Rosa's DopplerBench

Percentage Accurate: 72.8% → 98.1%

Time: 18.4s

Alternatives: 17

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: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return ((t1 * (v / (t1 + u))) / -1.0) / (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))) / (-1.0d0)) / (t1 + u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 74.1%

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

   2. distribute-lft-neg-out 74.1%

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

   3. distribute-rgt-neg-in 74.1%

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

   4. associate-/r* 80.3%

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

   5. distribute-neg-frac2 80.3%

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

3. Simplified 80.3%

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

   1. associate-*r/ 97.1%

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

   2. neg-mul-1 97.1%

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

   3. associate-/r* 97.1%

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

6. Applied egg-rr 97.1%

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

Alternative 2: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -8.8 \cdot 10^{+83}:\\ \;\;\;\;\frac{u \cdot \frac{v}{t1} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;t1 \cdot \frac{t\_1}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-1 + \frac{u}{t1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))))
   (if (<= t1 -8.8e+83)
     (/ (- (* u (/ v t1)) v) (+ t1 u))
     (if (<= t1 5.2e+163)
       (* t1 (/ t_1 (- (- u) t1)))
       (* t_1 (+ -1.0 (/ u t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (t1 <= -8.8e+83) {
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	} else if (t1 <= 5.2e+163) {
		tmp = t1 * (t_1 / (-u - t1));
	} else {
		tmp = t_1 * (-1.0 + (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (t1 + u)
    if (t1 <= (-8.8d+83)) then
        tmp = ((u * (v / t1)) - v) / (t1 + u)
    else if (t1 <= 5.2d+163) then
        tmp = t1 * (t_1 / (-u - t1))
    else
        tmp = t_1 * ((-1.0d0) + (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (t1 <= -8.8e+83) {
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	} else if (t1 <= 5.2e+163) {
		tmp = t1 * (t_1 / (-u - t1));
	} else {
		tmp = t_1 * (-1.0 + (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (t1 + u)
	tmp = 0
	if t1 <= -8.8e+83:
		tmp = ((u * (v / t1)) - v) / (t1 + u)
	elif t1 <= 5.2e+163:
		tmp = t1 * (t_1 / (-u - t1))
	else:
		tmp = t_1 * (-1.0 + (u / t1))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -8.8e+83)
		tmp = Float64(Float64(Float64(u * Float64(v / t1)) - v) / Float64(t1 + u));
	elseif (t1 <= 5.2e+163)
		tmp = Float64(t1 * Float64(t_1 / Float64(Float64(-u) - t1)));
	else
		tmp = Float64(t_1 * Float64(-1.0 + Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -8.8e+83)
		tmp = ((u * (v / t1)) - v) / (t1 + u);
	elseif (t1 <= 5.2e+163)
		tmp = t1 * (t_1 / (-u - t1));
	else
		tmp = t_1 * (-1.0 + (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -8.8e+83], N[(N[(N[(u * N[(v / t1), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.2e+163], N[(t1 * N[(t$95$1 / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(-1.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.79999999999999995e83

