Rosa's DopplerBench

Percentage Accurate: 72.5% → 98.1%

Time: 13.7s

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: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

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(t1 * Float64(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 * N[(v / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

   1. distribute-lft-neg-out 73.3%

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

   2. *-commutative 73.3%

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

   3. distribute-lft-neg-out 73.3%

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

   4. associate-*l/ 71.8%

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

   5. *-commutative 71.8%

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

3. Simplified 71.8%

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

   1. associate-/r* 80.5%

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

   2. associate-*r/ 97.1%

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

   3. remove-double-neg 97.1%

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

   4. frac-2neg 97.1%

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

   5. +-commutative 97.1%

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

   6. distribute-neg-in 97.1%

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

   7. unsub-neg 97.1%

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

5. Applied egg-rr 97.1%

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

6. Final simplification 97.1%

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

Alternative 2: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{-8} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{-1}{t1 + u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -2e+37)
     t_1
     (if (<= t1 1.1e-84)
       (/ (/ (- t1) u) (/ u v))
       (if (or (<= t1 1.2e-8) (not (<= t1 4.2e+86)))
         t_1
         (* t1 (/ (/ -1.0 (+ t1 u)) (/ u v))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2e+37) {
		tmp = t_1;
	} else if (t1 <= 1.1e-84) {
		tmp = (-t1 / u) / (u / v);
	} else if ((t1 <= 1.2e-8) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = t1 * ((-1.0 / (t1 + u)) / (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-2d+37)) then
        tmp = t_1
    else if (t1 <= 1.1d-84) then
        tmp = (-t1 / u) / (u / v)
    else if ((t1 <= 1.2d-8) .or. (.not. (t1 <= 4.2d+86))) then
        tmp = t_1
    else
        tmp = t1 * (((-1.0d0) / (t1 + u)) / (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -2e+37) {
		tmp = t_1;
	} else if (t1 <= 1.1e-84) {
		tmp = (-t1 / u) / (u / v);
	} else if ((t1 <= 1.2e-8) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = t1 * ((-1.0 / (t1 + u)) / (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -2e+37:
		tmp = t_1
	elif t1 <= 1.1e-84:
		tmp = (-t1 / u) / (u / v)
	elif (t1 <= 1.2e-8) or not (t1 <= 4.2e+86):
		tmp = t_1
	else:
		tmp = t1 * ((-1.0 / (t1 + u)) / (u / v))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -2e+37)
		tmp = t_1;
	elseif (t1 <= 1.1e-84)
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	elseif ((t1 <= 1.2e-8) || !(t1 <= 4.2e+86))
		tmp = t_1;
	else
		tmp = Float64(t1 * Float64(Float64(-1.0 / Float64(t1 + u)) / Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -2e+37)
		tmp = t_1;
	elseif (t1 <= 1.1e-84)
		tmp = (-t1 / u) / (u / v);
	elseif ((t1 <= 1.2e-8) || ~((t1 <= 4.2e+86)))
		tmp = t_1;
	else
		tmp = t1 * ((-1.0 / (t1 + u)) / (u / v));
	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, -2e+37], t$95$1, If[LessEqual[t1, 1.1e-84], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 1.2e-8], N[Not[LessEqual[t1, 4.2e+86]], $MachinePrecision]], t$95$1, N[(t1 * N[(N[(-1.0 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.99999999999999991e37 or 1.0999999999999999e-84 < t1 < 1.19999999999999999e-8 or 4.1999999999999998e86 < t1

   1. Initial program 67.3%

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

      1. distribute-lft-neg-out 67.3%

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

      2. *-commutative 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-*l/ 66.6%

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

      5. *-commutative 66.6%

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

   3. Simplified 66.6%

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

      1. associate-*r/ 67.3%

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

      2. distribute-rgt-neg-in 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-/r* 81.2%

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

      5. *-commutative 81.2%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in t1 around inf 92.3%

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

      1. mul-1-neg 92.3%

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

      2. unsub-neg 92.3%

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

      3. *-commutative 92.3%

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

   8. Simplified 92.3%

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

   if -1.99999999999999991e37 < t1 < 1.0999999999999999e-84

   1. Initial program 79.8%

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

      1. distribute-lft-neg-out 79.8%

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

      2. *-commutative 79.8%

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

      3. distribute-lft-neg-out 79.8%

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

      4. associate-*l/ 76.1%

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

      5. *-commutative 76.1%

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

   3. Simplified 76.1%

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

      1. clear-num 76.1%

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

      2. un-div-inv 78.6%

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

      3. neg-mul-1 78.6%

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

      4. times-frac 83.8%

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

      5. associate-/r* 93.7%

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

      6. div-inv 93.7%

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

      7. metadata-eval 93.7%

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

      8. *-commutative 93.7%

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

      9. neg-mul-1 93.7%

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

      10. +-commutative 93.7%

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

      11. distribute-neg-in 93.7%

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

      12. unsub-neg 93.7%

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

   5. Applied egg-rr 93.7%

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

   6. Taylor expanded in t1 around 0 83.5%

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

   7. Taylor expanded in t1 around 0 83.5%

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

      1. associate-*r/ 83.5%

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

      2. neg-mul-1 83.5%

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

   9. Simplified 83.5%

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

   if 1.19999999999999999e-8 < t1 < 4.1999999999999998e86

   1. Initial program 77.9%

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

      1. distribute-lft-neg-out 77.9%

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

      2. *-commutative 77.9%

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

      3. distribute-lft-neg-out 77.9%

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

      4. associate-*l/ 83.6%

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

      5. *-commutative 83.6%

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

   3. Simplified 83.6%

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

      1. clear-num 83.5%

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

      2. un-div-inv 83.5%

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

      3. neg-mul-1 83.5%

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

      4. times-frac 99.3%

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

      5. associate-/r* 99.5%

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

      6. div-inv 99.5%

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

      7. metadata-eval 99.5%

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

      8. *-commutative 99.5%

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

      9. neg-mul-1 99.5%

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

      10. +-commutative 99.5%

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

      11. distribute-neg-in 99.5%

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

      12. unsub-neg 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in t1 around 0 72.5%

