Rosa's DopplerBench

Percentage Accurate: 72.2% → 98.2%

Time: 10.7s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= (/ (* t1 (- v)) (* (+ t1 u) (+ t1 u))) -4e-154)
   (/ v (* (+ t1 u) (- -1.0 (/ u t1))))
   (* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (((t1 * -v) / ((t1 + u) * (t1 + u))) <= -4e-154) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (-t1 / (t1 + u)) * (v / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (((t1 * -v) / ((t1 + u) * (t1 + u))) <= (-4d-154)) then
        tmp = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
    else
        tmp = (-t1 / (t1 + u)) * (v / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (((t1 * -v) / ((t1 + u) * (t1 + u))) <= -4e-154) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (-t1 / (t1 + u)) * (v / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if ((t1 * -v) / ((t1 + u) * (t1 + u))) <= -4e-154:
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)))
	else:
		tmp = (-t1 / (t1 + u)) * (v / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (Float64(Float64(t1 * Float64(-v)) / Float64(Float64(t1 + u) * Float64(t1 + u))) <= -4e-154)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 - Float64(u / t1))));
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(t1 + u)) * Float64(v / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (((t1 * -v) / ((t1 + u) * (t1 + u))) <= -4e-154)
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	else
		tmp = (-t1 / (t1 + u)) * (v / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[N[(N[(t1 * (-v)), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-154], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u))) < -3.9999999999999999e-154

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. times-frac 88.7%

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

      3. neg-mul-1 88.7%

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

      4. associate-/l* 88.7%

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

      5. associate-*r/ 88.7%

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

      6. associate-/l* 88.7%

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

      7. associate-/l/ 88.7%

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

      8. neg-mul-1 88.7%

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

      9. *-lft-identity 88.7%

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

      10. metadata-eval 88.7%

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

      11. times-frac 88.7%

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

      12. neg-mul-1 88.7%

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

      13. remove-double-neg 88.7%

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

      14. neg-mul-1 88.7%

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

      15. sub0-neg 88.7%

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

      16. associate--r+ 88.7%

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

      17. neg-sub0 88.7%

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

      18. div-sub 88.7%

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

      19. distribute-frac-neg 88.7%

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

      20. *-inverses 88.7%

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

      21. metadata-eval 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in v around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. neg-mul-1 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.5%

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

      3. remove-double-neg 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. sub-neg 99.7%

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

   10. Simplified 99.7%

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

   if -3.9999999999999999e-154 < (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 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 98.2%

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

   3. Simplified 98.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq -4 \cdot 10^{-154}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}\\ \end{array} \]

Alternative 2: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -110000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -8.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq -3.8 \cdot 10^{-85} \lor \neg \left(t1 \leq 6.2 \cdot 10^{-89}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -110000.0)
     t_1
     (if (<= t1 -8.5e-64)
       (* (/ (- v) u) (/ t1 u))
       (if (or (<= t1 -3.8e-85) (not (<= t1 6.2e-89)))
         t_1
         (* (- v) (/ (/ t1 u) u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -110000.0) {
		tmp = t_1;
	} else if (t1 <= -8.5e-64) {
		tmp = (-v / u) * (t1 / u);
	} else if ((t1 <= -3.8e-85) || !(t1 <= 6.2e-89)) {
		tmp = t_1;
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-110000.0d0)) then
        tmp = t_1
    else if (t1 <= (-8.5d-64)) then
        tmp = (-v / u) * (t1 / u)
    else if ((t1 <= (-3.8d-85)) .or. (.not. (t1 <= 6.2d-89))) then
        tmp = t_1
    else
        tmp = -v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -110000.0) {
		tmp = t_1;
	} else if (t1 <= -8.5e-64) {
		tmp = (-v / u) * (t1 / u);
	} else if ((t1 <= -3.8e-85) || !(t1 <= 6.2e-89)) {
		tmp = t_1;
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -110000.0:
		tmp = t_1
	elif t1 <= -8.5e-64:
		tmp = (-v / u) * (t1 / u)
	elif (t1 <= -3.8e-85) or not (t1 <= 6.2e-89):
		tmp = t_1
	else:
		tmp = -v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -110000.0)
		tmp = t_1;
	elseif (t1 <= -8.5e-64)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	elseif ((t1 <= -3.8e-85) || !(t1 <= 6.2e-89))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -110000.0)
		tmp = t_1;
	elseif (t1 <= -8.5e-64)
		tmp = (-v / u) * (t1 / u);
	elseif ((t1 <= -3.8e-85) || ~((t1 <= 6.2e-89)))
		tmp = t_1;
	else
		tmp = -v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -110000.0], t$95$1, If[LessEqual[t1, -8.5e-64], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -3.8e-85], N[Not[LessEqual[t1, 6.2e-89]], $MachinePrecision]], t$95$1, N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.1e5 or -8.49999999999999996e-64 < t1 < -3.7999999999999999e-85 or 6.19999999999999993e-89 < t1

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 97.0%

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

      1. associate-*r/ 97.0%

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

      2. neg-mul-1 97.0%

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

      3. +-commutative 97.0%

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

   6. Simplified 97.0%

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

      1. frac-2neg 97.0%

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

      2. div-inv 96.8%

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

      3. remove-double-neg 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. associate-*r/ 97.0%

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

      2. *-rgt-identity 97.0%

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

      3. distribute-rgt-neg-in 97.0%

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

      4. +-commutative 97.0%

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

      5. distribute-neg-in 97.0%

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

      6. metadata-eval 97.0%

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

      7. sub-neg 97.0%

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

   10. Simplified 97.0%

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

   11. Taylor expanded in t1 around inf 81.3%

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

       1. +-commutative 81.3%

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

       2. mul-1-neg 81.3%

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

       3. unsub-neg 81.3%

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

       4. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -1.1e5 < t1 < -8.49999999999999996e-64

   1. Initial program 92.3%

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

      1. *-commutative 92.3%

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

      2. times-frac 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-*r/ 99.8%

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

      6. associate-/l* 99.8%

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

      7. associate-/l/ 99.8%

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

      8. neg-mul-1 99.8%

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

      9. *-lft-identity 99.8%

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

      10. metadata-eval 99.8%

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

      11. times-frac 99.8%

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

      12. neg-mul-1 99.8%

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

      13. remove-double-neg 99.8%

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

      14. neg-mul-1 99.8%

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

      15. sub0-neg 99.8%

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

      16. associate--r+ 99.8%

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

      17. neg-sub0 99.8%

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

      18. div-sub 99.8%

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

      19. distribute-frac-neg 99.8%

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

      20. *-inverses 99.8%

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

      21. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in v around 0 92.7%

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

      1. associate-*r/ 92.7%

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

      2. neg-mul-1 92.7%

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

      3. +-commutative 92.7%

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

   6. Simplified 92.7%

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

      1. frac-2neg 92.7%

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

      2. div-inv 92.3%

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

      3. remove-double-neg 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. associate-*r/ 92.7%

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

      2. *-rgt-identity 92.7%

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

      3. distribute-rgt-neg-in 92.7%

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

      4. +-commutative 92.7%

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

      5. distribute-neg-in 92.7%

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

      6. metadata-eval 92.7%

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

      7. sub-neg 92.7%

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

   10. Simplified 92.7%

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

   11. Taylor expanded in t1 around 0 63.7%

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

       1. mul-1-neg 63.7%

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

       2. *-commutative 63.7%

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

       3. unpow2 63.7%

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

       4. times-frac 70.8%

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

   13. Simplified 70.8%

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

   if -3.7999999999999999e-85 < t1 < 6.19999999999999993e-89

   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. *-commutative 79.4%

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

      2. times-frac 91.4%

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

      3. neg-mul-1 91.4%

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

      4. associate-/l* 91.4%

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

      5. associate-*r/ 91.4%

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

      6. associate-/l* 91.4%

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

      7. associate-/l/ 91.4%

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

      8. neg-mul-1 91.4%

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

      9. *-lft-identity 91.4%

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

      10. metadata-eval 91.4%

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

      11. times-frac 91.4%

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

      12. neg-mul-1 91.4%

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

      13. remove-double-neg 91.4%

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

      14. neg-mul-1 91.4%

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

      15. sub0-neg 91.4%

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

      16. associate--r+ 91.4%

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

      17. neg-sub0 91.4%

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

      18. div-sub 91.4%

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

      19. distribute-frac-neg 91.4%

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

      20. *-inverses 91.4%

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

      21. metadata-eval 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in v around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

      3. +-commutative 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in t1 around 0 75.0%