   1. Initial program 55.4%

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

      1. associate-/l* 49.6%

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

      2. distribute-lft-neg-out 49.6%

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

      3. distribute-rgt-neg-in 49.6%

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

      4. associate-/r* 55.9%

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

      5. distribute-neg-frac2 55.9%

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

   3. Simplified 55.9%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 93.8%

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

      1. +-commutative 93.8%

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

      2. mul-1-neg 93.8%

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

      3. sub-neg 93.8%

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

      4. associate-/l* 96.2%

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

   9. Simplified 96.2%

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

   if -8.79999999999999995e83 < t1 < 5.2000000000000003e163

   1. Initial program 82.7%

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

      1. associate-/l* 83.0%

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

      2. distribute-lft-neg-out 83.0%

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

      3. distribute-rgt-neg-in 83.0%

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

      4. associate-/r* 87.3%

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

      5. distribute-neg-frac2 87.3%

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

   3. Simplified 87.3%

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

   if 5.2000000000000003e163 < t1

   1. Initial program 45.6%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 94.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8.8 \cdot 10^{+83}:\\ \;\;\;\;\frac{u \cdot \frac{v}{t1} - v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{+163}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-1 + \frac{u}{t1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{u}{t1}}{\frac{t1 + u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2e-30)
   (/ v (- (- u) t1))
   (if (<= t1 4.6e-96)
     (/ (/ (- t1) (/ u v)) u)
     (/ (+ -1.0 (/ u t1)) (/ (+ t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2e-30) {
		tmp = v / (-u - t1);
	} else if (t1 <= 4.6e-96) {
		tmp = (-t1 / (u / v)) / u;
	} else {
		tmp = (-1.0 + (u / t1)) / ((t1 + u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2d-30)) then
        tmp = v / (-u - t1)
    else if (t1 <= 4.6d-96) then
        tmp = (-t1 / (u / v)) / u
    else
        tmp = ((-1.0d0) + (u / t1)) / ((t1 + u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2e-30) {
		tmp = v / (-u - t1);
	} else if (t1 <= 4.6e-96) {
		tmp = (-t1 / (u / v)) / u;
	} else {
		tmp = (-1.0 + (u / t1)) / ((t1 + u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2e-30:
		tmp = v / (-u - t1)
	elif t1 <= 4.6e-96:
		tmp = (-t1 / (u / v)) / u
	else:
		tmp = (-1.0 + (u / t1)) / ((t1 + u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2e-30)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= 4.6e-96)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / u);
	else
		tmp = Float64(Float64(-1.0 + Float64(u / t1)) / Float64(Float64(t1 + u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2e-30)
		tmp = v / (-u - t1);
	elseif (t1 <= 4.6e-96)
		tmp = (-t1 / (u / v)) / u;
	else
		tmp = (-1.0 + (u / t1)) / ((t1 + u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2e-30], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.6e-96], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(N[(-1.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2e-30

   1. Initial program 65.8%

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

      1. associate-/l* 64.8%

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

      2. distribute-lft-neg-out 64.8%

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

      3. distribute-rgt-neg-in 64.8%

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

      4. associate-/r* 70.1%

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

      5. distribute-neg-frac2 70.1%

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

   3. Simplified 70.1%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 87.0%

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

      1. mul-1-neg 87.0%

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

   9. Simplified 87.0%

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

   if -2e-30 < t1 < 4.6e-96

   1. Initial program 82.5%

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

      1. times-frac 92.3%

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

      2. distribute-frac-neg 92.3%

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

      3. distribute-neg-frac2 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-neg-in 92.3%

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

      6. unsub-neg 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in t1 around 0 79.2%

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

      1. associate-*l/ 81.2%

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

      2. frac-2neg 81.2%

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

      3. clear-num 81.1%

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

      4. un-div-inv 81.9%

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

      5. neg-sub0 81.9%

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

      6. add-sqr-sqrt 37.3%

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

      7. sqrt-unprod 81.9%

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

      8. sqr-neg 81.9%

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

      9. sqrt-unprod 44.5%

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

      10. add-sqr-sqrt 82.1%

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

      11. add-sqr-sqrt 36.1%

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

      12. sqrt-unprod 57.6%

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

      13. sqr-neg 57.6%

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

      14. sqrt-unprod 22.2%

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

      15. add-sqr-sqrt 39.9%

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

      16. associate-+l- 39.9%

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

      17. neg-sub0 39.9%

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

      18. add-sqr-sqrt 17.7%

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

      19. sqrt-unprod 58.4%

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

      20. sqr-neg 58.4%

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

      21. sqrt-unprod 45.8%

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

      22. add-sqr-sqrt 82.1%

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

      23. add-sqr-sqrt 44.5%

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

      24. sqrt-unprod 81.9%

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

      25. sqr-neg 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. distribute-neg-frac 81.9%

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

   9. Simplified 81.9%

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

   10. Taylor expanded in t1 around 0 82.2%

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

   if 4.6e-96 < t1

   1. Initial program 71.0%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around inf 81.2%

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

      1. clear-num 82.3%

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

      2. un-div-inv 82.3%

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

      3. sub-neg 82.3%

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

      4. metadata-eval 82.3%

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

   7. Applied egg-rr 82.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{u}{t1}}{\frac{t1 + u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{-30}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -4e-30)
   (/ v (- (- u) t1))
   (if (<= t1 2.5e-96) (/ (/ (- t1) (/ u v)) u) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4e-30) {
		tmp = v / (-u - t1);
	} else if (t1 <= 2.5e-96) {
		tmp = (-t1 / (u / v)) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-4d-30)) then
        tmp = v / (-u - t1)
    else if (t1 <= 2.5d-96) then
        tmp = (-t1 / (u / v)) / u
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -4e-30) {
		tmp = v / (-u - t1);
	} else if (t1 <= 2.5e-96) {
		tmp = (-t1 / (u / v)) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -4e-30:
		tmp = v / (-u - t1)
	elif t1 <= 2.5e-96:
		tmp = (-t1 / (u / v)) / u
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -4e-30)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= 2.5e-96)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / u);
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -4e-30)
		tmp = v / (-u - t1);
	elseif (t1 <= 2.5e-96)
		tmp = (-t1 / (u / v)) / u;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -4e-30], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.5e-96], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4e-30

