
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -7.5e+15], N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[u, 4.3e-79], N[((-v) / t1), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.5e15

   1. Initial program 77.1%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in t1 around 0 80.7%

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

      1. mul-1-neg 80.7%

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

   6. Simplified 80.7%

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

      1. distribute-neg-frac 80.7%

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

      2. associate-*l/ 84.1%

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

      3. add-sqr-sqrt 37.6%

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

      4. sqrt-unprod 55.8%

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

      5. sqr-neg 55.8%

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

      6. sqrt-unprod 32.0%

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

      7. add-sqr-sqrt 58.0%

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

      8. associate-*l/ 58.1%

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

      9. frac-2neg 58.1%

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

      10. associate-*l/ 58.0%

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

      11. add-sqr-sqrt 26.0%

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

      12. sqrt-unprod 58.6%

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

      13. sqr-neg 58.6%

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

      14. sqrt-unprod 46.3%

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

      15. add-sqr-sqrt 84.1%

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

      16. +-commutative 84.1%

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

   8. Applied egg-rr 84.1%

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

      1. associate-*r/ 78.6%

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

      2. associate-/l* 85.4%

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

      3. +-commutative 85.4%

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

   10. Applied egg-rr 85.4%

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

   11. Taylor expanded in t1 around 0 82.9%

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

   if -7.5e15 < u < 4.29999999999999982e-79

   1. Initial program 69.7%

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

      1. times-frac 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 79.9%

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

      1. associate-*r/ 79.9%

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

      2. neg-mul-1 79.9%

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

   6. Simplified 79.9%

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

   if 4.29999999999999982e-79 < u

   1. Initial program 74.8%

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

      1. times-frac 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around 0 79.4%

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

      1. mul-1-neg 79.4%

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

   6. Simplified 79.4%

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

      1. distribute-neg-frac 79.4%

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

      2. associate-*l/ 79.9%

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

      3. add-sqr-sqrt 51.8%

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

      4. sqrt-unprod 57.0%

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

      5. sqr-neg 57.0%

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

      6. sqrt-unprod 17.3%

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

      7. add-sqr-sqrt 52.7%

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

      8. associate-*l/ 52.7%

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

      9. frac-2neg 52.7%

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

      10. associate-*l/ 52.7%

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

      11. add-sqr-sqrt 35.3%

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

      12. sqrt-unprod 52.4%

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

      13. sqr-neg 52.4%

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

      14. sqrt-unprod 28.0%

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

      15. add-sqr-sqrt 79.9%

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

      16. +-commutative 79.9%

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

   8. Applied egg-rr 79.9%

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

   9. Taylor expanded in t1 around 0 73.1%

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

       1. associate-*r/ 76.9%

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

   11. Simplified 76.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 12: 66.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+89} \lor \neg \left(u \leq 3.2 \cdot 10^{+22}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.2e+89) (not (<= u 3.2e+22)))
   (* v (/ (/ t1 u) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e+89) || !(u <= 3.2e+22)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.2d+89)) .or. (.not. (u <= 3.2d+22))) then
        tmp = v * ((t1 / u) / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.2e+89) || !(u <= 3.2e+22)) {
		tmp = v * ((t1 / u) / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.2e+89) or not (u <= 3.2e+22):
		tmp = v * ((t1 / u) / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.2e+89) || !(u <= 3.2e+22))
		tmp = Float64(v * Float64(Float64(t1 / u) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.2e+89) || ~((u <= 3.2e+22)))
		tmp = v * ((t1 / u) / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.2e+89], N[Not[LessEqual[u, 3.2e+22]], $MachinePrecision]], N[(v * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.19999999999999987e89 or 3.2e22 < u

   1. Initial program 75.9%

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

      1. associate-/l* 76.5%

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

      2. associate-/l* 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in t1 around 0 75.0%

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

      1. unpow2 75.0%

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

   6. Simplified 75.0%

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

      1. expm1-log1p-u 73.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-t1}{\frac{u \cdot u}{v}}\right)\right)} \]

      2. expm1-udef 66.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-t1}{\frac{u \cdot u}{v}}\right)} - 1} \]

      3. associate-/r/ 66.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-t1}{u \cdot u} \cdot v}\right)} - 1 \]

      4. *-commutative 66.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{v \cdot \frac{-t1}{u \cdot u}}\right)} - 1 \]

      5. add-sqr-sqrt 37.1%

         \[\leadsto e^{\mathsf{log1p}\left(v \cdot \frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}{u \cdot u}\right)} - 1 \]