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

      1. mul-1-neg 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*l/ 80.3%

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

      4. *-commutative 80.3%

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

      5. distribute-rgt-neg-in 80.3%

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

      6. associate-/r* 84.8%

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

   9. Simplified 84.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -110000:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -8.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq -3.8 \cdot 10^{-85} \lor \neg \left(t1 \leq 6.2 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 3: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -850000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{elif}\;t1 \leq -6.5 \cdot 10^{-85} \lor \neg \left(t1 \leq 2.8 \cdot 10^{-90}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -850000.0)
     t_1
     (if (<= t1 -2.2e-63)
       (/ (/ (- t1) (/ u v)) u)
       (if (or (<= t1 -6.5e-85) (not (<= t1 2.8e-90)))
         t_1
         (* (- v) (/ (/ t1 u) u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -850000.0) {
		tmp = t_1;
	} else if (t1 <= -2.2e-63) {
		tmp = (-t1 / (u / v)) / u;
	} else if ((t1 <= -6.5e-85) || !(t1 <= 2.8e-90)) {
		tmp = t_1;
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-850000.0d0)) then
        tmp = t_1
    else if (t1 <= (-2.2d-63)) then
        tmp = (-t1 / (u / v)) / u
    else if ((t1 <= (-6.5d-85)) .or. (.not. (t1 <= 2.8d-90))) then
        tmp = t_1
    else
        tmp = -v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -850000.0) {
		tmp = t_1;
	} else if (t1 <= -2.2e-63) {
		tmp = (-t1 / (u / v)) / u;
	} else if ((t1 <= -6.5e-85) || !(t1 <= 2.8e-90)) {
		tmp = t_1;
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -850000.0:
		tmp = t_1
	elif t1 <= -2.2e-63:
		tmp = (-t1 / (u / v)) / u
	elif (t1 <= -6.5e-85) or not (t1 <= 2.8e-90):
		tmp = t_1
	else:
		tmp = -v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -850000.0)
		tmp = t_1;
	elseif (t1 <= -2.2e-63)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / u);
	elseif ((t1 <= -6.5e-85) || !(t1 <= 2.8e-90))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -850000.0)
		tmp = t_1;
	elseif (t1 <= -2.2e-63)
		tmp = (-t1 / (u / v)) / u;
	elseif ((t1 <= -6.5e-85) || ~((t1 <= 2.8e-90)))
		tmp = t_1;
	else
		tmp = -v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -850000.0], t$95$1, If[LessEqual[t1, -2.2e-63], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], If[Or[LessEqual[t1, -6.5e-85], N[Not[LessEqual[t1, 2.8e-90]], $MachinePrecision]], t$95$1, N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.5e5 or -2.2e-63 < t1 < -6.5e-85 or 2.7999999999999999e-90 < t1

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 97.0%

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

      1. associate-*r/ 97.0%

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

      2. neg-mul-1 97.0%

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

      3. +-commutative 97.0%

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

   6. Simplified 97.0%

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

      1. frac-2neg 97.0%

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

      2. div-inv 96.8%

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

      3. remove-double-neg 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. associate-*r/ 97.0%

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

      2. *-rgt-identity 97.0%

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

      3. distribute-rgt-neg-in 97.0%

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

      4. +-commutative 97.0%

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

      5. distribute-neg-in 97.0%

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

      6. metadata-eval 97.0%

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

      7. sub-neg 97.0%

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

   10. Simplified 97.0%

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

   11. Taylor expanded in t1 around inf 81.3%

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

       1. +-commutative 81.3%

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

       2. mul-1-neg 81.3%

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

       3. unsub-neg 81.3%

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

       4. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -8.5e5 < t1 < -2.2e-63

   1. Initial program 92.3%

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

      1. associate-/l* 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in t1 around 0 63.6%

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

      1. unpow2 63.6%

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

   6. Simplified 63.6%

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

      1. div-inv 63.7%

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

      2. add-sqr-sqrt 63.6%

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

      3. sqrt-unprod 63.7%

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

      4. sqr-neg 63.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 24.8%

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

      7. clear-num 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. *-commutative 24.8%

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

      2. associate-/r* 24.7%

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

      3. associate-*l/ 24.7%

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

      4. add-sqr-sqrt 0.8%

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

      5. sqrt-prod 41.3%

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

      6. sqr-neg 41.3%

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

      7. sqrt-unprod 40.3%

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

      8. add-sqr-sqrt 70.8%

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

      9. associate-*l/ 71.0%

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

      10. associate-/r/ 70.8%

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

      11. div-inv 70.8%

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

      12. associate-/r* 70.6%

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

      13. add-sqr-sqrt 40.2%

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

      14. sqrt-unprod 41.2%

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

      15. sqr-neg 41.2%

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

      16. sqrt-prod 0.8%

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

      17. add-sqr-sqrt 24.7%

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

      18. associate-/r* 24.8%

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

      19. div-inv 24.8%

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

      20. pow2 24.8%

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

      21. pow-flip 24.8%

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

      22. metadata-eval 24.8%

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

      23. frac-2neg 24.8%

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

      24. metadata-eval 24.8%

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

      25. add-sqr-sqrt 24.8%

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

      26. sqrt-unprod 24.8%

          \[\leadsto \frac{v \cdot {u}^{-2}}{\frac{-1}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}} \]

      27. sqr-neg 24.8%

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

   10. Applied egg-rr 63.6%

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

       1. sqr-pow 63.5%

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

       2. associate-*r* 70.6%

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

       3. /-rgt-identity 70.6%

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

       4. metadata-eval 70.6%

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

       5. unpow-1 70.6%

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

       6. times-frac 70.8%

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

       7. *-rgt-identity 70.8%

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

       8. associate-/l/ 70.8%

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

       9. metadata-eval 70.8%

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

       10. unpow-1 70.8%

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

       11. times-frac 70.6%

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

       12. *-rgt-identity 70.6%

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

       13. *-commutative 70.6%

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

       14. rem-square-sqrt 31.3%

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

       15. times-frac 31.2%

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

       16. /-rgt-identity 31.2%

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

       17. associate-*r/ 31.3%

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

       18. associate-*l/ 31.3%

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

   12. Simplified 71.1%

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

   13. Taylor expanded in v around 0 71.0%

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

       1. mul-1-neg 71.0%

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

       2. associate-/l* 71.0%

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

   15. Simplified 71.0%

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

   if -6.5e-85 < t1 < 2.7999999999999999e-90

   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. *-commutative 79.4%

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

      2. times-frac 91.4%

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

      3. neg-mul-1 91.4%

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

      4. associate-/l* 91.4%

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

      5. associate-*r/ 91.4%

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

      6. associate-/l* 91.4%

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

      7. associate-/l/ 91.4%

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

      8. neg-mul-1 91.4%

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

      9. *-lft-identity 91.4%

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

      10. metadata-eval 91.4%

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

      11. times-frac 91.4%

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

      12. neg-mul-1 91.4%

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

      13. remove-double-neg 91.4%

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

      14. neg-mul-1 91.4%

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

      15. sub0-neg 91.4%

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

      16. associate--r+ 91.4%

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

      17. neg-sub0 91.4%

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

      18. div-sub 91.4%

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

      19. distribute-frac-neg 91.4%

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

      20. *-inverses 91.4%

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

      21. metadata-eval 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in v around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

      3. +-commutative 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in t1 around 0 75.0%

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

      1. mul-1-neg 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*l/ 80.3%

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

      4. *-commutative 80.3%

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

      5. distribute-rgt-neg-in 80.3%

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

      6. associate-/r* 84.8%

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

   9. Simplified 84.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -850000:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -2.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{elif}\;t1 \leq -6.5 \cdot 10^{-85} \lor \neg \left(t1 \leq 2.8 \cdot 10^{-90}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 4: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -52000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{v}{-\frac{u}{t1}}}{u}\\ \mathbf{elif}\;t1 \leq -8.5 \cdot 10^{-86} \lor \neg \left(t1 \leq 8.5 \cdot 10^{-89}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -52000.0)
     t_1
     (if (<= t1 -3.5e-63)
       (/ (/ v (- (/ u t1))) u)
       (if (or (<= t1 -8.5e-86) (not (<= t1 8.5e-89)))
         t_1
         (* (- v) (/ (/ t1 u) u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -52000.0) {
		tmp = t_1;
	} else if (t1 <= -3.5e-63) {
		tmp = (v / -(u / t1)) / u;
	} else if ((t1 <= -8.5e-86) || !(t1 <= 8.5e-89)) {
		tmp = t_1;
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-52000.0d0)) then
        tmp = t_1
    else if (t1 <= (-3.5d-63)) then
        tmp = (v / -(u / t1)) / u
    else if ((t1 <= (-8.5d-86)) .or. (.not. (t1 <= 8.5d-89))) then
        tmp = t_1
    else
        tmp = -v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -52000.0) {
		tmp = t_1;
	} else if (t1 <= -3.5e-63) {
		tmp = (v / -(u / t1)) / u;
	} else if ((t1 <= -8.5e-86) || !(t1 <= 8.5e-89)) {
		tmp = t_1;
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -52000.0:
		tmp = t_1
	elif t1 <= -3.5e-63:
		tmp = (v / -(u / t1)) / u
	elif (t1 <= -8.5e-86) or not (t1 <= 8.5e-89):
		tmp = t_1
	else:
		tmp = -v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -52000.0)
		tmp = t_1;
	elseif (t1 <= -3.5e-63)
		tmp = Float64(Float64(v / Float64(-Float64(u / t1))) / u);
	elseif ((t1 <= -8.5e-86) || !(t1 <= 8.5e-89))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -52000.0)
		tmp = t_1;
	elseif (t1 <= -3.5e-63)
		tmp = (v / -(u / t1)) / u;
	elseif ((t1 <= -8.5e-86) || ~((t1 <= 8.5e-89)))
		tmp = t_1;
	else
		tmp = -v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -52000.0], t$95$1, If[LessEqual[t1, -3.5e-63], N[(N[(v / (-N[(u / t1), $MachinePrecision])), $MachinePrecision] / u), $MachinePrecision], If[Or[LessEqual[t1, -8.5e-86], N[Not[LessEqual[t1, 8.5e-89]], $MachinePrecision]], t$95$1, N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -52000 or -3.50000000000000003e-63 < t1 < -8.499999999999999e-86 or 8.49999999999999937e-89 < t1

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 97.0%

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

      1. associate-*r/ 97.0%

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

      2. neg-mul-1 97.0%

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

      3. +-commutative 97.0%

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

   6. Simplified 97.0%

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

      1. frac-2neg 97.0%

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

      2. div-inv 96.8%

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

      3. remove-double-neg 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. associate-*r/ 97.0%

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

      2. *-rgt-identity 97.0%

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

      3. distribute-rgt-neg-in 97.0%

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

      4. +-commutative 97.0%

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

      5. distribute-neg-in 97.0%

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

      6. metadata-eval 97.0%

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

      7. sub-neg 97.0%

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

   10. Simplified 97.0%

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

   11. Taylor expanded in t1 around inf 81.3%

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

       1. +-commutative 81.3%

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

       2. mul-1-neg 81.3%

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

       3. unsub-neg 81.3%

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

       4. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -52000 < t1 < -3.50000000000000003e-63