   1. Initial program 65.8%

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

      1. associate-/l* 64.8%

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

      2. distribute-lft-neg-out 64.8%

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

      3. distribute-rgt-neg-in 64.8%

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

      4. associate-/r* 70.1%

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

      5. distribute-neg-frac2 70.1%

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

   3. Simplified 70.1%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 87.0%

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

      1. mul-1-neg 87.0%

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

   9. Simplified 87.0%

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

   if -4e-30 < t1 < 2.49999999999999997e-96

   1. Initial program 82.5%

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

      1. times-frac 92.3%

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

      2. distribute-frac-neg 92.3%

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

      3. distribute-neg-frac2 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-neg-in 92.3%

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

      6. unsub-neg 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in t1 around 0 79.2%

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

      1. associate-*l/ 81.2%

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

      2. frac-2neg 81.2%

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

      3. clear-num 81.1%

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

      4. un-div-inv 81.9%

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

      5. neg-sub0 81.9%

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

      6. add-sqr-sqrt 37.3%

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

      7. sqrt-unprod 81.9%

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

      8. sqr-neg 81.9%

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

      9. sqrt-unprod 44.5%

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

      10. add-sqr-sqrt 82.1%

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

      11. add-sqr-sqrt 36.1%

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

      12. sqrt-unprod 57.6%

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

      13. sqr-neg 57.6%

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

      14. sqrt-unprod 22.2%

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

      15. add-sqr-sqrt 39.9%

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

      16. associate-+l- 39.9%

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

      17. neg-sub0 39.9%

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

      18. add-sqr-sqrt 17.7%

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

      19. sqrt-unprod 58.4%

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

      20. sqr-neg 58.4%

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

      21. sqrt-unprod 45.8%

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

      22. add-sqr-sqrt 82.1%

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

      23. add-sqr-sqrt 44.5%

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

      24. sqrt-unprod 81.9%

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

      25. sqr-neg 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. distribute-neg-frac 81.9%

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

   9. Simplified 81.9%

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

   10. Taylor expanded in t1 around 0 82.2%

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

   if 2.49999999999999997e-96 < t1

   1. Initial program 71.0%

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

      1. associate-/l* 72.1%

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

      2. distribute-lft-neg-out 72.1%

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

      3. distribute-rgt-neg-in 72.1%

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

      4. associate-/r* 80.2%

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

      5. distribute-neg-frac2 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. associate-/r* 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in t1 around inf 82.2%

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

      1. mul-1-neg 82.2%

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

   9. Simplified 82.2%

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

       1. add-sqr-sqrt 34.0%

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

       2. sqrt-unprod 86.1%

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

       3. sqr-neg 86.1%

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

       4. sqrt-unprod 48.5%

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

       5. add-sqr-sqrt 82.2%

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

       6. sub-neg 82.2%

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

   11. Applied egg-rr 82.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{-30}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{-31}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -6e-31)
   (/ v (- (- u) t1))
   (if (<= t1 4.6e-96) (* (/ (- t1) u) (/ v u)) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -6e-31) {
		tmp = v / (-u - t1);
	} else if (t1 <= 4.6e-96) {
		tmp = (-t1 / u) * (v / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-6d-31)) then
        tmp = v / (-u - t1)
    else if (t1 <= 4.6d-96) then
        tmp = (-t1 / u) * (v / u)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -6e-31) {
		tmp = v / (-u - t1);
	} else if (t1 <= 4.6e-96) {
		tmp = (-t1 / u) * (v / u);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -6e-31:
		tmp = v / (-u - t1)
	elif t1 <= 4.6e-96:
		tmp = (-t1 / u) * (v / u)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -6e-31)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= 4.6e-96)
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -6e-31)
		tmp = v / (-u - t1);
	elseif (t1 <= 4.6e-96)
		tmp = (-t1 / u) * (v / u);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -6e-31], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.6e-96], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -5.99999999999999962e-31