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

      1. frac-2neg 72.5%

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

      2. div-inv 72.5%

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

      3. distribute-lft-neg-in 72.5%

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

      4. neg-mul-1 72.5%

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

      5. times-frac 72.6%

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

      6. frac-2neg 72.6%

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

      7. remove-double-neg 72.6%

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

      8. metadata-eval 72.6%

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

      9. /-rgt-identity 72.6%

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

      10. sub-neg 72.6%

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

      11. distribute-neg-in 72.6%

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

      12. +-commutative 72.6%

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

      13. metadata-eval 72.6%

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

      14. frac-2neg 72.6%

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

      15. metadata-eval 72.6%

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

      16. metadata-eval 72.6%

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

      17. remove-double-neg 72.6%

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

   8. Applied egg-rr 72.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{-8} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{-1}{t1 + u}}{\frac{u}{v}}\\ \end{array} \]

Alternative 3: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.52 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\left(-t1\right) - u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.52e+37)
     t_1
     (if (<= t1 5.8e-88)
       (/ (/ (- t1) u) (/ u v))
       (if (or (<= t1 1.25e-9) (not (<= t1 4.2e+86)))
         t_1
         (/ (/ t1 (- (- t1) u)) (/ u v)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.52e+37) {
		tmp = t_1;
	} else if (t1 <= 5.8e-88) {
		tmp = (-t1 / u) / (u / v);
	} else if ((t1 <= 1.25e-9) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = (t1 / (-t1 - u)) / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-1.52d+37)) then
        tmp = t_1
    else if (t1 <= 5.8d-88) then
        tmp = (-t1 / u) / (u / v)
    else if ((t1 <= 1.25d-9) .or. (.not. (t1 <= 4.2d+86))) then
        tmp = t_1
    else
        tmp = (t1 / (-t1 - u)) / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.52e+37) {
		tmp = t_1;
	} else if (t1 <= 5.8e-88) {
		tmp = (-t1 / u) / (u / v);
	} else if ((t1 <= 1.25e-9) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = (t1 / (-t1 - u)) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.52e+37:
		tmp = t_1
	elif t1 <= 5.8e-88:
		tmp = (-t1 / u) / (u / v)
	elif (t1 <= 1.25e-9) or not (t1 <= 4.2e+86):
		tmp = t_1
	else:
		tmp = (t1 / (-t1 - u)) / (u / v)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.52e+37)
		tmp = t_1;
	elseif (t1 <= 5.8e-88)
		tmp = Float64(Float64(Float64(-t1) / u) / Float64(u / v));
	elseif ((t1 <= 1.25e-9) || !(t1 <= 4.2e+86))
		tmp = t_1;
	else
		tmp = Float64(Float64(t1 / Float64(Float64(-t1) - u)) / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.52e+37)
		tmp = t_1;
	elseif (t1 <= 5.8e-88)
		tmp = (-t1 / u) / (u / v);
	elseif ((t1 <= 1.25e-9) || ~((t1 <= 4.2e+86)))
		tmp = t_1;
	else
		tmp = (t1 / (-t1 - u)) / (u / v);
	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, -1.52e+37], t$95$1, If[LessEqual[t1, 5.8e-88], N[(N[((-t1) / u), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 1.25e-9], N[Not[LessEqual[t1, 4.2e+86]], $MachinePrecision]], t$95$1, N[(N[(t1 / N[((-t1) - u), $MachinePrecision]), $MachinePrecision] / N[(u / v), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.5200000000000001e37 or 5.8000000000000003e-88 < t1 < 1.25e-9 or 4.1999999999999998e86 < t1

   1. Initial program 67.3%

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

      1. distribute-lft-neg-out 67.3%

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

      2. *-commutative 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-*l/ 66.6%

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

      5. *-commutative 66.6%

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

   3. Simplified 66.6%

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

      1. associate-*r/ 67.3%

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

      2. distribute-rgt-neg-in 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-/r* 81.2%

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

      5. *-commutative 81.2%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in t1 around inf 92.3%