   1. Initial program 92.3%

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

      1. associate-/l* 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in t1 around 0 63.6%

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

      1. unpow2 63.6%

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

   6. Simplified 63.6%

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

      1. div-inv 63.7%

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

      2. add-sqr-sqrt 63.6%

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

      3. sqrt-unprod 63.7%

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

      4. sqr-neg 63.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 24.8%

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

      7. clear-num 24.8%

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

   8. Applied egg-rr 24.8%

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

      1. *-commutative 24.8%

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

      2. associate-/r* 24.7%

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

      3. associate-*l/ 24.7%

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

      4. add-sqr-sqrt 0.8%

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

      5. sqrt-prod 41.3%

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

      6. sqr-neg 41.3%

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

      7. sqrt-unprod 40.3%

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

      8. add-sqr-sqrt 70.8%

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

      9. associate-*l/ 71.0%

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

      10. associate-/r/ 70.8%

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

      11. div-inv 70.8%

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

      12. associate-/r* 70.6%

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

      13. add-sqr-sqrt 40.2%

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

      14. sqrt-unprod 41.2%

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

      15. sqr-neg 41.2%

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

      16. sqrt-prod 0.8%

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

      17. add-sqr-sqrt 24.7%

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

      18. associate-/r* 24.8%

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

      19. div-inv 24.8%

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

      20. pow2 24.8%

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

      21. pow-flip 24.8%

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

      22. metadata-eval 24.8%

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

      23. frac-2neg 24.8%

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

      24. metadata-eval 24.8%

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

      25. add-sqr-sqrt 24.8%

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

      26. sqrt-unprod 24.8%

          \[\leadsto \frac{v \cdot {u}^{-2}}{\frac{-1}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}} \]

      27. sqr-neg 24.8%

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

   10. Applied egg-rr 63.6%

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

       1. sqr-pow 63.5%

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

       2. associate-*r* 70.6%

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

       3. /-rgt-identity 70.6%

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

       4. metadata-eval 70.6%

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

       5. unpow-1 70.6%

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

       6. times-frac 70.8%

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

       7. *-rgt-identity 70.8%

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

       8. associate-/l/ 70.8%

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

       9. metadata-eval 70.8%

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

       10. unpow-1 70.8%

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

       11. times-frac 70.6%

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

       12. *-rgt-identity 70.6%

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

       13. *-commutative 70.6%

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

       14. rem-square-sqrt 31.3%

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

       15. times-frac 31.2%

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

       16. /-rgt-identity 31.2%

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

       17. associate-*r/ 31.3%

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

       18. associate-*l/ 31.3%

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

   12. Simplified 71.1%

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

   if -8.499999999999999e-86 < t1 < 8.49999999999999937e-89

   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. *-commutative 79.4%

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

      2. times-frac 91.4%

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

      3. neg-mul-1 91.4%

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

      4. associate-/l* 91.4%

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

      5. associate-*r/ 91.4%

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

      6. associate-/l* 91.4%

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

      7. associate-/l/ 91.4%

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

      8. neg-mul-1 91.4%

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

      9. *-lft-identity 91.4%

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

      10. metadata-eval 91.4%

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

      11. times-frac 91.4%

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

      12. neg-mul-1 91.4%

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

      13. remove-double-neg 91.4%

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

      14. neg-mul-1 91.4%

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

      15. sub0-neg 91.4%

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

      16. associate--r+ 91.4%

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

      17. neg-sub0 91.4%

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

      18. div-sub 91.4%

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

      19. distribute-frac-neg 91.4%

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

      20. *-inverses 91.4%

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

      21. metadata-eval 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in v around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

      3. +-commutative 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in t1 around 0 75.0%

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

      1. mul-1-neg 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*l/ 80.3%

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

      4. *-commutative 80.3%

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

      5. distribute-rgt-neg-in 80.3%

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

      6. associate-/r* 84.8%

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

   9. Simplified 84.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -52000:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{v}{-\frac{u}{t1}}}{u}\\ \mathbf{elif}\;t1 \leq -8.5 \cdot 10^{-86} \lor \neg \left(t1 \leq 8.5 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 5: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -850000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -1.9 \cdot 10^{-85} \lor \neg \left(t1 \leq 7.8 \cdot 10^{-89}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -850000.0)
     t_1
     (if (<= t1 -2e-62)
       (/ (/ v u) (- -1.0 (/ u t1)))
       (if (or (<= t1 -1.9e-85) (not (<= t1 7.8e-89)))
         t_1
         (* (- v) (/ (/ t1 u) u)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -850000.0) {
		tmp = t_1;
	} else if (t1 <= -2e-62) {
		tmp = (v / u) / (-1.0 - (u / t1));
	} else if ((t1 <= -1.9e-85) || !(t1 <= 7.8e-89)) {
		tmp = t_1;
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-850000.0d0)) then
        tmp = t_1
    else if (t1 <= (-2d-62)) then
        tmp = (v / u) / ((-1.0d0) - (u / t1))
    else if ((t1 <= (-1.9d-85)) .or. (.not. (t1 <= 7.8d-89))) then
        tmp = t_1
    else
        tmp = -v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -850000.0) {
		tmp = t_1;
	} else if (t1 <= -2e-62) {
		tmp = (v / u) / (-1.0 - (u / t1));
	} else if ((t1 <= -1.9e-85) || !(t1 <= 7.8e-89)) {
		tmp = t_1;
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -850000.0:
		tmp = t_1
	elif t1 <= -2e-62:
		tmp = (v / u) / (-1.0 - (u / t1))
	elif (t1 <= -1.9e-85) or not (t1 <= 7.8e-89):
		tmp = t_1
	else:
		tmp = -v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -850000.0)
		tmp = t_1;
	elseif (t1 <= -2e-62)
		tmp = Float64(Float64(v / u) / Float64(-1.0 - Float64(u / t1)));
	elseif ((t1 <= -1.9e-85) || !(t1 <= 7.8e-89))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -850000.0)
		tmp = t_1;
	elseif (t1 <= -2e-62)
		tmp = (v / u) / (-1.0 - (u / t1));
	elseif ((t1 <= -1.9e-85) || ~((t1 <= 7.8e-89)))
		tmp = t_1;
	else
		tmp = -v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -850000.0], t$95$1, If[LessEqual[t1, -2e-62], N[(N[(v / u), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, -1.9e-85], N[Not[LessEqual[t1, 7.8e-89]], $MachinePrecision]], t$95$1, N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.5e5 or -2.0000000000000001e-62 < t1 < -1.8999999999999999e-85 or 7.79999999999999957e-89 < t1

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 97.0%

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

      1. associate-*r/ 97.0%

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

      2. neg-mul-1 97.0%

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

      3. +-commutative 97.0%

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

   6. Simplified 97.0%

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

      1. frac-2neg 97.0%

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

      2. div-inv 96.8%

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

      3. remove-double-neg 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. associate-*r/ 97.0%

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

      2. *-rgt-identity 97.0%

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

      3. distribute-rgt-neg-in 97.0%

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

      4. +-commutative 97.0%

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

      5. distribute-neg-in 97.0%

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

      6. metadata-eval 97.0%

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

      7. sub-neg 97.0%

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

   10. Simplified 97.0%

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

   11. Taylor expanded in t1 around inf 81.3%

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

       1. +-commutative 81.3%

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

       2. mul-1-neg 81.3%

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

       3. unsub-neg 81.3%

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

       4. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -8.5e5 < t1 < -2.0000000000000001e-62

   1. Initial program 92.3%

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

      1. *-commutative 92.3%

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

      2. times-frac 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-*r/ 99.8%

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

      6. associate-/l* 99.8%

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

      7. associate-/l/ 99.8%

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

      8. neg-mul-1 99.8%

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

      9. *-lft-identity 99.8%

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

      10. metadata-eval 99.8%

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

      11. times-frac 99.8%

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

      12. neg-mul-1 99.8%

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

      13. remove-double-neg 99.8%

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

      14. neg-mul-1 99.8%

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

      15. sub0-neg 99.8%

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

      16. associate--r+ 99.8%

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

      17. neg-sub0 99.8%

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

      18. div-sub 99.8%

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

      19. distribute-frac-neg 99.8%

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

      20. *-inverses 99.8%

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

      21. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 71.2%

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

   if -1.8999999999999999e-85 < t1 < 7.79999999999999957e-89

   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. *-commutative 79.4%

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

      2. times-frac 91.4%

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

      3. neg-mul-1 91.4%

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

      4. associate-/l* 91.4%

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

      5. associate-*r/ 91.4%

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

      6. associate-/l* 91.4%

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

      7. associate-/l/ 91.4%

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

      8. neg-mul-1 91.4%

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

      9. *-lft-identity 91.4%

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

      10. metadata-eval 91.4%

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

      11. times-frac 91.4%

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

      12. neg-mul-1 91.4%

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

      13. remove-double-neg 91.4%

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

      14. neg-mul-1 91.4%

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

      15. sub0-neg 91.4%

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

      16. associate--r+ 91.4%

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

      17. neg-sub0 91.4%

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

      18. div-sub 91.4%

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

      19. distribute-frac-neg 91.4%

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

      20. *-inverses 91.4%

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

      21. metadata-eval 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in v around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

      3. +-commutative 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in t1 around 0 75.0%