   1. Initial program 65.8%

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

      1. associate-/l* 64.8%

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

      2. distribute-lft-neg-out 64.8%

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

      3. distribute-rgt-neg-in 64.8%

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

      4. associate-/r* 70.1%

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

      5. distribute-neg-frac2 70.1%

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

   3. Simplified 70.1%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 87.0%

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

      1. mul-1-neg 87.0%

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

   9. Simplified 87.0%

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

   if -5.99999999999999962e-31 < t1 < 4.6e-96

   1. Initial program 82.5%

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

      1. times-frac 92.3%

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

      2. distribute-frac-neg 92.3%

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

      3. distribute-neg-frac2 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-neg-in 92.3%

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

      6. unsub-neg 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in t1 around 0 79.2%

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

   6. Taylor expanded in t1 around 0 79.5%

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

      1. associate-*r/ 79.5%

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

      2. mul-1-neg 79.5%

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

   8. Simplified 79.5%

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

   if 4.6e-96 < t1

   1. Initial program 71.0%

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

      1. associate-/l* 72.1%

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

      2. distribute-lft-neg-out 72.1%

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

      3. distribute-rgt-neg-in 72.1%

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

      4. associate-/r* 80.2%

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

      5. distribute-neg-frac2 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. associate-/r* 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in t1 around inf 82.2%

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

      1. mul-1-neg 82.2%

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

   9. Simplified 82.2%

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

       1. add-sqr-sqrt 34.0%

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

       2. sqrt-unprod 86.1%

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

       3. sqr-neg 86.1%

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

       4. sqrt-unprod 48.5%

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

       5. add-sqr-sqrt 82.2%

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

       6. sub-neg 82.2%

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

   11. Applied egg-rr 82.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{-31}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{-30}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-94}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -8e-30)
   (/ v (- (- u) t1))
   (if (<= t1 8.5e-94) (* t1 (/ (- v) (* u u))) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -8e-30) {
		tmp = v / (-u - t1);
	} else if (t1 <= 8.5e-94) {
		tmp = t1 * (-v / (u * u));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-8d-30)) then
        tmp = v / (-u - t1)
    else if (t1 <= 8.5d-94) then
        tmp = t1 * (-v / (u * u))
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -8e-30) {
		tmp = v / (-u - t1);
	} else if (t1 <= 8.5e-94) {
		tmp = t1 * (-v / (u * u));
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -8e-30:
		tmp = v / (-u - t1)
	elif t1 <= 8.5e-94:
		tmp = t1 * (-v / (u * u))
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -8e-30)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= 8.5e-94)
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(u * u)));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -8e-30)
		tmp = v / (-u - t1);
	elseif (t1 <= 8.5e-94)
		tmp = t1 * (-v / (u * u));
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -8e-30], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 8.5e-94], N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -8.000000000000001e-30

   1. Initial program 65.8%

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

      1. associate-/l* 64.8%

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

      2. distribute-lft-neg-out 64.8%

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

      3. distribute-rgt-neg-in 64.8%

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

      4. associate-/r* 70.1%

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

      5. distribute-neg-frac2 70.1%

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

   3. Simplified 70.1%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

      3. associate-/r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t1 around inf 87.0%

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

      1. mul-1-neg 87.0%

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

   9. Simplified 87.0%

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

   if -8.000000000000001e-30 < t1 < 8.50000000000000003e-94

   1. Initial program 82.5%

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

      1. associate-/l* 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in t1 around 0 72.6%

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

   6. Taylor expanded in t1 around 0 72.9%

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

   if 8.50000000000000003e-94 < t1

   1. Initial program 71.0%

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

      1. associate-/l* 72.1%

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

      2. distribute-lft-neg-out 72.1%

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

      3. distribute-rgt-neg-in 72.1%

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

      4. associate-/r* 80.2%

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

      5. distribute-neg-frac2 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. associate-/r* 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in t1 around inf 82.2%