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

      1. mul-1-neg 92.3%

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

      2. unsub-neg 92.3%

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

      3. *-commutative 92.3%

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

   8. Simplified 92.3%

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

   if -1.5200000000000001e37 < t1 < 5.8000000000000003e-88

   1. Initial program 79.8%

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

      1. distribute-lft-neg-out 79.8%

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

      2. *-commutative 79.8%

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

      3. distribute-lft-neg-out 79.8%

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

      4. associate-*l/ 76.1%

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

      5. *-commutative 76.1%

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

   3. Simplified 76.1%

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

      1. clear-num 76.1%

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

      2. un-div-inv 78.6%

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

      3. neg-mul-1 78.6%

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

      4. times-frac 83.8%

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

      5. associate-/r* 93.7%

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

      6. div-inv 93.7%

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

      7. metadata-eval 93.7%

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

      8. *-commutative 93.7%

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

      9. neg-mul-1 93.7%

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

      10. +-commutative 93.7%

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

      11. distribute-neg-in 93.7%

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

      12. unsub-neg 93.7%

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

   5. Applied egg-rr 93.7%

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

   6. Taylor expanded in t1 around 0 83.5%

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

   7. Taylor expanded in t1 around 0 83.5%

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

      1. associate-*r/ 83.5%

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

      2. neg-mul-1 83.5%

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

   9. Simplified 83.5%

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

   if 1.25e-9 < t1 < 4.1999999999999998e86

   1. Initial program 77.9%

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

      1. distribute-lft-neg-out 77.9%

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

      2. *-commutative 77.9%

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

      3. distribute-lft-neg-out 77.9%

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

      4. associate-*l/ 83.6%

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

      5. *-commutative 83.6%

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

   3. Simplified 83.6%

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

      1. clear-num 83.5%

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

      2. un-div-inv 83.5%

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

      3. neg-mul-1 83.5%

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

      4. times-frac 99.3%

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

      5. associate-/r* 99.5%

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

      6. div-inv 99.5%

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

      7. metadata-eval 99.5%

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

      8. *-commutative 99.5%

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

      9. neg-mul-1 99.5%

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

      10. +-commutative 99.5%

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

      11. distribute-neg-in 99.5%

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

      12. unsub-neg 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in t1 around 0 72.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.52 \cdot 10^{+37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\left(-t1\right) - u}}{\frac{u}{v}}\\ \end{array} \]

Alternative 4: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 2.05 \cdot 10^{-162}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+75}:\\ \;\;\;\;t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.22e+37)
     t_1
     (if (<= t1 2.05e-162)
       (* v (/ (/ t1 u) (- (- t1) u)))
       (if (<= t1 3.2e+75) (* t1 (/ (- v) (* (+ t1 u) (+ t1 u)))) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.22e+37) {
		tmp = t_1;
	} else if (t1 <= 2.05e-162) {
		tmp = v * ((t1 / u) / (-t1 - u));
	} else if (t1 <= 3.2e+75) {
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	} else {
		tmp = 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 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-1.22d+37)) then
        tmp = t_1
    else if (t1 <= 2.05d-162) then
        tmp = v * ((t1 / u) / (-t1 - u))
    else if (t1 <= 3.2d+75) then
        tmp = t1 * (-v / ((t1 + u) * (t1 + u)))
    else
        tmp = t_1
    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 <= -1.22e+37) {
		tmp = t_1;
	} else if (t1 <= 2.05e-162) {
		tmp = v * ((t1 / u) / (-t1 - u));
	} else if (t1 <= 3.2e+75) {
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.22e+37:
		tmp = t_1
	elif t1 <= 2.05e-162:
		tmp = v * ((t1 / u) / (-t1 - u))
	elif t1 <= 3.2e+75:
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.22e+37)
		tmp = t_1;
	elseif (t1 <= 2.05e-162)
		tmp = Float64(v * Float64(Float64(t1 / u) / Float64(Float64(-t1) - u)));
	elseif (t1 <= 3.2e+75)
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(Float64(t1 + u) * Float64(t1 + u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.22e+37)
		tmp = t_1;
	elseif (t1 <= 2.05e-162)
		tmp = v * ((t1 / u) / (-t1 - u));
	elseif (t1 <= 3.2e+75)
		tmp = t1 * (-v / ((t1 + u) * (t1 + u)));
	else
		tmp = t_1;
	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, -1.22e+37], t$95$1, If[LessEqual[t1, 2.05e-162], N[(v * N[(N[(t1 / u), $MachinePrecision] / N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.2e+75], N[(t1 * N[((-v) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.22e37 or 3.19999999999999985e75 < t1

   1. Initial program 62.6%

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

      1. distribute-lft-neg-out 62.6%

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

      2. *-commutative 62.6%

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

      3. distribute-lft-neg-out 62.6%

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

      4. associate-*l/ 61.8%

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

      5. *-commutative 61.8%

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

   3. Simplified 61.8%

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

      1. associate-*r/ 62.6%

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

      2. distribute-rgt-neg-in 62.6%

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

      3. distribute-lft-neg-out 62.6%

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

      4. associate-/r* 78.2%

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

      5. *-commutative 78.2%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in t1 around inf 92.8%

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

      1. mul-1-neg 92.8%

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

      2. unsub-neg 92.8%

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

      3. *-commutative 92.8%

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

   8. Simplified 92.8%

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

   if -1.22e37 < t1 < 2.0500000000000001e-162

   1. Initial program 78.8%

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

      1. sqr-neg 78.8%

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

      2. times-frac 91.9%

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

      3. frac-2neg 91.9%

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

      4. associate-*r/ 92.4%

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

      5. associate-/l* 92.9%

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

      6. associate-/r/ 93.8%

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

      7. +-commutative 93.8%

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

      8. distribute-neg-in 93.8%

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

      9. unsub-neg 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in t1 around 0 84.0%