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

      1. mul-1-neg 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*l/ 80.3%

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

      4. *-commutative 80.3%

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

      5. distribute-rgt-neg-in 80.3%

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

      6. associate-/r* 84.8%

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

   9. Simplified 84.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -850000:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -1.9 \cdot 10^{-85} \lor \neg \left(t1 \leq 7.8 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 6: 95.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 5.8 \cdot 10^{+179}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 5.8e+179)
   (/ v (* (+ t1 u) (- -1.0 (/ u t1))))
   (/ (/ (- t1) (/ u v)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 5.8e+179) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (-t1 / (u / v)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 5.8d+179) then
        tmp = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
    else
        tmp = (-t1 / (u / v)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 5.8e+179) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (-t1 / (u / v)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 5.8e+179:
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)))
	else:
		tmp = (-t1 / (u / v)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 5.8e+179)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 - Float64(u / t1))));
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 5.8e+179)
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	else
		tmp = (-t1 / (u / v)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 5.8e+179], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 5.80000000000000038e179

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. times-frac 96.6%

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

      3. neg-mul-1 96.6%

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

      4. associate-/l* 96.6%

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

      5. associate-*r/ 96.6%

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

      6. associate-/l* 96.6%

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

      7. associate-/l/ 96.6%

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

      8. neg-mul-1 96.6%

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

      9. *-lft-identity 96.6%

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

      10. metadata-eval 96.6%

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

      11. times-frac 96.6%

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

      12. neg-mul-1 96.6%

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

      13. remove-double-neg 96.6%

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

      14. neg-mul-1 96.6%

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

      15. sub0-neg 96.6%

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

      16. associate--r+ 96.6%

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

      17. neg-sub0 96.6%

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

      18. div-sub 96.6%

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

      19. distribute-frac-neg 96.6%

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

      20. *-inverses 96.6%

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

      21. metadata-eval 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in v around 0 98.2%

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

      1. associate-*r/ 98.2%

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

      2. neg-mul-1 98.2%

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

      3. +-commutative 98.2%

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

   6. Simplified 98.2%

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

      1. frac-2neg 98.2%

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

      2. div-inv 98.0%

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

      3. remove-double-neg 98.0%

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

   8. Applied egg-rr 98.0%

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

      1. associate-*r/ 98.2%

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

      2. *-rgt-identity 98.2%

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

      3. distribute-rgt-neg-in 98.2%

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

      4. +-commutative 98.2%

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

      5. distribute-neg-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. sub-neg 98.2%

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

   10. Simplified 98.2%

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

   if 5.80000000000000038e179 < u

   1. Initial program 65.3%

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

      1. associate-/l* 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in t1 around 0 66.1%

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

      1. unpow2 66.1%

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

   6. Simplified 66.1%

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

      1. div-inv 66.1%

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

      2. add-sqr-sqrt 29.3%

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

      3. sqrt-unprod 61.2%

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

      4. sqr-neg 61.2%

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

      5. sqrt-unprod 36.8%

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

      6. add-sqr-sqrt 66.1%

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

      7. clear-num 66.1%

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

   8. Applied egg-rr 66.1%

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

      1. *-commutative 66.1%

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

      2. associate-/r* 65.8%

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

      3. associate-*l/ 65.4%

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

      4. add-sqr-sqrt 65.4%

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

      5. sqrt-prod 66.1%

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

      6. sqr-neg 66.1%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 96.6%

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

      9. associate-*l/ 82.3%

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

      10. associate-/r/ 93.6%

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

      11. div-inv 93.5%

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

      12. associate-/r* 82.3%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 66.1%

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

      15. sqr-neg 66.1%

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

      16. sqrt-prod 65.8%

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

      17. add-sqr-sqrt 65.8%

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

      18. associate-/r* 66.1%

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

      19. div-inv 66.1%

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

      20. pow2 66.1%

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

      21. pow-flip 66.1%

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

      22. metadata-eval 66.1%

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

      23. frac-2neg 66.1%

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

      24. metadata-eval 66.1%

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

      25. add-sqr-sqrt 29.3%

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

      26. sqrt-unprod 61.2%

          \[\leadsto \frac{v \cdot {u}^{-2}}{\frac{-1}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}} \]

      27. sqr-neg 61.2%

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

   10. Applied egg-rr 66.1%

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

       1. sqr-pow 66.1%

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

       2. associate-*r* 82.2%

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

       3. /-rgt-identity 82.2%

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

       4. metadata-eval 82.2%

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

       5. unpow-1 82.2%

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

       6. times-frac 82.2%

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

       7. *-rgt-identity 82.2%

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

       8. associate-/l/ 82.2%

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

       9. metadata-eval 82.2%

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

       10. unpow-1 82.2%

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

       11. times-frac 82.3%

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

       12. *-rgt-identity 82.3%

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

       13. *-commutative 82.3%

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

       14. rem-square-sqrt 55.5%

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

       15. times-frac 55.5%

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

       16. /-rgt-identity 55.5%

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

       17. associate-*r/ 57.4%

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

       18. associate-*l/ 61.9%

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

   12. Simplified 91.4%

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

   13. Taylor expanded in v around 0 77.8%

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

       1. mul-1-neg 77.8%

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

       2. associate-/l* 96.6%

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

   15. Simplified 96.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 5.8 \cdot 10^{+179}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \end{array} \]

Alternative 7: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 - \frac{u}{t1}\\ \mathbf{if}\;u \leq 1.05 \cdot 10^{-34}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- -1.0 (/ u t1))))
   (if (<= u 1.05e-34) (/ v (* (+ t1 u) t_1)) (/ (/ v (+ t1 u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -1.0 - (u / t1);
	double tmp;
	if (u <= 1.05e-34) {
		tmp = v / ((t1 + u) * t_1);
	} else {
		tmp = (v / (t1 + u)) / t_1;
	}
	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 = (-1.0d0) - (u / t1)
    if (u <= 1.05d-34) then
        tmp = v / ((t1 + u) * t_1)
    else
        tmp = (v / (t1 + u)) / t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -1.0 - (u / t1);
	double tmp;
	if (u <= 1.05e-34) {
		tmp = v / ((t1 + u) * t_1);
	} else {
		tmp = (v / (t1 + u)) / t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -1.0 - (u / t1)
	tmp = 0
	if u <= 1.05e-34:
		tmp = v / ((t1 + u) * t_1)
	else:
		tmp = (v / (t1 + u)) / t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(-1.0 - Float64(u / t1))
	tmp = 0.0
	if (u <= 1.05e-34)
		tmp = Float64(v / Float64(Float64(t1 + u) * t_1));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -1.0 - (u / t1);
	tmp = 0.0;
	if (u <= 1.05e-34)
		tmp = v / ((t1 + u) * t_1);
	else
		tmp = (v / (t1 + u)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, 1.05e-34], N[(v / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < 1.05e-34

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. times-frac 96.5%

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

      3. neg-mul-1 96.5%

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

      4. associate-/l* 96.5%

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

      5. associate-*r/ 96.5%

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

      6. associate-/l* 96.5%

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

      7. associate-/l/ 96.5%

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

      8. neg-mul-1 96.5%

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

      9. *-lft-identity 96.5%

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

      10. metadata-eval 96.5%

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

      11. times-frac 96.5%

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

      12. neg-mul-1 96.5%

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

      13. remove-double-neg 96.5%

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

      14. neg-mul-1 96.5%

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

      15. sub0-neg 96.5%

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

      16. associate--r+ 96.5%

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

      17. neg-sub0 96.5%

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

      18. div-sub 96.5%

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

      19. distribute-frac-neg 96.5%

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

      20. *-inverses 96.5%

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

      21. metadata-eval 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in v around 0 98.9%

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

      1. associate-*r/ 98.9%

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

      2. neg-mul-1 98.9%

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

      3. +-commutative 98.9%

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

   6. Simplified 98.9%

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

      1. frac-2neg 98.9%

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

      2. div-inv 98.7%

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

      3. remove-double-neg 98.7%

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

   8. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

      3. distribute-rgt-neg-in 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. metadata-eval 98.9%

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

      7. sub-neg 98.9%

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

   10. Simplified 98.9%

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

   if 1.05e-34 < u

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

      2. times-frac 96.9%

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

      3. neg-mul-1 96.9%

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

      4. associate-/l* 96.8%

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

      5. associate-*r/ 96.9%

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

      6. associate-/l* 96.9%

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

      7. associate-/l/ 96.9%

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

      8. neg-mul-1 96.9%

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

      9. *-lft-identity 96.9%

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

      10. metadata-eval 96.9%

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

      11. times-frac 96.9%

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

      12. neg-mul-1 96.9%

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

      13. remove-double-neg 96.9%

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

      14. neg-mul-1 96.9%

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

      15. sub0-neg 96.9%

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

      16. associate--r+ 96.9%

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

      17. neg-sub0 96.9%

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

      18. div-sub 96.9%

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

      19. distribute-frac-neg 96.9%

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

      20. *-inverses 96.9%

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

      21. metadata-eval 96.9%

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

   3. Simplified 96.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 1.05 \cdot 10^{-34}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \end{array} \]

Alternative 8: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -55000 \lor \neg \left(t1 \leq 2.8 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -55000.0) (not (<= t1 2.8e-89)))
   (/ v (- (* u -2.0) t1))
   (* (/ (- v) u) (/ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -55000.0) || !(t1 <= 2.8e-89)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-v / u) * (t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-55000.0d0)) .or. (.not. (t1 <= 2.8d-89))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (-v / u) * (t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -55000.0) || !(t1 <= 2.8e-89)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-v / u) * (t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -55000.0) or not (t1 <= 2.8e-89):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-v / u) * (t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -55000.0) || !(t1 <= 2.8e-89))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -55000.0) || ~((t1 <= 2.8e-89)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-v / u) * (t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -55000.0], N[Not[LessEqual[t1, 2.8e-89]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -55000 or 2.7999999999999999e-89 < t1

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

      2. times-frac 99.9%

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

      3. neg-mul-1 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-/l* 99.9%

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

      7. associate-/l/ 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-lft-identity 99.9%