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

      1. mul-1-neg 82.2%

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

   9. Simplified 82.2%

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

       1. add-sqr-sqrt 34.0%

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

       2. sqrt-unprod 86.1%

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

       3. sqr-neg 86.1%

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

       4. sqrt-unprod 48.5%

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

       5. add-sqr-sqrt 82.2%

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

       6. sub-neg 82.2%

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

   11. Applied egg-rr 82.2%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{-30}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-94}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+54} \lor \neg \left(u \leq 2.35 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.15e+54) (not (<= u 2.35e+67)))
   (/ t1 (* u (/ u v)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.15e+54) || !(u <= 2.35e+67)) {
		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 <= (-1.15d+54)) .or. (.not. (u <= 2.35d+67))) 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 <= -1.15e+54) || !(u <= 2.35e+67)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.15e+54) or not (u <= 2.35e+67):
		tmp = t1 / (u * (u / v))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.15e+54) || !(u <= 2.35e+67))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.15e+54) || ~((u <= 2.35e+67)))
		tmp = t1 / (u * (u / v));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.15e+54], N[Not[LessEqual[u, 2.35e+67]], $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 < -1.14999999999999997e54 or 2.35000000000000009e67 < u

   1. Initial program 78.3%

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

      1. times-frac 97.1%

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

      2. distribute-frac-neg 97.1%

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

      3. distribute-neg-frac2 97.1%

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

      4. +-commutative 97.1%

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

      5. distribute-neg-in 97.1%

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

      6. unsub-neg 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in t1 around 0 85.2%

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

   6. Taylor expanded in t1 around 0 85.2%

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

      1. associate-*r/ 85.2%

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

      2. mul-1-neg 85.2%

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

   8. Simplified 85.2%

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

      1. *-commutative 85.2%

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

      2. clear-num 85.2%

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

      3. frac-times 82.4%

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

      4. *-un-lft-identity 82.4%

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

      5. add-sqr-sqrt 42.3%

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

      6. sqrt-unprod 65.4%

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

      7. sqr-neg 65.4%

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

      8. sqrt-unprod 37.0%

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

      9. add-sqr-sqrt 70.5%

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

   10. Applied egg-rr 70.5%

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

   if -1.14999999999999997e54 < u < 2.35000000000000009e67

   1. Initial program 72.5%

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

      1. associate-/l* 71.4%

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

      2. distribute-lft-neg-out 71.4%

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

      3. distribute-rgt-neg-in 71.4%

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

      4. associate-/r* 77.6%

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

      5. distribute-neg-frac2 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in t1 around inf 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   7. Simplified 70.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+54} \lor \neg \left(u \leq 2.35 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+148} \lor \neg \left(u \leq 3.2 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.3e+148) (not (<= u 3.2e+71))) (* (/ v u) -0.5) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e+148) || !(u <= 3.2e+71)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.3d+148)) .or. (.not. (u <= 3.2d+71))) then
        tmp = (v / u) * (-0.5d0)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e+148) || !(u <= 3.2e+71)) {
		tmp = (v / u) * -0.5;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.3e+148) or not (u <= 3.2e+71):
		tmp = (v / u) * -0.5
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.3e+148) || !(u <= 3.2e+71))
		tmp = Float64(Float64(v / u) * -0.5);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.3e+148) || ~((u <= 3.2e+71)))
		tmp = (v / u) * -0.5;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.3e+148], N[Not[LessEqual[u, 3.2e+71]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.3000000000000001e148 or 3.20000000000000023e71 < u

   1. Initial program 77.3%

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

   3. Taylor expanded in t1 around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. associate-*r* 65.4%

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

      4. distribute-rgt-in 77.3%

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

      5. *-commutative 77.3%

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

   5. Simplified 77.3%

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

   6. Taylor expanded in t1 around inf 37.7%

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

      1. *-commutative 37.7%

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

   8. Simplified 37.7%

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

   if -2.3000000000000001e148 < u < 3.20000000000000023e71

   1. Initial program 73.5%

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

      1. associate-/l* 72.5%

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

      2. distribute-lft-neg-out 72.5%

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

      3. distribute-rgt-neg-in 72.5%

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

      4. associate-/r* 78.7%

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

      5. distribute-neg-frac2 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in t1 around inf 67.0%

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

      1. associate-*r/ 67.0%

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

      2. neg-mul-1 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+148} \lor \neg \left(u \leq 3.2 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 3.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.2e+54)
   (/ 1.0 (/ u v))
   (if (<= u 3.2e+71) (/ v (- t1)) (* (/ v u) -0.5))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.2e+54) {
		tmp = 1.0 / (u / v);
	} else if (u <= 3.2e+71) {
		tmp = v / -t1;
	} else {
		tmp = (v / u) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.2d+54)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 3.2d+71) then
        tmp = v / -t1
    else
        tmp = (v / u) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.2e+54) {
		tmp = 1.0 / (u / v);
	} else if (u <= 3.2e+71) {
		tmp = v / -t1;
	} else {
		tmp = (v / u) * -0.5;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.2e+54:
		tmp = 1.0 / (u / v)
	elif u <= 3.2e+71:
		tmp = v / -t1
	else:
		tmp = (v / u) * -0.5
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.2e+54)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 3.2e+71)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(v / u) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.2e+54)
		tmp = 1.0 / (u / v);
	elseif (u <= 3.2e+71)
		tmp = v / -t1;
	else
		tmp = (v / u) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.2e+54], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.2e+71], N[(v / (-t1)), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.1999999999999999e54