      6. sqrt-unprod 56.8%

         \[\leadsto e^{\mathsf{log1p}\left(v \cdot \frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}{u \cdot u}\right)} - 1 \]

      7. sqr-neg 56.8%

         \[\leadsto e^{\mathsf{log1p}\left(v \cdot \frac{\sqrt{\color{blue}{t1 \cdot t1}}}{u \cdot u}\right)} - 1 \]

      8. sqrt-unprod 29.2%

         \[\leadsto e^{\mathsf{log1p}\left(v \cdot \frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}}}{u \cdot u}\right)} - 1 \]

      9. add-sqr-sqrt 65.6%

         \[\leadsto e^{\mathsf{log1p}\left(v \cdot \frac{\color{blue}{t1}}{u \cdot u}\right)} - 1 \]

   8. Applied egg-rr 65.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(v \cdot \frac{t1}{u \cdot u}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 65.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \frac{t1}{u \cdot u}\right)\right)} \]

      2. expm1-log1p 65.5%

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

      3. associate-/r* 63.3%

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

   10. Simplified 63.3%

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

   if -3.19999999999999987e89 < u < 3.2e22

   1. Initial program 71.0%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 71.2%

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

      1. associate-*r/ 71.2%

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

      2. neg-mul-1 71.2%

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

   6. Simplified 71.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+89} \lor \neg \left(u \leq 3.2 \cdot 10^{+22}\right):\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 13: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.1 \cdot 10^{+88} \lor \neg \left(u \leq 4.6 \cdot 10^{+22}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.1e+88) (not (<= u 4.6e+22)))
   (* v (/ t1 (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.1e+88) || !(u <= 4.6e+22)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.1d+88)) .or. (.not. (u <= 4.6d+22))) then
        tmp = v * (t1 / (u * u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.1e+88) || !(u <= 4.6e+22)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.1e+88) or not (u <= 4.6e+22):
		tmp = v * (t1 / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.1e+88) || !(u <= 4.6e+22))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.1e+88) || ~((u <= 4.6e+22)))
		tmp = v * (t1 / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.1e+88], N[Not[LessEqual[u, 4.6e+22]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.0999999999999997e88 or 4.6000000000000004e22 < u

   1. Initial program 75.9%

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

      1. associate-/l* 76.5%

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

      2. associate-/l* 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in t1 around 0 75.0%

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

      1. unpow2 75.0%

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

   6. Simplified 75.0%

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

      1. add-sqr-sqrt 40.7%

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

      2. sqrt-unprod 59.6%

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

      3. sqr-neg 59.6%

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

      4. sqrt-unprod 29.2%

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

      5. add-sqr-sqrt 65.5%

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

      6. associate-/r/ 65.5%

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

   8. Applied egg-rr 65.5%

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

   if -5.0999999999999997e88 < u < 4.6000000000000004e22

   1. Initial program 71.0%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 71.2%

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

      1. associate-*r/ 71.2%

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

      2. neg-mul-1 71.2%

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

   6. Simplified 71.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.1 \cdot 10^{+88} \lor \neg \left(u \leq 4.6 \cdot 10^{+22}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 14: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. *-commutative 73.1%

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

   2. times-frac 98.4%

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

   3. neg-mul-1 98.4%

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

   4. associate-/l* 98.0%

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

   5. associate-*r/ 98.1%

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

   6. associate-/l* 98.1%

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

   7. associate-/l/ 98.1%

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

   8. neg-mul-1 98.1%

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

   9. *-lft-identity 98.1%

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

   10. metadata-eval 98.1%

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

   11. times-frac 98.1%

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

   12. neg-mul-1 98.1%

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

   13. remove-double-neg 98.1%

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

   14. neg-mul-1 98.1%

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

   15. sub0-neg 98.1%

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

   16. associate--r+ 98.1%

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

   17. neg-sub0 98.1%

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

   18. div-sub 98.1%

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

   19. distribute-frac-neg 98.1%

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

   20. *-inverses 98.1%

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

   21. metadata-eval 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 15: 58.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+184} \lor \neg \left(u \leq 2.05 \cdot 10^{+189}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.4e+184) (not (<= u 2.05e+189))) (/ v (+ t1 u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.4e+184) || !(u <= 2.05e+189)) {
		tmp = v / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-7.4d+184)) .or. (.not. (u <= 2.05d+189))) then
        tmp = v / (t1 + u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.4e+184) || !(u <= 2.05e+189)) {
		tmp = v / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -7.4e+184) or not (u <= 2.05e+189):
		tmp = v / (t1 + u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -7.4e+184) || !(u <= 2.05e+189))
		tmp = Float64(v / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -7.4e+184) || ~((u <= 2.05e+189)))
		tmp = v / (t1 + u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -7.4e+184], N[Not[LessEqual[u, 2.05e+189]], $MachinePrecision]], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -7.3999999999999995e184 or 2.0500000000000001e189 < u