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

      10. metadata-eval 99.9%

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

      11. times-frac 99.9%

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

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. neg-mul-1 99.9%

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

      15. sub0-neg 99.9%

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

      16. associate--r+ 99.9%

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

      17. neg-sub0 99.9%

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

      18. div-sub 99.9%

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

      19. distribute-frac-neg 99.9%

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

      20. *-inverses 99.9%

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

      21. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 96.9%

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

      1. associate-*r/ 96.9%

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

      2. neg-mul-1 96.9%

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

      3. +-commutative 96.9%

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

   6. Simplified 96.9%

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

      1. frac-2neg 96.9%

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

      2. div-inv 96.6%

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

      3. remove-double-neg 96.6%

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

   8. Applied egg-rr 96.6%

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

      1. associate-*r/ 96.9%

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

      2. *-rgt-identity 96.9%

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

      3. distribute-rgt-neg-in 96.9%

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

      4. +-commutative 96.9%

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

      5. distribute-neg-in 96.9%

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

      6. metadata-eval 96.9%

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

      7. sub-neg 96.9%

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

   10. Simplified 96.9%

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

   11. Taylor expanded in t1 around inf 80.7%

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

       1. +-commutative 80.7%

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

       2. mul-1-neg 80.7%

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

       3. unsub-neg 80.7%

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

       4. *-commutative 80.7%

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

   13. Simplified 80.7%

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

   if -55000 < t1 < 2.7999999999999999e-89

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

      2. times-frac 92.8%

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

      3. neg-mul-1 92.8%

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

      4. associate-/l* 92.8%

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

      5. associate-*r/ 92.8%

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

      6. associate-/l* 92.8%

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

      7. associate-/l/ 92.8%

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

      8. neg-mul-1 92.8%

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

      9. *-lft-identity 92.8%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{1 \cdot \frac{t1 + u}{-t1}}} \]

      10. metadata-eval 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{t1 + u}{-t1}} \]

      11. times-frac 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1 \cdot \left(t1 + u\right)}{-1 \cdot \left(-t1\right)}}} \]

      12. neg-mul-1 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{-\left(-t1\right)}}} \]

      13. remove-double-neg 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{t1}}} \]

      14. neg-mul-1 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-\left(t1 + u\right)}}{t1}} \]

      15. sub0-neg 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(-t1\right)} - u}{t1}} \]

      18. div-sub 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-t1}{t1} - \frac{u}{t1}}} \]

      19. distribute-frac-neg 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\left(-\frac{t1}{t1}\right)} - \frac{u}{t1}} \]

      20. *-inverses 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\left(-\color{blue}{1}\right) - \frac{u}{t1}} \]

      21. metadata-eval 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{-1} - \frac{u}{t1}} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \]

   4. Taylor expanded in v around 0 95.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 95.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)}} \]

      2. neg-mul-1 95.8%

         \[\leadsto \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)} \]

      3. +-commutative 95.8%

         \[\leadsto \frac{-v}{\left(t1 + u\right) \cdot \color{blue}{\left(\frac{u}{t1} + 1\right)}} \]

   6. Simplified 95.8%

      \[\leadsto \color{blue}{\frac{-v}{\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]
   7. Step-by-step derivation

      1. frac-2neg 95.8%

         \[\leadsto \color{blue}{\frac{-\left(-v\right)}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      2. div-inv 95.6%

         \[\leadsto \color{blue}{\left(-\left(-v\right)\right) \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      3. remove-double-neg 95.6%

         \[\leadsto \color{blue}{v} \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)} \]

   8. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{v \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 95.8%

         \[\leadsto \color{blue}{\frac{v \cdot 1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      2. *-rgt-identity 95.8%

         \[\leadsto \frac{\color{blue}{v}}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)} \]

      3. distribute-rgt-neg-in 95.8%

         \[\leadsto \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(-\left(\frac{u}{t1} + 1\right)\right)}} \]

      4. +-commutative 95.8%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(-\color{blue}{\left(1 + \frac{u}{t1}\right)}\right)} \]

      5. distribute-neg-in 95.8%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{u}{t1}\right)\right)}} \]

      6. metadata-eval 95.8%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(\color{blue}{-1} + \left(-\frac{u}{t1}\right)\right)} \]

      7. sub-neg 95.8%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(-1 - \frac{u}{t1}\right)}} \]

   10. Simplified 95.8%

       \[\leadsto \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}} \]

   11. Taylor expanded in t1 around 0 72.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   12. Step-by-step derivation

       1. mul-1-neg 72.1%

          \[\leadsto \color{blue}{-\frac{t1 \cdot v}{{u}^{2}}} \]

       2. *-commutative 72.1%

          \[\leadsto -\frac{\color{blue}{v \cdot t1}}{{u}^{2}} \]

       3. unpow2 72.1%

          \[\leadsto -\frac{v \cdot t1}{\color{blue}{u \cdot u}} \]

       4. times-frac 76.8%

          \[\leadsto -\color{blue}{\frac{v}{u} \cdot \frac{t1}{u}} \]

   13. Simplified 76.8%

       \[\leadsto \color{blue}{-\frac{v}{u} \cdot \frac{t1}{u}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -55000 \lor \neg \left(t1 \leq 2.8 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]

Alternative 9: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -52000 \lor \neg \left(t1 \leq 6.2 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -52000.0) (not (<= t1 6.2e-89)))
   (/ v (- (* u -2.0) t1))
   (/ (/ (- t1) u) (/ u v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -52000.0) || !(t1 <= 6.2e-89)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-52000.0d0)) .or. (.not. (t1 <= 6.2d-89))) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else
        tmp = (-t1 / u) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -52000.0) || !(t1 <= 6.2e-89)) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -52000.0) or not (t1 <= 6.2e-89):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (-t1 / u) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -52000.0) || !(t1 <= 6.2e-89))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -52000.0) || ~((t1 <= 6.2e-89)))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (-t1 / u) / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -52000.0], N[Not[LessEqual[t1, 6.2e-89]], $MachinePrecision]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -52000 or 6.19999999999999993e-89 < t1

   1. Initial program 72.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. *-commutative 72.0%

         \[\leadsto \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      2. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

      3. neg-mul-1 99.9%

         \[\leadsto \frac{v}{t1 + u} \cdot \frac{\color{blue}{-1 \cdot t1}}{t1 + u} \]

      4. associate-/l* 99.9%

         \[\leadsto \frac{v}{t1 + u} \cdot \color{blue}{\frac{-1}{\frac{t1 + u}{t1}}} \]

      5. associate-*r/ 99.9%

         \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u} \cdot -1}{\frac{t1 + u}{t1}}} \]

      6. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{-1}}} \]

      7. associate-/l/ 99.9%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{t1 + u}{-1 \cdot t1}}} \]

      8. neg-mul-1 99.9%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-t1}}} \]

      9. *-lft-identity 99.9%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{1 \cdot \frac{t1 + u}{-t1}}} \]

      10. metadata-eval 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{t1 + u}{-t1}} \]

      11. times-frac 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1 \cdot \left(t1 + u\right)}{-1 \cdot \left(-t1\right)}}} \]

      12. neg-mul-1 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{-\left(-t1\right)}}} \]

      13. remove-double-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{t1}}} \]

      14. neg-mul-1 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-\left(t1 + u\right)}}{t1}} \]

      15. sub0-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(-t1\right)} - u}{t1}} \]

      18. div-sub 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-t1}{t1} - \frac{u}{t1}}} \]

      19. distribute-frac-neg 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\left(-\frac{t1}{t1}\right)} - \frac{u}{t1}} \]

      20. *-inverses 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\left(-\color{blue}{1}\right) - \frac{u}{t1}} \]

      21. metadata-eval 99.9%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{-1} - \frac{u}{t1}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \]

   4. Taylor expanded in v around 0 96.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 96.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)}} \]

      2. neg-mul-1 96.9%

         \[\leadsto \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)} \]

      3. +-commutative 96.9%

         \[\leadsto \frac{-v}{\left(t1 + u\right) \cdot \color{blue}{\left(\frac{u}{t1} + 1\right)}} \]

   6. Simplified 96.9%

      \[\leadsto \color{blue}{\frac{-v}{\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]
   7. Step-by-step derivation

      1. frac-2neg 96.9%

         \[\leadsto \color{blue}{\frac{-\left(-v\right)}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      2. div-inv 96.6%

         \[\leadsto \color{blue}{\left(-\left(-v\right)\right) \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      3. remove-double-neg 96.6%

         \[\leadsto \color{blue}{v} \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)} \]

   8. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{v \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 96.9%

         \[\leadsto \color{blue}{\frac{v \cdot 1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      2. *-rgt-identity 96.9%

         \[\leadsto \frac{\color{blue}{v}}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)} \]

      3. distribute-rgt-neg-in 96.9%

         \[\leadsto \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(-\left(\frac{u}{t1} + 1\right)\right)}} \]

      4. +-commutative 96.9%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(-\color{blue}{\left(1 + \frac{u}{t1}\right)}\right)} \]

      5. distribute-neg-in 96.9%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{u}{t1}\right)\right)}} \]

      6. metadata-eval 96.9%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(\color{blue}{-1} + \left(-\frac{u}{t1}\right)\right)} \]

      7. sub-neg 96.9%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(-1 - \frac{u}{t1}\right)}} \]

   10. Simplified 96.9%

       \[\leadsto \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}} \]

   11. Taylor expanded in t1 around inf 80.7%

       \[\leadsto \frac{v}{\color{blue}{-1 \cdot t1 + -2 \cdot u}} \]
   12. Step-by-step derivation

       1. +-commutative 80.7%

          \[\leadsto \frac{v}{\color{blue}{-2 \cdot u + -1 \cdot t1}} \]

       2. mul-1-neg 80.7%

          \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

       3. unsub-neg 80.7%

          \[\leadsto \frac{v}{\color{blue}{-2 \cdot u - t1}} \]

       4. *-commutative 80.7%

          \[\leadsto \frac{v}{\color{blue}{u \cdot -2} - t1} \]