   1. Initial program 80.8%

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

      1. associate-/l* 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in t1 around 0 80.9%

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

   6. Taylor expanded in t1 around inf 37.4%

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

      1. associate-*r/ 37.4%

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

      2. mul-1-neg 37.4%

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

   8. Simplified 37.4%

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

      1. add-sqr-sqrt 18.1%

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

      2. sqrt-unprod 38.5%

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

      3. sqr-neg 38.5%

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

      4. sqrt-unprod 19.0%

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

      5. add-sqr-sqrt 37.4%

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

      6. clear-num 39.4%

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

      7. inv-pow 39.4%

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

   10. Applied egg-rr 39.4%

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

       1. unpow-1 39.4%

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

   12. Simplified 39.4%

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

   if -2.1999999999999999e54 < u < 3.20000000000000023e71

   1. Initial program 72.5%

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

      1. associate-/l* 71.4%

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

      2. distribute-lft-neg-out 71.4%

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

      3. distribute-rgt-neg-in 71.4%

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

      4. associate-/r* 77.6%

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

      5. distribute-neg-frac2 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in t1 around inf 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   7. Simplified 70.2%

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

   if 3.20000000000000023e71 < u

   1. Initial program 76.1%

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

   3. Taylor expanded in t1 around 0 67.5%

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

      1. +-commutative 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*r* 67.5%

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

      4. distribute-rgt-in 76.1%

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

      5. *-commutative 76.1%

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

   5. Simplified 76.1%

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

   6. Taylor expanded in t1 around inf 36.6%

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

      1. *-commutative 36.6%

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

   8. Simplified 36.6%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 3.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+148} \lor \neg \left(u \leq 1.85 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -9.4e+148) (not (<= u 1.85e+71))) (/ (- v) u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.4e+148) || !(u <= 1.85e+71)) {
		tmp = -v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-9.4d+148)) .or. (.not. (u <= 1.85d+71))) then
        tmp = -v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -9.4e+148) || !(u <= 1.85e+71)) {
		tmp = -v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -9.4e+148) or not (u <= 1.85e+71):
		tmp = -v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -9.4e+148) || !(u <= 1.85e+71))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -9.4e+148) || ~((u <= 1.85e+71)))
		tmp = -v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -9.4e+148], N[Not[LessEqual[u, 1.85e+71]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -9.3999999999999994e148 or 1.85e71 < u

   1. Initial program 77.3%

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

      1. associate-/l* 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in t1 around 0 77.8%

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

   6. Taylor expanded in t1 around inf 37.6%

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

      1. associate-*r/ 37.6%

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

      2. mul-1-neg 37.6%

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

   8. Simplified 37.6%

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

   if -9.3999999999999994e148 < u < 1.85e71

   1. Initial program 73.5%

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

      1. associate-/l* 72.5%

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

      2. distribute-lft-neg-out 72.5%

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

      3. distribute-rgt-neg-in 72.5%

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

      4. associate-/r* 78.7%

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

      5. distribute-neg-frac2 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in t1 around inf 67.0%

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

      1. associate-*r/ 67.0%

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

      2. neg-mul-1 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.4 \cdot 10^{+148} \lor \neg \left(u \leq 1.85 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+150} \lor \neg \left(u \leq 4.5 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.9e+150) (not (<= u 4.5e+62))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+150) || !(u <= 4.5e+62)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.9d+150)) .or. (.not. (u <= 4.5d+62))) then
        tmp = v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+150) || !(u <= 4.5e+62)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.9e+150) or not (u <= 4.5e+62):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.9e+150) || !(u <= 4.5e+62))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.9e+150) || ~((u <= 4.5e+62)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.9e+150], N[Not[LessEqual[u, 4.5e+62]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.89999999999999995e150 or 4.49999999999999999e62 < u