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

   if 2.0500000000000001e-162 < t1 < 3.19999999999999985e75

   1. Initial program 89.0%

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

      1. distribute-lft-neg-out 89.0%

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

      2. *-commutative 89.0%

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

      3. distribute-lft-neg-out 89.0%

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

      4. associate-*l/ 91.5%

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

      5. *-commutative 91.5%

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

   3. Simplified 91.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 2.05 \cdot 10^{-162}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{\left(-t1\right) - u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{+75}:\\ \;\;\;\;t1 \cdot \frac{-v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 5: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37} \lor \neg \left(t1 \leq 10^{-84}\right) \land \left(t1 \leq 1.25 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right)\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.22e+37)
         (and (not (<= t1 1e-84)) (or (<= t1 1.25e-9) (not (<= t1 4.2e+86)))))
   (/ v (- (* u -2.0) t1))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.22e+37) || (!(t1 <= 1e-84) && ((t1 <= 1.25e-9) || !(t1 <= 4.2e+86)))) {
		tmp = v / ((u * -2.0) - 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 ((t1 <= (-1.22d+37)) .or. (.not. (t1 <= 1d-84)) .and. (t1 <= 1.25d-9) .or. (.not. (t1 <= 4.2d+86))) then
        tmp = v / ((u * (-2.0d0)) - 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 ((t1 <= -1.22e+37) || (!(t1 <= 1e-84) && ((t1 <= 1.25e-9) || !(t1 <= 4.2e+86)))) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.22e+37) or (not (t1 <= 1e-84) and ((t1 <= 1.25e-9) or not (t1 <= 4.2e+86))):
		tmp = v / ((u * -2.0) - t1)
	else:
		tmp = (t1 / u) * (-v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.22e+37) || (!(t1 <= 1e-84) && ((t1 <= 1.25e-9) || !(t1 <= 4.2e+86))))
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.22e+37) || (~((t1 <= 1e-84)) && ((t1 <= 1.25e-9) || ~((t1 <= 4.2e+86)))))
		tmp = v / ((u * -2.0) - t1);
	else
		tmp = (t1 / u) * (-v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.22e+37], And[N[Not[LessEqual[t1, 1e-84]], $MachinePrecision], Or[LessEqual[t1, 1.25e-9], N[Not[LessEqual[t1, 4.2e+86]], $MachinePrecision]]]], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.22e37 or 1e-84 < t1 < 1.25e-9 or 4.1999999999999998e86 < t1

   1. Initial program 67.3%

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

      1. distribute-lft-neg-out 67.3%

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

      2. *-commutative 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-*l/ 66.6%

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

      5. *-commutative 66.6%

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

   3. Simplified 66.6%

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

      1. associate-*r/ 67.3%

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

      2. distribute-rgt-neg-in 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-/r* 81.2%

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

      5. *-commutative 81.2%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in t1 around inf 92.3%

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

      1. mul-1-neg 92.3%

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

      2. unsub-neg 92.3%

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

      3. *-commutative 92.3%

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

   8. Simplified 92.3%

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

   if -1.22e37 < t1 < 1e-84 or 1.25e-9 < t1 < 4.1999999999999998e86

   1. Initial program 79.6%

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

      1. distribute-lft-neg-out 79.6%

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

      2. *-commutative 79.6%

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

      3. distribute-lft-neg-out 79.6%

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

      4. associate-*l/ 77.1%

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

      5. *-commutative 77.1%

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

   3. Simplified 77.1%

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

      1. associate-*r/ 79.6%

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

      2. distribute-rgt-neg-in 79.6%

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

      3. distribute-lft-neg-out 79.6%

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

      4. associate-/r* 85.8%

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

      5. *-commutative 85.8%

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

      6. associate-/l* 94.3%

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

      7. associate-/l/ 91.6%

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

   5. Applied egg-rr 91.6%

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

   6. Taylor expanded in t1 around 0 72.2%

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

      1. *-commutative 72.2%

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

      2. unpow2 72.2%

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

      3. associate-*r/ 72.2%

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

      4. neg-mul-1 72.2%

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

      5. distribute-rgt-neg-in 72.2%

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

      6. times-frac 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37} \lor \neg \left(t1 \leq 10^{-84}\right) \land \left(t1 \leq 1.25 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right)\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 6: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+39} \lor \neg \left(t1 \leq 10^{-85}\right) \land \left(t1 \leq 4.5 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right)\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.5e+39)
         (and (not (<= t1 1e-85)) (or (<= t1 4.5e-9) (not (<= t1 4.2e+86)))))
   (/ v (- (* u -2.0) t1))
   (/ (/ (- t1) u) (/ u v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.5e+39) || (!(t1 <= 1e-85) && ((t1 <= 4.5e-9) || !(t1 <= 4.2e+86)))) {
		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 <= (-1.5d+39)) .or. (.not. (t1 <= 1d-85)) .and. (t1 <= 4.5d-9) .or. (.not. (t1 <= 4.2d+86))) 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 <= -1.5e+39) || (!(t1 <= 1e-85) && ((t1 <= 4.5e-9) || !(t1 <= 4.2e+86)))) {
		tmp = v / ((u * -2.0) - t1);
	} else {
		tmp = (-t1 / u) / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.5e+39) or (not (t1 <= 1e-85) and ((t1 <= 4.5e-9) or not (t1 <= 4.2e+86))):
		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 <= -1.5e+39) || (!(t1 <= 1e-85) && ((t1 <= 4.5e-9) || !(t1 <= 4.2e+86))))
		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 <= -1.5e+39) || (~((t1 <= 1e-85)) && ((t1 <= 4.5e-9) || ~((t1 <= 4.2e+86)))))
		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, -1.5e+39], And[N[Not[LessEqual[t1, 1e-85]], $MachinePrecision], Or[LessEqual[t1, 4.5e-9], N[Not[LessEqual[t1, 4.2e+86]], $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 < -1.5e39 or 9.9999999999999998e-86 < t1 < 4.49999999999999976e-9 or 4.1999999999999998e86 < t1

   1. Initial program 67.3%

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

      1. distribute-lft-neg-out 67.3%

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

      2. *-commutative 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-*l/ 66.6%

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

      5. *-commutative 66.6%

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

   3. Simplified 66.6%

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

      1. associate-*r/ 67.3%

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

      2. distribute-rgt-neg-in 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-/r* 81.2%