   13. Simplified 80.7%

       \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]

   if -52000 < t1 < 6.19999999999999993e-89

   1. Initial program 81.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. *-commutative 81.2%

         \[\leadsto \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      2. times-frac 92.8%

         \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

      3. neg-mul-1 92.8%

         \[\leadsto \frac{v}{t1 + u} \cdot \frac{\color{blue}{-1 \cdot t1}}{t1 + u} \]

      4. associate-/l* 92.8%

         \[\leadsto \frac{v}{t1 + u} \cdot \color{blue}{\frac{-1}{\frac{t1 + u}{t1}}} \]

      5. associate-*r/ 92.8%

         \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u} \cdot -1}{\frac{t1 + u}{t1}}} \]

      6. associate-/l* 92.8%

         \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{-1}}} \]

      7. associate-/l/ 92.8%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{t1 + u}{-1 \cdot t1}}} \]

      8. neg-mul-1 92.8%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-t1}}} \]

      9. *-lft-identity 92.8%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{1 \cdot \frac{t1 + u}{-t1}}} \]

      10. metadata-eval 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{t1 + u}{-t1}} \]

      11. times-frac 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1 \cdot \left(t1 + u\right)}{-1 \cdot \left(-t1\right)}}} \]

      12. neg-mul-1 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{-\left(-t1\right)}}} \]

      13. remove-double-neg 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{t1}}} \]

      14. neg-mul-1 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-\left(t1 + u\right)}}{t1}} \]

      15. sub0-neg 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(-t1\right)} - u}{t1}} \]

      18. div-sub 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-t1}{t1} - \frac{u}{t1}}} \]

      19. distribute-frac-neg 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\left(-\frac{t1}{t1}\right)} - \frac{u}{t1}} \]

      20. *-inverses 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\left(-\color{blue}{1}\right) - \frac{u}{t1}} \]

      21. metadata-eval 92.8%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{-1} - \frac{u}{t1}} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \]

   4. Taylor expanded in v around 0 95.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 95.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)}} \]

      2. neg-mul-1 95.8%

         \[\leadsto \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)} \]

      3. +-commutative 95.8%

         \[\leadsto \frac{-v}{\left(t1 + u\right) \cdot \color{blue}{\left(\frac{u}{t1} + 1\right)}} \]

   6. Simplified 95.8%

      \[\leadsto \color{blue}{\frac{-v}{\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]
   7. Step-by-step derivation

      1. frac-2neg 95.8%

         \[\leadsto \color{blue}{\frac{-\left(-v\right)}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      2. div-inv 95.6%

         \[\leadsto \color{blue}{\left(-\left(-v\right)\right) \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      3. remove-double-neg 95.6%

         \[\leadsto \color{blue}{v} \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)} \]

   8. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{v \cdot \frac{1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 95.8%

         \[\leadsto \color{blue}{\frac{v \cdot 1}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)}} \]

      2. *-rgt-identity 95.8%

         \[\leadsto \frac{\color{blue}{v}}{-\left(t1 + u\right) \cdot \left(\frac{u}{t1} + 1\right)} \]

      3. distribute-rgt-neg-in 95.8%

         \[\leadsto \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(-\left(\frac{u}{t1} + 1\right)\right)}} \]

      4. +-commutative 95.8%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(-\color{blue}{\left(1 + \frac{u}{t1}\right)}\right)} \]

      5. distribute-neg-in 95.8%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(\left(-1\right) + \left(-\frac{u}{t1}\right)\right)}} \]

      6. metadata-eval 95.8%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \left(\color{blue}{-1} + \left(-\frac{u}{t1}\right)\right)} \]

      7. sub-neg 95.8%

         \[\leadsto \frac{v}{\left(t1 + u\right) \cdot \color{blue}{\left(-1 - \frac{u}{t1}\right)}} \]

   10. Simplified 95.8%

       \[\leadsto \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}} \]

   11. Taylor expanded in t1 around 0 72.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   12. Step-by-step derivation

       1. mul-1-neg 72.1%

          \[\leadsto \color{blue}{-\frac{t1 \cdot v}{{u}^{2}}} \]

       2. *-commutative 72.1%

          \[\leadsto -\frac{\color{blue}{v \cdot t1}}{{u}^{2}} \]

       3. unpow2 72.1%

          \[\leadsto -\frac{v \cdot t1}{\color{blue}{u \cdot u}} \]

       4. times-frac 76.8%

          \[\leadsto -\color{blue}{\frac{v}{u} \cdot \frac{t1}{u}} \]

   13. Simplified 76.8%

       \[\leadsto \color{blue}{-\frac{v}{u} \cdot \frac{t1}{u}} \]
   14. Step-by-step derivation

       1. *-commutative 76.8%

          \[\leadsto -\color{blue}{\frac{t1}{u} \cdot \frac{v}{u}} \]

       2. clear-num 76.7%

          \[\leadsto -\frac{t1}{u} \cdot \color{blue}{\frac{1}{\frac{u}{v}}} \]

       3. un-div-inv 77.8%

          \[\leadsto -\color{blue}{\frac{\frac{t1}{u}}{\frac{u}{v}}} \]

   15. Applied egg-rr 77.8%

       \[\leadsto -\color{blue}{\frac{\frac{t1}{u}}{\frac{u}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -52000 \lor \neg \left(t1 \leq 6.2 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \]

Alternative 10: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{-47}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.3e-47)
   (* t1 (/ (/ (- v) u) (+ t1 u)))
   (if (<= u 4.2e-63) (/ (- v) t1) (/ (/ (- t1) (/ u v)) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.3e-47) {
		tmp = t1 * ((-v / u) / (t1 + u));
	} else if (u <= 4.2e-63) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (u / v)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.3d-47)) then
        tmp = t1 * ((-v / u) / (t1 + u))
    else if (u <= 4.2d-63) then
        tmp = -v / t1
    else
        tmp = (-t1 / (u / v)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.3e-47) {
		tmp = t1 * ((-v / u) / (t1 + u));
	} else if (u <= 4.2e-63) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (u / v)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.3e-47:
		tmp = t1 * ((-v / u) / (t1 + u))
	elif u <= 4.2e-63:
		tmp = -v / t1
	else:
		tmp = (-t1 / (u / v)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.3e-47)
		tmp = Float64(t1 * Float64(Float64(Float64(-v) / u) / Float64(t1 + u)));
	elseif (u <= 4.2e-63)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.3e-47)
		tmp = t1 * ((-v / u) / (t1 + u));
	elseif (u <= 4.2e-63)
		tmp = -v / t1;
	else
		tmp = (-t1 / (u / v)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.3e-47], N[(t1 * N[(N[((-v) / u), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.2e-63], N[((-v) / t1), $MachinePrecision], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.29999999999999982e-47

   1. Initial program 89.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 87.0%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      2. neg-mul-1 87.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      3. *-commutative 87.0%

         \[\leadsto \frac{\color{blue}{t1 \cdot -1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]

      4. associate-*r/ 87.0%

         \[\leadsto \color{blue}{t1 \cdot \frac{-1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

      5. associate-/l* 87.5%

         \[\leadsto t1 \cdot \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

      6. neg-mul-1 87.5%

         \[\leadsto t1 \cdot \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      7. associate-/r* 91.6%

         \[\leadsto t1 \cdot \color{blue}{\frac{\frac{-v}{t1 + u}}{t1 + u}} \]

   3. Simplified 91.6%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{-v}{t1 + u}}{t1 + u}} \]

   4. Taylor expanded in t1 around 0 82.4%

      \[\leadsto t1 \cdot \frac{\color{blue}{-1 \cdot \frac{v}{u}}}{t1 + u} \]
   5. Step-by-step derivation

      1. associate-*r/ 82.4%

         \[\leadsto t1 \cdot \frac{\color{blue}{\frac{-1 \cdot v}{u}}}{t1 + u} \]

      2. neg-mul-1 82.4%

         \[\leadsto t1 \cdot \frac{\frac{\color{blue}{-v}}{u}}{t1 + u} \]

   6. Simplified 82.4%

      \[\leadsto t1 \cdot \frac{\color{blue}{\frac{-v}{u}}}{t1 + u} \]

   if -2.29999999999999982e-47 < u < 4.2e-63

   1. Initial program 71.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 95.7%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 80.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   5. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 80.1%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   6. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]

   if 4.2e-63 < u

   1. Initial program 70.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 70.2%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   4. Taylor expanded in t1 around 0 64.1%

      \[\leadsto \frac{-t1}{\color{blue}{\frac{{u}^{2}}{v}}} \]
   5. Step-by-step derivation

      1. unpow2 64.1%

         \[\leadsto \frac{-t1}{\frac{\color{blue}{u \cdot u}}{v}} \]

   6. Simplified 64.1%

      \[\leadsto \frac{-t1}{\color{blue}{\frac{u \cdot u}{v}}} \]
   7. Step-by-step derivation

      1. div-inv 63.4%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{1}{\frac{u \cdot u}{v}}} \]

      2. add-sqr-sqrt 24.9%

         \[\leadsto \color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      3. sqrt-unprod 42.8%

         \[\leadsto \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      4. sqr-neg 42.8%

         \[\leadsto \sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      5. sqrt-unprod 22.7%

         \[\leadsto \color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      6. add-sqr-sqrt 39.4%

         \[\leadsto \color{blue}{t1} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      7. clear-num 38.1%

         \[\leadsto t1 \cdot \color{blue}{\frac{v}{u \cdot u}} \]

   8. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{t1 \cdot \frac{v}{u \cdot u}} \]
   9. Step-by-step derivation

      1. *-commutative 38.1%

         \[\leadsto \color{blue}{\frac{v}{u \cdot u} \cdot t1} \]

      2. associate-/r* 37.9%

         \[\leadsto \color{blue}{\frac{\frac{v}{u}}{u}} \cdot t1 \]