   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. associate-/l* 78.3%

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

      2. distribute-lft-neg-out 78.3%

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

      3. distribute-rgt-neg-in 78.3%

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

      4. associate-/r* 84.2%

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

      5. distribute-neg-frac2 84.2%

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

   3. Simplified 84.2%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 44.2%

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

      1. mul-1-neg 44.2%

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

   9. Simplified 44.2%

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

       1. div-inv 44.2%

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

       2. add-sqr-sqrt 20.6%

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

       3. sqrt-unprod 40.0%

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

       4. sqr-neg 40.0%

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

       5. sqrt-unprod 20.0%

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

       6. add-sqr-sqrt 36.6%

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

   11. Applied egg-rr 36.6%

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

   12. Taylor expanded in t1 around 0 36.6%

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

   if -1.89999999999999995e150 < u < 4.49999999999999999e62

   1. Initial program 73.2%

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

      1. associate-/l* 72.2%

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

      2. distribute-lft-neg-out 72.2%

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

      3. distribute-rgt-neg-in 72.2%

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

      4. associate-/r* 78.5%

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

      5. distribute-neg-frac2 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in t1 around inf 67.7%

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

      1. associate-*r/ 67.7%

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

      2. neg-mul-1 67.7%

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

   7. Simplified 67.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+150} \lor \neg \left(u \leq 4.5 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 23.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+114} \lor \neg \left(t1 \leq 1.25 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.6e+114) (not (<= t1 1.25e+100))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.6e+114) || !(t1 <= 1.25e+100)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.6d+114)) .or. (.not. (t1 <= 1.25d+100))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.6e+114) || !(t1 <= 1.25e+100)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.6e+114) or not (t1 <= 1.25e+100):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.6e+114) || !(t1 <= 1.25e+100))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.6e+114) || ~((t1 <= 1.25e+100)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.6e+114], N[Not[LessEqual[t1, 1.25e+100]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.6e114 or 1.25e100 < t1

   1. Initial program 54.8%

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

      1. associate-/l* 51.0%

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

      2. distribute-lft-neg-out 51.0%

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

      3. distribute-rgt-neg-in 51.0%

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

      4. associate-/r* 62.1%

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

      5. distribute-neg-frac2 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in t1 around inf 85.9%

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

      1. associate-*r/ 85.9%

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

      2. neg-mul-1 85.9%

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

   7. Simplified 85.9%

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

      1. *-un-lft-identity 85.9%

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

      2. *-commutative 85.9%

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

      3. add-sqr-sqrt 40.8%

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

      4. sqrt-unprod 51.8%

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

      5. sqr-neg 51.8%

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

      6. sqrt-unprod 17.2%

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

      7. add-sqr-sqrt 36.4%

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

   9. Applied egg-rr 36.4%

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

       1. *-rgt-identity 36.4%

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

   11. Simplified 36.4%

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

   if -2.6e114 < t1 < 1.25e100

   1. Initial program 83.2%

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

      1. associate-/l* 84.1%

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

      2. distribute-lft-neg-out 84.1%

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

      3. distribute-rgt-neg-in 84.1%

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

      4. associate-/r* 88.1%

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

      5. distribute-neg-frac2 88.1%

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

   3. Simplified 88.1%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

      3. associate-/r* 95.9%

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

   6. Applied egg-rr 95.9%

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

   7. Taylor expanded in t1 around inf 48.0%

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

      1. mul-1-neg 48.0%

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

   9. Simplified 48.0%

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

       1. div-inv 47.9%

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

       2. add-sqr-sqrt 19.2%

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

       3. sqrt-unprod 25.1%

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

       4. sqr-neg 25.1%

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

       5. sqrt-unprod 10.7%

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

       6. add-sqr-sqrt 18.9%

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

   11. Applied egg-rr 18.9%

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

   12. Taylor expanded in t1 around 0 19.2%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{+114} \lor \neg \left(t1 \leq 1.25 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.2% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 74.1%