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

      5. *-commutative 81.2%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in t1 around inf 92.3%

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

      1. mul-1-neg 92.3%

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

      2. unsub-neg 92.3%

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

      3. *-commutative 92.3%

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

   8. Simplified 92.3%

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

   if -1.5e39 < t1 < 9.9999999999999998e-86 or 4.49999999999999976e-9 < t1 < 4.1999999999999998e86

   1. Initial program 79.6%

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

      1. distribute-lft-neg-out 79.6%

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

      2. *-commutative 79.6%

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

      3. distribute-lft-neg-out 79.6%

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

      4. associate-*l/ 77.1%

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

      5. *-commutative 77.1%

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

   3. Simplified 77.1%

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

      1. clear-num 77.1%

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

      2. un-div-inv 79.3%

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

      3. neg-mul-1 79.3%

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

      4. times-frac 86.0%

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

      5. associate-/r* 94.6%

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

      6. div-inv 94.6%

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

      7. metadata-eval 94.6%

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

      8. *-commutative 94.6%

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

      9. neg-mul-1 94.6%

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

      10. +-commutative 94.6%

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

      11. distribute-neg-in 94.6%

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

      12. unsub-neg 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in t1 around 0 81.9%

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

   7. Taylor expanded in t1 around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. neg-mul-1 81.9%

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

   9. Simplified 81.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{+39} \lor \neg \left(t1 \leq 10^{-85}\right) \land \left(t1 \leq 4.5 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right)\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{u}{v}}\\ \end{array} \]

Alternative 7: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 10^{-85}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-10} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.22e+37)
     t_1
     (if (<= t1 1e-85)
       (* (/ t1 u) (/ (- v) u))
       (if (or (<= t1 4.6e-10) (not (<= t1 4.2e+86)))
         t_1
         (/ (- t1) (* u (/ u v))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.22e+37) {
		tmp = t_1;
	} else if (t1 <= 1e-85) {
		tmp = (t1 / u) * (-v / u);
	} else if ((t1 <= 4.6e-10) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = -t1 / (u * (u / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-1.22d+37)) then
        tmp = t_1
    else if (t1 <= 1d-85) then
        tmp = (t1 / u) * (-v / u)
    else if ((t1 <= 4.6d-10) .or. (.not. (t1 <= 4.2d+86))) then
        tmp = t_1
    else
        tmp = -t1 / (u * (u / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.22e+37) {
		tmp = t_1;
	} else if (t1 <= 1e-85) {
		tmp = (t1 / u) * (-v / u);
	} else if ((t1 <= 4.6e-10) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = -t1 / (u * (u / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.22e+37:
		tmp = t_1
	elif t1 <= 1e-85:
		tmp = (t1 / u) * (-v / u)
	elif (t1 <= 4.6e-10) or not (t1 <= 4.2e+86):
		tmp = t_1
	else:
		tmp = -t1 / (u * (u / v))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.22e+37)
		tmp = t_1;
	elseif (t1 <= 1e-85)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	elseif ((t1 <= 4.6e-10) || !(t1 <= 4.2e+86))
		tmp = t_1;
	else
		tmp = Float64(Float64(-t1) / Float64(u * Float64(u / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.22e+37)
		tmp = t_1;
	elseif (t1 <= 1e-85)
		tmp = (t1 / u) * (-v / u);
	elseif ((t1 <= 4.6e-10) || ~((t1 <= 4.2e+86)))
		tmp = t_1;
	else
		tmp = -t1 / (u * (u / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.22e+37], t$95$1, If[LessEqual[t1, 1e-85], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 4.6e-10], N[Not[LessEqual[t1, 4.2e+86]], $MachinePrecision]], t$95$1, N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.22e37 or 9.9999999999999998e-86 < t1 < 4.60000000000000014e-10 or 4.1999999999999998e86 < t1

   1. Initial program 67.3%

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

      1. distribute-lft-neg-out 67.3%

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

      2. *-commutative 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-*l/ 66.6%

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

      5. *-commutative 66.6%

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

   3. Simplified 66.6%

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

      1. associate-*r/ 67.3%

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

      2. distribute-rgt-neg-in 67.3%

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

      3. distribute-lft-neg-out 67.3%

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

      4. associate-/r* 81.2%

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

      5. *-commutative 81.2%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in t1 around inf 92.3%