      3. associate-*l/ 37.8%

         \[\leadsto \color{blue}{\frac{\frac{v}{u} \cdot t1}{u}} \]

      4. add-sqr-sqrt 37.8%

         \[\leadsto \frac{\frac{v}{u} \cdot t1}{\color{blue}{\sqrt{u} \cdot \sqrt{u}}} \]

      5. sqrt-prod 38.1%

         \[\leadsto \frac{\frac{v}{u} \cdot t1}{\color{blue}{\sqrt{u \cdot u}}} \]

      6. sqr-neg 38.1%

         \[\leadsto \frac{\frac{v}{u} \cdot t1}{\sqrt{\color{blue}{\left(-u\right) \cdot \left(-u\right)}}} \]

      7. sqrt-unprod 0.0%

         \[\leadsto \frac{\frac{v}{u} \cdot t1}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}}} \]

      8. add-sqr-sqrt 75.0%

         \[\leadsto \frac{\frac{v}{u} \cdot t1}{\color{blue}{-u}} \]

      9. associate-*l/ 68.6%

         \[\leadsto \color{blue}{\frac{\frac{v}{u}}{-u} \cdot t1} \]

      10. associate-/r/ 73.9%

          \[\leadsto \color{blue}{\frac{\frac{v}{u}}{\frac{-u}{t1}}} \]

      11. div-inv 73.8%

          \[\leadsto \frac{\frac{v}{u}}{\color{blue}{\left(-u\right) \cdot \frac{1}{t1}}} \]

      12. associate-/r* 68.5%

          \[\leadsto \color{blue}{\frac{\frac{\frac{v}{u}}{-u}}{\frac{1}{t1}}} \]

      13. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\frac{\frac{v}{u}}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}}}}{\frac{1}{t1}} \]

      14. sqrt-unprod 38.1%

          \[\leadsto \frac{\frac{\frac{v}{u}}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}}}{\frac{1}{t1}} \]

      15. sqr-neg 38.1%

          \[\leadsto \frac{\frac{\frac{v}{u}}{\sqrt{\color{blue}{u \cdot u}}}}{\frac{1}{t1}} \]

      16. sqrt-prod 37.9%

          \[\leadsto \frac{\frac{\frac{v}{u}}{\color{blue}{\sqrt{u} \cdot \sqrt{u}}}}{\frac{1}{t1}} \]

      17. add-sqr-sqrt 37.9%

          \[\leadsto \frac{\frac{\frac{v}{u}}{\color{blue}{u}}}{\frac{1}{t1}} \]

      18. associate-/r* 38.1%

          \[\leadsto \frac{\color{blue}{\frac{v}{u \cdot u}}}{\frac{1}{t1}} \]

      19. div-inv 38.1%

          \[\leadsto \frac{\color{blue}{v \cdot \frac{1}{u \cdot u}}}{\frac{1}{t1}} \]

      20. pow2 38.1%

          \[\leadsto \frac{v \cdot \frac{1}{\color{blue}{{u}^{2}}}}{\frac{1}{t1}} \]

      21. pow-flip 38.1%

          \[\leadsto \frac{v \cdot \color{blue}{{u}^{\left(-2\right)}}}{\frac{1}{t1}} \]

      22. metadata-eval 38.1%

          \[\leadsto \frac{v \cdot {u}^{\color{blue}{-2}}}{\frac{1}{t1}} \]

      23. frac-2neg 38.1%

          \[\leadsto \frac{v \cdot {u}^{-2}}{\color{blue}{\frac{-1}{-t1}}} \]

      24. metadata-eval 38.1%

          \[\leadsto \frac{v \cdot {u}^{-2}}{\frac{\color{blue}{-1}}{-t1}} \]

      25. add-sqr-sqrt 15.4%

          \[\leadsto \frac{v \cdot {u}^{-2}}{\frac{-1}{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}}}} \]

      26. sqrt-unprod 46.9%

          \[\leadsto \frac{v \cdot {u}^{-2}}{\frac{-1}{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}}}} \]

      27. sqr-neg 46.9%

          \[\leadsto \frac{v \cdot {u}^{-2}}{\frac{-1}{\sqrt{\color{blue}{t1 \cdot t1}}}} \]

   10. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{\frac{v \cdot {u}^{-2}}{\frac{-1}{t1}}} \]
   11. Step-by-step derivation

       1. sqr-pow 62.0%

          \[\leadsto \frac{v \cdot \color{blue}{\left({u}^{\left(\frac{-2}{2}\right)} \cdot {u}^{\left(\frac{-2}{2}\right)}\right)}}{\frac{-1}{t1}} \]

       2. associate-*r* 68.4%

          \[\leadsto \frac{\color{blue}{\left(v \cdot {u}^{\left(\frac{-2}{2}\right)}\right) \cdot {u}^{\left(\frac{-2}{2}\right)}}}{\frac{-1}{t1}} \]

       3. /-rgt-identity 68.4%

          \[\leadsto \frac{\left(\color{blue}{\frac{v}{1}} \cdot {u}^{\left(\frac{-2}{2}\right)}\right) \cdot {u}^{\left(\frac{-2}{2}\right)}}{\frac{-1}{t1}} \]

       4. metadata-eval 68.4%

          \[\leadsto \frac{\left(\frac{v}{1} \cdot {u}^{\color{blue}{-1}}\right) \cdot {u}^{\left(\frac{-2}{2}\right)}}{\frac{-1}{t1}} \]

       5. unpow-1 68.4%

          \[\leadsto \frac{\left(\frac{v}{1} \cdot \color{blue}{\frac{1}{u}}\right) \cdot {u}^{\left(\frac{-2}{2}\right)}}{\frac{-1}{t1}} \]

       6. times-frac 68.5%

          \[\leadsto \frac{\color{blue}{\frac{v \cdot 1}{1 \cdot u}} \cdot {u}^{\left(\frac{-2}{2}\right)}}{\frac{-1}{t1}} \]

       7. *-rgt-identity 68.5%

          \[\leadsto \frac{\frac{\color{blue}{v}}{1 \cdot u} \cdot {u}^{\left(\frac{-2}{2}\right)}}{\frac{-1}{t1}} \]

       8. associate-/l/ 68.5%

          \[\leadsto \frac{\color{blue}{\frac{\frac{v}{u}}{1}} \cdot {u}^{\left(\frac{-2}{2}\right)}}{\frac{-1}{t1}} \]

       9. metadata-eval 68.5%

          \[\leadsto \frac{\frac{\frac{v}{u}}{1} \cdot {u}^{\color{blue}{-1}}}{\frac{-1}{t1}} \]

       10. unpow-1 68.5%

           \[\leadsto \frac{\frac{\frac{v}{u}}{1} \cdot \color{blue}{\frac{1}{u}}}{\frac{-1}{t1}} \]

       11. times-frac 68.5%

           \[\leadsto \frac{\color{blue}{\frac{\frac{v}{u} \cdot 1}{1 \cdot u}}}{\frac{-1}{t1}} \]

       12. *-rgt-identity 68.5%

           \[\leadsto \frac{\frac{\color{blue}{\frac{v}{u}}}{1 \cdot u}}{\frac{-1}{t1}} \]

       13. *-commutative 68.5%

           \[\leadsto \frac{\frac{\frac{v}{u}}{\color{blue}{u \cdot 1}}}{\frac{-1}{t1}} \]

       14. rem-square-sqrt 37.6%

           \[\leadsto \frac{\frac{\color{blue}{\sqrt{\frac{v}{u}} \cdot \sqrt{\frac{v}{u}}}}{u \cdot 1}}{\frac{-1}{t1}} \]

       15. times-frac 37.6%

           \[\leadsto \frac{\color{blue}{\frac{\sqrt{\frac{v}{u}}}{u} \cdot \frac{\sqrt{\frac{v}{u}}}{1}}}{\frac{-1}{t1}} \]

       16. /-rgt-identity 37.6%

           \[\leadsto \frac{\frac{\sqrt{\frac{v}{u}}}{u} \cdot \color{blue}{\sqrt{\frac{v}{u}}}}{\frac{-1}{t1}} \]

       17. associate-*r/ 39.7%

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{v}{u}}}{u} \cdot \frac{\sqrt{\frac{v}{u}}}{\frac{-1}{t1}}} \]

       18. associate-*l/ 41.4%

           \[\leadsto \color{blue}{\frac{\sqrt{\frac{v}{u}} \cdot \frac{\sqrt{\frac{v}{u}}}{\frac{-1}{t1}}}{u}} \]

   12. Simplified 73.0%

       \[\leadsto \color{blue}{\frac{\frac{v}{\frac{-u}{t1}}}{u}} \]

   13. Taylor expanded in v around 0 68.0%

       \[\leadsto \frac{\color{blue}{-1 \cdot \frac{t1 \cdot v}{u}}}{u} \]
   14. Step-by-step derivation

       1. mul-1-neg 68.0%

          \[\leadsto \frac{\color{blue}{-\frac{t1 \cdot v}{u}}}{u} \]

       2. associate-/l* 75.0%

          \[\leadsto \frac{-\color{blue}{\frac{t1}{\frac{u}{v}}}}{u} \]

   15. Simplified 75.0%

       \[\leadsto \frac{\color{blue}{-\frac{t1}{\frac{u}{v}}}}{u} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{-47}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{t1 + u}\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \end{array} \]

Alternative 11: 66.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+23} \lor \neg \left(u \leq 2.3 \cdot 10^{+181}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.7e+23) (not (<= u 2.3e+181)))
   (* t1 (/ v (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7e+23) || !(u <= 2.3e+181)) {
		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 <= (-1.7d+23)) .or. (.not. (u <= 2.3d+181))) 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 <= -1.7e+23) || !(u <= 2.3e+181)) {
		tmp = t1 * (v / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.7e+23) or not (u <= 2.3e+181):
		tmp = t1 * (v / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.7e+23) || !(u <= 2.3e+181))
		tmp = Float64(t1 * Float64(v / 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 <= -1.7e+23) || ~((u <= 2.3e+181)))
		tmp = t1 * (v / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.7e+23], N[Not[LessEqual[u, 2.3e+181]], $MachinePrecision]], N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.69999999999999996e23 or 2.2999999999999999e181 < u