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

   2. distribute-lft-neg-out 74.1%

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

   3. distribute-rgt-neg-in 74.1%

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

   4. associate-/r* 80.3%

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

   5. distribute-neg-frac2 80.3%

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

3. Simplified 80.3%

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

   1. distribute-frac-neg2 80.3%

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

   2. associate-/r* 74.1%

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

   3. distribute-rgt-neg-in 74.1%

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

   4. distribute-lft-neg-out 74.1%

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

   5. associate-*r/ 74.7%

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

   6. times-frac 96.2%

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

   7. frac-2neg 96.2%

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

   8. associate-*r/ 97.1%

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

   9. add-sqr-sqrt 48.7%

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

   10. sqrt-unprod 45.5%

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

   11. sqr-neg 45.5%

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

   12. sqrt-unprod 19.2%

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

   13. add-sqr-sqrt 36.1%

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

   14. add-sqr-sqrt 17.6%

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

   15. sqrt-unprod 56.8%

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

   16. sqr-neg 56.8%

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

   17. sqrt-prod 48.5%

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

   18. add-sqr-sqrt 97.1%

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

6. Applied egg-rr 97.1%

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

7. Final simplification 97.1%

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

Alternative 14: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. times-frac 96.2%

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

   2. distribute-frac-neg 96.2%

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

   3. distribute-neg-frac2 96.2%

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

   4. +-commutative 96.2%

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

   5. distribute-neg-in 96.2%

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

   6. unsub-neg 96.2%

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

3. Simplified 96.2%

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

5. Final simplification 96.2%

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

Alternative 15: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 1.4e+74) (/ v (- (- u) t1)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1.4e+74) {
		tmp = v / (-u - t1);
	} 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 (v <= 1.4d+74) then
        tmp = v / (-u - t1)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1.4e+74) {
		tmp = v / (-u - t1);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 1.4e+74:
		tmp = v / (-u - t1)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 1.4e+74)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 1.4e+74)
		tmp = v / (-u - t1);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 1.4e+74], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.40000000000000001e74

   1. Initial program 74.9%

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

      1. associate-/l* 75.3%

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

      2. distribute-lft-neg-out 75.3%

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

      3. distribute-rgt-neg-in 75.3%

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

      4. associate-/r* 80.9%

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

      5. distribute-neg-frac2 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r/ 97.4%

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

      2. neg-mul-1 97.4%

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

      3. associate-/r* 97.4%

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in t1 around inf 62.1%

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

      1. mul-1-neg 62.1%

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

   9. Simplified 62.1%

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

   if 1.40000000000000001e74 < v

   1. Initial program 74.0%

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

      1. associate-/l* 69.1%

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

      2. distribute-lft-neg-out 69.1%

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

      3. distribute-rgt-neg-in 69.1%

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

      4. associate-/r* 77.5%

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

      5. distribute-neg-frac2 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in t1 around inf 54.9%

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

      1. associate-*r/ 54.9%

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

      2. neg-mul-1 54.9%

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

   7. Simplified 54.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 74.1%

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

   2. distribute-lft-neg-out 74.1%

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

   3. distribute-rgt-neg-in 74.1%

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

   4. associate-/r* 80.3%

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

   5. distribute-neg-frac2 80.3%

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

3. Simplified 80.3%

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

   1. associate-*r/ 97.1%

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

   2. neg-mul-1 97.1%

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

   3. associate-/r* 97.1%

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

6. Applied egg-rr 97.1%

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

7. Taylor expanded in t1 around inf 59.9%

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

   1. mul-1-neg 59.9%

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

9. Simplified 59.9%

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

    1. add-sqr-sqrt 27.5%

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

    2. sqrt-unprod 68.5%

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

    3. sqr-neg 68.5%

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

    4. sqrt-unprod 32.7%

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

    5. add-sqr-sqrt 60.7%

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

    6. sub-neg 60.7%

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

11. Applied egg-rr 60.7%

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

12. Final simplification 60.7%

    \[\leadsto \frac{v}{u - t1} \]
13. Add Preprocessing

Alternative 17: 14.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

   1. associate-/l* 74.1%

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

   2. distribute-lft-neg-out 74.1%

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

   3. distribute-rgt-neg-in 74.1%

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

   4. associate-/r* 80.3%

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

   5. distribute-neg-frac2 80.3%

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

3. Simplified 80.3%

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

5. Taylor expanded in t1 around inf 52.6%

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

   1. associate-*r/ 52.6%

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

   2. neg-mul-1 52.6%

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

7. Simplified 52.6%

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

   1. *-un-lft-identity 52.6%

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

   2. *-commutative 52.6%

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

   3. add-sqr-sqrt 22.6%

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

   4. sqrt-unprod 30.1%

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

   5. sqr-neg 30.1%

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

   6. sqrt-unprod 6.9%

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

   7. add-sqr-sqrt 13.3%

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

9. Applied egg-rr 13.3%

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

    1. *-rgt-identity 13.3%

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

11. Simplified 13.3%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))