   1. Initial program 79.4%

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

      1. associate-*r/ 100.0%

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

      2. add-sqr-sqrt 49.9%

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

      3. sqrt-unprod 70.0%

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

      4. sqr-neg 70.0%

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

      5. sqrt-unprod 38.7%

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

      6. add-sqr-sqrt 79.6%

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

      7. frac-2neg 79.6%

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

      8. add-sqr-sqrt 40.9%

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

      9. sqrt-unprod 70.0%

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

      10. sqr-neg 70.0%

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

      11. sqrt-unprod 49.9%

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

      12. add-sqr-sqrt 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. add-sqr-sqrt 50.0%

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

      15. sqrt-unprod 94.2%

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

      16. sqr-neg 94.2%

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

      17. sqrt-unprod 47.7%

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

      18. add-sqr-sqrt 97.7%

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

      19. sub-neg 97.7%

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

   5. Applied egg-rr 97.7%

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

   6. Taylor expanded in t1 around inf 50.4%

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

   if -7.3999999999999995e184 < u < 2.0500000000000001e189

   1. Initial program 71.4%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around inf 60.7%

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

      1. associate-*r/ 60.7%

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

      2. neg-mul-1 60.7%

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

   6. Simplified 60.7%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+184} \lor \neg \left(u \leq 2.05 \cdot 10^{+189}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 16: 58.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.6 \cdot 10^{+184} \lor \neg \left(u \leq 1.18 \cdot 10^{+188}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.6e+184) (not (<= u 1.18e+188))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.6e+184) || !(u <= 1.18e+188)) {
		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 <= (-6.6d+184)) .or. (.not. (u <= 1.18d+188))) 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 <= -6.6e+184) || !(u <= 1.18e+188)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.6e+184) or not (u <= 1.18e+188):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.6e+184) || !(u <= 1.18e+188))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.6e+184) || ~((u <= 1.18e+188)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.6e+184], N[Not[LessEqual[u, 1.18e+188]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.5999999999999996e184 or 1.18e188 < u

   1. Initial program 79.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 0 96.3%

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

      1. mul-1-neg 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in t1 around inf 49.0%

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

      1. associate-*r/ 49.0%

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

      2. neg-mul-1 49.0%

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

   9. Simplified 49.0%

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

   if -6.5999999999999996e184 < u < 1.18e188

   1. Initial program 71.4%

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

      1. times-frac 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around inf 60.7%

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

      1. associate-*r/ 60.7%

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

      2. neg-mul-1 60.7%

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

   6. Simplified 60.7%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.6 \cdot 10^{+184} \lor \neg \left(u \leq 1.18 \cdot 10^{+188}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 17: 61.1% 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 73.1%

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

   1. times-frac 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in t1 around inf 61.6%

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

5. Final simplification 61.6%

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

Alternative 18: 53.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in t1 around inf 52.8%

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

   1. associate-*r/ 52.8%

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

   2. neg-mul-1 52.8%

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

6. Simplified 52.8%

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

7. Final simplification 52.8%

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

Alternative 19: 14.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 98.4%

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

3. Simplified 98.4%

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

   1. associate-*r/ 98.2%

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

   2. add-sqr-sqrt 54.7%

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

   3. sqrt-unprod 46.0%

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

   4. sqr-neg 46.0%

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

   5. sqrt-unprod 17.1%

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

   6. add-sqr-sqrt 39.1%

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

   7. frac-2neg 39.1%

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

   8. add-sqr-sqrt 22.0%

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

   9. sqrt-unprod 44.5%

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

   10. sqr-neg 44.5%

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

   11. sqrt-unprod 43.1%

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

   12. add-sqr-sqrt 98.2%

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

   13. distribute-neg-in 98.2%

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

   14. add-sqr-sqrt 54.8%

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

   15. sqrt-unprod 71.4%

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

   16. sqr-neg 71.4%

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

   17. sqrt-unprod 26.7%

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

   18. add-sqr-sqrt 60.3%

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

   19. sub-neg 60.3%

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

5. Applied egg-rr 60.3%

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

6. Taylor expanded in t1 around inf 15.4%

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

7. Final simplification 15.4%

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

Reproduce

herbie shell --seed 2023260 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))