   1. Initial program 82.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.3%

         \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]

   4. Taylor expanded in t1 around 0 76.9%

      \[\leadsto \frac{-t1}{\color{blue}{\frac{{u}^{2}}{v}}} \]
   5. Step-by-step derivation

      1. unpow2 76.9%

         \[\leadsto \frac{-t1}{\frac{\color{blue}{u \cdot u}}{v}} \]

   6. Simplified 76.9%

      \[\leadsto \frac{-t1}{\color{blue}{\frac{u \cdot u}{v}}} \]
   7. Step-by-step derivation

      1. div-inv 76.9%

         \[\leadsto \color{blue}{\left(-t1\right) \cdot \frac{1}{\frac{u \cdot u}{v}}} \]

      2. add-sqr-sqrt 38.1%

         \[\leadsto \color{blue}{\left(\sqrt{-t1} \cdot \sqrt{-t1}\right)} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      3. sqrt-unprod 61.9%

         \[\leadsto \color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      4. sqr-neg 61.9%

         \[\leadsto \sqrt{\color{blue}{t1 \cdot t1}} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      5. sqrt-unprod 37.6%

         \[\leadsto \color{blue}{\left(\sqrt{t1} \cdot \sqrt{t1}\right)} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      6. add-sqr-sqrt 72.2%

         \[\leadsto \color{blue}{t1} \cdot \frac{1}{\frac{u \cdot u}{v}} \]

      7. clear-num 72.2%

         \[\leadsto t1 \cdot \color{blue}{\frac{v}{u \cdot u}} \]

   8. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{t1 \cdot \frac{v}{u \cdot u}} \]

   if -1.69999999999999996e23 < u < 2.2999999999999999e181

   1. Initial program 73.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 96.1%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 65.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   5. Step-by-step derivation

      1. associate-*r/ 65.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 65.8%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   6. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+23} \lor \neg \left(u \leq 2.3 \cdot 10^{+181}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+152} \lor \neg \left(u \leq 7.5 \cdot 10^{+181}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.6e+152) (not (<= u 7.5e+181))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.6e+152) || !(u <= 7.5e+181)) {
		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 <= (-2.6d+152)) .or. (.not. (u <= 7.5d+181))) 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 <= -2.6e+152) || !(u <= 7.5e+181)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.6e+152) or not (u <= 7.5e+181):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.6e+152) || !(u <= 7.5e+181))
		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 <= -2.6e+152) || ~((u <= 7.5e+181)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.6e+152], N[Not[LessEqual[u, 7.5e+181]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.6000000000000001e152 or 7.5000000000000005e181 < u

   1. Initial program 76.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. *-commutative 76.8%

         \[\leadsto \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]

      2. times-frac 98.3%

         \[\leadsto \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]

      3. neg-mul-1 98.3%

         \[\leadsto \frac{v}{t1 + u} \cdot \frac{\color{blue}{-1 \cdot t1}}{t1 + u} \]

      4. associate-/l* 98.3%

         \[\leadsto \frac{v}{t1 + u} \cdot \color{blue}{\frac{-1}{\frac{t1 + u}{t1}}} \]

      5. associate-*r/ 98.3%

         \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u} \cdot -1}{\frac{t1 + u}{t1}}} \]

      6. associate-/l* 98.3%

         \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{-1}}} \]

      7. associate-/l/ 98.3%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{t1 + u}{-1 \cdot t1}}} \]

      8. neg-mul-1 98.3%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-t1}}} \]

      9. *-lft-identity 98.3%

         \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{1 \cdot \frac{t1 + u}{-t1}}} \]

      10. metadata-eval 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{t1 + u}{-t1}} \]

      11. times-frac 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1 \cdot \left(t1 + u\right)}{-1 \cdot \left(-t1\right)}}} \]

      12. neg-mul-1 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{-\left(-t1\right)}}} \]

      13. remove-double-neg 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{t1}}} \]

      14. neg-mul-1 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-\left(t1 + u\right)}}{t1}} \]

      15. sub0-neg 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(-t1\right)} - u}{t1}} \]

      18. div-sub 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-t1}{t1} - \frac{u}{t1}}} \]

      19. distribute-frac-neg 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{\left(-\frac{t1}{t1}\right)} - \frac{u}{t1}} \]

      20. *-inverses 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\left(-\color{blue}{1}\right) - \frac{u}{t1}} \]

      21. metadata-eval 98.3%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\color{blue}{-1} - \frac{u}{t1}} \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \]

   4. Taylor expanded in t1 around 0 96.6%

      \[\leadsto \frac{\color{blue}{\frac{v}{u}}}{-1 - \frac{u}{t1}} \]

   5. Taylor expanded in u around 0 47.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{u}} \]
   6. Step-by-step derivation

      1. neg-mul-1 47.8%

         \[\leadsto \color{blue}{-\frac{v}{u}} \]

      2. distribute-neg-frac 47.8%

         \[\leadsto \color{blue}{\frac{-v}{u}} \]

   7. Simplified 47.8%

      \[\leadsto \color{blue}{\frac{-v}{u}} \]

   if -2.6000000000000001e152 < u < 7.5000000000000005e181

   1. Initial program 76.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Step-by-step derivation

      1. times-frac 96.2%

         \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

   4. Taylor expanded in t1 around inf 62.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   5. Step-by-step derivation

      1. associate-*r/ 62.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

      2. neg-mul-1 62.7%

         \[\leadsto \frac{\color{blue}{-v}}{t1} \]

   6. Simplified 62.7%

      \[\leadsto \color{blue}{\frac{-v}{t1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+152} \lor \neg \left(u \leq 7.5 \cdot 10^{+181}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 13: 61.5% 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 76.3%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 96.6%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

4. Taylor expanded in t1 around inf 60.6%

   \[\leadsto \color{blue}{-1} \cdot \frac{v}{t1 + u} \]

5. Final simplification 60.6%

   \[\leadsto \frac{-v}{t1 + u} \]

Alternative 14: 53.8% 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 76.3%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 96.6%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

4. Taylor expanded in t1 around inf 52.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
5. Step-by-step derivation

   1. associate-*r/ 52.9%

      \[\leadsto \color{blue}{\frac{-1 \cdot v}{t1}} \]

   2. neg-mul-1 52.9%

      \[\leadsto \frac{\color{blue}{-v}}{t1} \]

6. Simplified 52.9%

   \[\leadsto \color{blue}{\frac{-v}{t1}} \]

7. Final simplification 52.9%

   \[\leadsto \frac{-v}{t1} \]

Alternative 15: 13.9% 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 76.3%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Step-by-step derivation

   1. times-frac 96.6%

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
4. Step-by-step derivation

   1. clear-num 96.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{t1 + u}{-t1}}} \cdot \frac{v}{t1 + u} \]

   2. frac-times 96.3%

      \[\leadsto \color{blue}{\frac{1 \cdot v}{\frac{t1 + u}{-t1} \cdot \left(t1 + u\right)}} \]

   3. *-un-lft-identity 96.3%

      \[\leadsto \frac{\color{blue}{v}}{\frac{t1 + u}{-t1} \cdot \left(t1 + u\right)} \]

   4. frac-2neg 96.3%

      \[\leadsto \frac{v}{\color{blue}{\frac{-\left(t1 + u\right)}{-\left(-t1\right)}} \cdot \left(t1 + u\right)} \]

   5. distribute-neg-in 96.3%

      \[\leadsto \frac{v}{\frac{\color{blue}{\left(-t1\right) + \left(-u\right)}}{-\left(-t1\right)} \cdot \left(t1 + u\right)} \]

   6. add-sqr-sqrt 44.8%

      \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{-t1} \cdot \sqrt{-t1}} + \left(-u\right)}{-\left(-t1\right)} \cdot \left(t1 + u\right)} \]

   7. sqrt-unprod 71.2%

      \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{\left(-t1\right) \cdot \left(-t1\right)}} + \left(-u\right)}{-\left(-t1\right)} \cdot \left(t1 + u\right)} \]

   8. sqr-neg 71.2%

      \[\leadsto \frac{v}{\frac{\sqrt{\color{blue}{t1 \cdot t1}} + \left(-u\right)}{-\left(-t1\right)} \cdot \left(t1 + u\right)} \]

   9. sqrt-unprod 31.1%

      \[\leadsto \frac{v}{\frac{\color{blue}{\sqrt{t1} \cdot \sqrt{t1}} + \left(-u\right)}{-\left(-t1\right)} \cdot \left(t1 + u\right)} \]

   10. add-sqr-sqrt 60.4%

       \[\leadsto \frac{v}{\frac{\color{blue}{t1} + \left(-u\right)}{-\left(-t1\right)} \cdot \left(t1 + u\right)} \]

   11. sub-neg 60.4%

       \[\leadsto \frac{v}{\frac{\color{blue}{t1 - u}}{-\left(-t1\right)} \cdot \left(t1 + u\right)} \]

   12. remove-double-neg 60.4%

       \[\leadsto \frac{v}{\frac{t1 - u}{\color{blue}{t1}} \cdot \left(t1 + u\right)} \]

5. Applied egg-rr 60.4%

   \[\leadsto \color{blue}{\frac{v}{\frac{t1 - u}{t1} \cdot \left(t1 + u\right)}} \]

6. Taylor expanded in t1 around inf 16.4%

   \[\leadsto \color{blue}{\frac{v}{t1}} \]

7. Final simplification 16.4%

   \[\leadsto \frac{v}{t1} \]

Reproduce

herbie shell --seed 2023208 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))