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

      1. mul-1-neg 92.3%

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

      2. unsub-neg 92.3%

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

      3. *-commutative 92.3%

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

   8. Simplified 92.3%

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

   if -1.22e37 < t1 < 9.9999999999999998e-86

   1. Initial program 79.8%

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

      1. distribute-lft-neg-out 79.8%

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

      2. *-commutative 79.8%

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

      3. distribute-lft-neg-out 79.8%

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

      4. associate-*l/ 76.1%

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

      5. *-commutative 76.1%

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

   3. Simplified 76.1%

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

      1. associate-*r/ 79.8%

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

      2. distribute-rgt-neg-in 79.8%

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

      3. distribute-lft-neg-out 79.8%

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

      4. associate-/r* 86.2%

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

      5. *-commutative 86.2%

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

      6. associate-/l* 93.4%

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

      7. associate-/l/ 92.0%

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

   5. Applied egg-rr 92.0%

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

   6. Taylor expanded in t1 around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. unpow2 75.4%

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

      3. associate-*r/ 75.4%

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

      4. neg-mul-1 75.4%

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

      5. distribute-rgt-neg-in 75.4%

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

      6. times-frac 82.7%

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

   8. Simplified 82.7%

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

   if 4.60000000000000014e-10 < t1 < 4.1999999999999998e86

   1. Initial program 77.9%

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

      1. distribute-lft-neg-out 77.9%

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

      2. *-commutative 77.9%

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

      3. distribute-lft-neg-out 77.9%

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

      4. associate-*l/ 83.6%

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

      5. *-commutative 83.6%

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

   3. Simplified 83.6%

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

      1. associate-/r* 99.6%

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

      2. associate-*r/ 99.5%

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

      3. remove-double-neg 99.5%

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

      4. frac-2neg 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-neg-in 99.5%

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

      7. unsub-neg 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r/ 83.6%

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

      2. frac-2neg 83.6%

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

      3. distribute-lft-neg-out 83.6%

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

      4. sub-neg 83.6%

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

      5. distribute-neg-in 83.6%

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

      6. +-commutative 83.6%

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

      7. remove-double-neg 83.6%

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

      8. associate-*r/ 99.5%

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

      9. clear-num 99.3%

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

      10. div-inv 99.4%

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

      11. associate-/r* 99.5%

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

      12. associate-*r/ 99.7%

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

      13. associate-*r/ 99.7%

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

      14. *-rgt-identity 99.7%

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

      15. frac-2neg 99.7%

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

      16. remove-double-neg 99.7%

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

      17. +-commutative 99.7%

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

      18. distribute-neg-in 99.7%

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

      19. sub-neg 99.7%

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

      20. frac-2neg 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-/l/ 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in t1 around 0 53.3%

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

       1. mul-1-neg 53.3%

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

       2. associate-/l* 56.5%

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

       3. unpow2 56.5%

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

       4. associate-*r/ 72.4%

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

       5. distribute-frac-neg 72.4%

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

   12. Simplified 72.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 10^{-85}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-10} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 8: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{-v}{u \cdot u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.52 \cdot 10^{+37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 18.5:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (- v) (* u u)))) (t_2 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.52e+37)
     t_2
     (if (<= t1 1.15e-87)
       t_1
       (if (<= t1 1.6e-8)
         t_2
         (if (<= t1 18.5) t_1 (/ -1.0 (/ (+ t1 u) v))))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (-v / (u * u));
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.52e+37) {
		tmp = t_2;
	} else if (t1 <= 1.15e-87) {
		tmp = t_1;
	} else if (t1 <= 1.6e-8) {
		tmp = t_2;
	} else if (t1 <= 18.5) {
		tmp = t_1;
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t1 * (-v / (u * u))
    t_2 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-1.52d+37)) then
        tmp = t_2
    else if (t1 <= 1.15d-87) then
        tmp = t_1
    else if (t1 <= 1.6d-8) then
        tmp = t_2
    else if (t1 <= 18.5d0) then
        tmp = t_1
    else
        tmp = (-1.0d0) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * (-v / (u * u));
	double t_2 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.52e+37) {
		tmp = t_2;
	} else if (t1 <= 1.15e-87) {
		tmp = t_1;
	} else if (t1 <= 1.6e-8) {
		tmp = t_2;
	} else if (t1 <= 18.5) {
		tmp = t_1;
	} else {
		tmp = -1.0 / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * (-v / (u * u))
	t_2 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.52e+37:
		tmp = t_2
	elif t1 <= 1.15e-87:
		tmp = t_1
	elif t1 <= 1.6e-8:
		tmp = t_2
	elif t1 <= 18.5:
		tmp = t_1
	else:
		tmp = -1.0 / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(Float64(-v) / Float64(u * u)))
	t_2 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.52e+37)
		tmp = t_2;
	elseif (t1 <= 1.15e-87)
		tmp = t_1;
	elseif (t1 <= 1.6e-8)
		tmp = t_2;
	elseif (t1 <= 18.5)
		tmp = t_1;
	else
		tmp = Float64(-1.0 / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * (-v / (u * u));
	t_2 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.52e+37)
		tmp = t_2;
	elseif (t1 <= 1.15e-87)
		tmp = t_1;
	elseif (t1 <= 1.6e-8)
		tmp = t_2;
	elseif (t1 <= 18.5)
		tmp = t_1;
	else
		tmp = -1.0 / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.52e+37], t$95$2, If[LessEqual[t1, 1.15e-87], t$95$1, If[LessEqual[t1, 1.6e-8], t$95$2, If[LessEqual[t1, 18.5], t$95$1, N[(-1.0 / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.5200000000000001e37 or 1.1500000000000001e-87 < t1 < 1.6000000000000001e-8

   1. Initial program 72.9%

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

      1. distribute-lft-neg-out 72.9%

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

      2. *-commutative 72.9%

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

      3. distribute-lft-neg-out 72.9%

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

      4. associate-*l/ 70.7%

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

      5. *-commutative 70.7%

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

   3. Simplified 70.7%

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

      1. associate-*r/ 72.9%

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

      2. distribute-rgt-neg-in 72.9%

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

      3. distribute-lft-neg-out 72.9%

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

      4. associate-/r* 86.7%

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

      5. *-commutative 86.7%

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

      6. associate-/l* 100.0%

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

      7. associate-/l/ 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in t1 around inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. unsub-neg 90.9%

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

      3. *-commutative 90.9%

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

   8. Simplified 90.9%

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

   if -1.5200000000000001e37 < t1 < 1.1500000000000001e-87 or 1.6000000000000001e-8 < t1 < 18.5

   1. Initial program 80.7%

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

      1. distribute-lft-neg-out 80.7%

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

      2. *-commutative 80.7%

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

      3. distribute-lft-neg-out 80.7%

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

      4. associate-*l/ 77.1%

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

      5. *-commutative 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in t1 around 0 72.0%

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

      1. unpow2 72.0%

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

      2. associate-*r/ 72.0%

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

      3. neg-mul-1 72.0%

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

   6. Simplified 72.0%

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

   if 18.5 < t1

   1. Initial program 58.6%

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

      1. distribute-lft-neg-out 58.6%

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

      2. *-commutative 58.6%

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

      3. distribute-lft-neg-out 58.6%

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

      4. associate-*l/ 62.4%

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

      5. *-commutative 62.4%

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

   3. Simplified 62.4%

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

      1. clear-num 61.2%

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

      2. un-div-inv 61.3%

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

      3. neg-mul-1 61.3%

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

      4. times-frac 80.7%

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

      5. associate-/r* 99.6%

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

      6. div-inv 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. neg-mul-1 99.6%

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

      10. +-commutative 99.6%

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

      11. distribute-neg-in 99.6%

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

      12. unsub-neg 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in t1 around inf 84.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.52 \cdot 10^{+37}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 1.15 \cdot 10^{-87}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 18.5:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t1 + u}{v}}\\ \end{array} \]

Alternative 9: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. distribute-lft-neg-out 73.3%

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

   2. *-commutative 73.3%

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

   3. distribute-lft-neg-out 73.3%

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

   4. times-frac 97.0%

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

3. Simplified 97.0%

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

4. Final simplification 97.0%

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

Alternative 10: 67.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{+135} \lor \neg \left(u \leq 4.5 \cdot 10^{+221}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -8.6e+135) (not (<= u 4.5e+221)))
   (* (/ t1 u) (/ v u))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.6e+135) || !(u <= 4.5e+221)) {
		tmp = (t1 / u) * (v / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-8.6d+135)) .or. (.not. (u <= 4.5d+221))) then
        tmp = (t1 / u) * (v / u)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.6e+135) || !(u <= 4.5e+221)) {
		tmp = (t1 / u) * (v / u);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -8.6e+135) or not (u <= 4.5e+221):
		tmp = (t1 / u) * (v / u)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -8.6e+135) || !(u <= 4.5e+221))
		tmp = Float64(Float64(t1 / u) * Float64(v / u));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -8.6e+135) || ~((u <= 4.5e+221)))
		tmp = (t1 / u) * (v / u);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -8.6e+135], N[Not[LessEqual[u, 4.5e+221]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -8.59999999999999945e135 or 4.5000000000000002e221 < 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. distribute-lft-neg-out 82.3%

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

      2. *-commutative 82.3%

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

      3. distribute-lft-neg-out 82.3%

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

      4. associate-*l/ 82.8%

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

      5. *-commutative 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in t1 around 0 82.8%

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

      1. unpow2 82.8%

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

      2. associate-*r/ 82.8%

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

      3. neg-mul-1 82.8%

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

   6. Simplified 82.8%

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

   8. Applied egg-rr 76.8%

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

      1. associate-*r/ 77.1%

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

      2. associate-/l* 76.8%

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

      3. associate-/r/ 75.0%

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

      4. *-commutative 75.0%

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

   10. Applied egg-rr 75.0%

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

   if -8.59999999999999945e135 < u < 4.5000000000000002e221

   1. Initial program 71.2%

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

      1. distribute-lft-neg-out 71.2%

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

      2. *-commutative 71.2%

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

      3. distribute-lft-neg-out 71.2%

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

      4. associate-*l/ 69.1%

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

      5. *-commutative 69.1%

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

   3. Simplified 69.1%

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

      1. associate-/r* 77.1%

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

      2. associate-*r/ 96.4%

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

      3. remove-double-neg 96.4%

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

      4. frac-2neg 96.4%

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

      5. +-commutative 96.4%

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

      6. distribute-neg-in 96.4%

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

      7. unsub-neg 96.4%

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

   5. Applied egg-rr 96.4%

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

   6. Taylor expanded in t1 around inf 65.5%

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

      1. mul-1-neg 65.5%

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

   8. Simplified 65.5%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{+135} \lor \neg \left(u \leq 4.5 \cdot 10^{+221}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 11: 68.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+78} \lor \neg \left(u \leq 2.15 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.75e+78) (not (<= u 2.15e+83)))
   (/ t1 (* u (/ u v)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+78) || !(u <= 2.15e+83)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.75d+78)) .or. (.not. (u <= 2.15d+83))) then
        tmp = t1 / (u * (u / v))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.75e+78) || !(u <= 2.15e+83)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.75e+78) or not (u <= 2.15e+83):
		tmp = t1 / (u * (u / v))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.75e+78) || !(u <= 2.15e+83))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.75e+78) || ~((u <= 2.15e+83)))
		tmp = t1 / (u * (u / v));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.75e+78], N[Not[LessEqual[u, 2.15e+83]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.7500000000000001e78 or 2.15e83 < u

   1. Initial program 79.5%

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

      1. distribute-lft-neg-out 79.5%

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

      2. *-commutative 79.5%

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

      3. distribute-lft-neg-out 79.5%

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

      4. associate-*l/ 80.1%

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

      5. *-commutative 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in t1 around 0 79.1%

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

      1. unpow2 79.1%

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

      2. associate-*r/ 79.1%

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

      3. neg-mul-1 79.1%

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

   6. Simplified 79.1%

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

   8. Applied egg-rr 69.3%

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

   if -1.7500000000000001e78 < u < 2.15e83

   1. Initial program 69.9%

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

      1. distribute-lft-neg-out 69.9%

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

      2. *-commutative 69.9%

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

      3. distribute-lft-neg-out 69.9%

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

      4. associate-*l/ 67.2%

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

      5. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in t1 around inf 68.3%

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

      1. associate-*r/ 68.3%

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

      2. neg-mul-1 68.3%

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

   6. Simplified 68.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+78} \lor \neg \left(u \leq 2.15 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+221}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -6.5e+83) (/ v u) (if (<= u 4.5e+221) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.5e+83) {
		tmp = v / u;
	} else if (u <= 4.5e+221) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-6.5d+83)) then
        tmp = v / u
    else if (u <= 4.5d+221) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -6.5e+83) {
		tmp = v / u;
	} else if (u <= 4.5e+221) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -6.5e+83:
		tmp = v / u
	elif u <= 4.5e+221:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -6.5e+83)
		tmp = Float64(v / u);
	elseif (u <= 4.5e+221)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -6.5e+83)
		tmp = v / u;
	elseif (u <= 4.5e+221)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -6.5e+83], N[(v / u), $MachinePrecision], If[LessEqual[u, 4.5e+221], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -6.5000000000000003e83 or 4.5000000000000002e221 < u

   1. Initial program 83.0%

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

      1. distribute-lft-neg-out 83.0%

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

      2. *-commutative 83.0%

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

      3. distribute-lft-neg-out 83.0%

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

      4. times-frac 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in t1 around 0 90.0%

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

   6. Applied egg-rr 73.6%

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

   7. Taylor expanded in t1 around inf 45.3%

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

   if -6.5000000000000003e83 < u < 4.5000000000000002e221

   1. Initial program 70.5%

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

      1. distribute-lft-neg-out 70.5%

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

      2. *-commutative 70.5%

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

      3. distribute-lft-neg-out 70.5%

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

      4. associate-*l/ 68.3%

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

      5. *-commutative 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in t1 around inf 64.6%

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

      1. associate-*r/ 64.6%

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

      2. neg-mul-1 64.6%

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

   6. Simplified 64.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+221}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 13: 58.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+223}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.8e+156)
   (/ -1.0 (/ u v))
   (if (<= u 2.5e+223) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e+156) {
		tmp = -1.0 / (u / v);
	} else if (u <= 2.5e+223) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.8d+156)) then
        tmp = (-1.0d0) / (u / v)
    else if (u <= 2.5d+223) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e+156) {
		tmp = -1.0 / (u / v);
	} else if (u <= 2.5e+223) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.8e+156:
		tmp = -1.0 / (u / v)
	elif u <= 2.5e+223:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.8e+156)
		tmp = Float64(-1.0 / Float64(u / v));
	elseif (u <= 2.5e+223)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.8e+156)
		tmp = -1.0 / (u / v);
	elseif (u <= 2.5e+223)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.8e+156], N[(-1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.5e+223], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.8000000000000002e156

   1. Initial program 82.9%

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

      1. distribute-lft-neg-out 82.9%

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

      2. *-commutative 82.9%

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

      3. distribute-lft-neg-out 82.9%

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

      4. associate-*l/ 83.2%

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

      5. *-commutative 83.2%

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

   3. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. un-div-inv 83.2%

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

      3. neg-mul-1 83.2%

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

      4. times-frac 94.5%

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

      5. associate-/r* 99.9%

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

      6. div-inv 99.9%

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

      7. metadata-eval 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. unsub-neg 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in t1 around 0 91.6%

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

   7. Taylor expanded in t1 around inf 56.0%

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

   if -4.8000000000000002e156 < u < 2.49999999999999992e223

   1. Initial program 71.2%

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

      1. distribute-lft-neg-out 71.2%

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

      2. *-commutative 71.2%

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

      3. distribute-lft-neg-out 71.2%

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

      4. associate-*l/ 69.2%

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

      5. *-commutative 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in t1 around inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. neg-mul-1 62.0%

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

   6. Simplified 62.0%

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

   if 2.49999999999999992e223 < u

   1. Initial program 83.3%

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

      1. distribute-lft-neg-out 83.3%

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

      2. *-commutative 83.3%

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

      3. distribute-lft-neg-out 83.3%

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

      4. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 99.9%

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

   6. Applied egg-rr 83.8%

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

   7. Taylor expanded in t1 around inf 44.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+223}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 14: 61.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. distribute-lft-neg-out 73.3%

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

   2. *-commutative 73.3%

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

   3. distribute-lft-neg-out 73.3%

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

   4. associate-*l/ 71.8%

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

   5. *-commutative 71.8%

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

3. Simplified 71.8%

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

   1. associate-/r* 80.5%

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

   2. associate-*r/ 97.1%

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

   3. remove-double-neg 97.1%

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

   4. frac-2neg 97.1%

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

   5. +-commutative 97.1%

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

   6. distribute-neg-in 97.1%

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

   7. unsub-neg 97.1%

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

5. Applied egg-rr 97.1%

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

6. Taylor expanded in t1 around inf 63.1%

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

   1. mul-1-neg 63.1%

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

8. Simplified 63.1%

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

9. Final simplification 63.1%

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

Alternative 15: 17.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. distribute-lft-neg-out 73.3%

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

   2. *-commutative 73.3%

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

   3. distribute-lft-neg-out 73.3%

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

   4. times-frac 97.0%

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

3. Simplified 97.0%

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

4. Taylor expanded in t1 around 0 57.3%

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

6. Applied egg-rr 32.8%

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

7. Taylor expanded in t1 around inf 18.5%

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

8. Final simplification 18.5%

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

Reproduce

herbie shell --seed 2023297 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))