Rosa's DopplerBench

Percentage Accurate: 73.6% → 98.1%

Time: 19.0s

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: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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 70.6%

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

   1. associate-/l* 69.2%

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

   2. distribute-lft-neg-out 69.2%

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

   3. distribute-rgt-neg-in 69.2%

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

   4. associate-/r* 82.7%

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

   5. distribute-neg-frac2 82.7%

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

3. Simplified 82.7%

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

   1. associate-*r/ 97.9%

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

   2. neg-mul-1 97.9%

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

   3. associate-/r* 97.9%

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

6. Applied egg-rr 97.9%

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

Alternative 2: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ t_2 := \frac{v}{t\_1}\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq 1.55 \cdot 10^{-211}:\\ \;\;\;\;t1 \cdot \frac{t\_2}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.3 \cdot 10^{+154}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)) (t_2 (/ v t_1)))
   (if (<= t1 -2e+122)
     t_2
     (if (<= t1 1.55e-211)
       (* t1 (/ t_2 (+ t1 u)))
       (if (<= t1 4.3e+154) (* v (/ t1 (* (+ t1 u) t_1))) (/ v (- t1)))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / t_1;
	double tmp;
	if (t1 <= -2e+122) {
		tmp = t_2;
	} else if (t1 <= 1.55e-211) {
		tmp = t1 * (t_2 / (t1 + u));
	} else if (t1 <= 4.3e+154) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -u - t1
    t_2 = v / t_1
    if (t1 <= (-2d+122)) then
        tmp = t_2
    else if (t1 <= 1.55d-211) then
        tmp = t1 * (t_2 / (t1 + u))
    else if (t1 <= 4.3d+154) then
        tmp = v * (t1 / ((t1 + u) * t_1))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double t_2 = v / t_1;
	double tmp;
	if (t1 <= -2e+122) {
		tmp = t_2;
	} else if (t1 <= 1.55e-211) {
		tmp = t1 * (t_2 / (t1 + u));
	} else if (t1 <= 4.3e+154) {
		tmp = v * (t1 / ((t1 + u) * t_1));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	t_2 = v / t_1
	tmp = 0
	if t1 <= -2e+122:
		tmp = t_2
	elif t1 <= 1.55e-211:
		tmp = t1 * (t_2 / (t1 + u))
	elif t1 <= 4.3e+154:
		tmp = v * (t1 / ((t1 + u) * t_1))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	t_2 = Float64(v / t_1)
	tmp = 0.0
	if (t1 <= -2e+122)
		tmp = t_2;
	elseif (t1 <= 1.55e-211)
		tmp = Float64(t1 * Float64(t_2 / Float64(t1 + u)));
	elseif (t1 <= 4.3e+154)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * t_1)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	t_2 = v / t_1;
	tmp = 0.0;
	if (t1 <= -2e+122)
		tmp = t_2;
	elseif (t1 <= 1.55e-211)
		tmp = t1 * (t_2 / (t1 + u));
	elseif (t1 <= 4.3e+154)
		tmp = v * (t1 / ((t1 + u) * t_1));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, Block[{t$95$2 = N[(v / t$95$1), $MachinePrecision]}, If[LessEqual[t1, -2e+122], t$95$2, If[LessEqual[t1, 1.55e-211], N[(t1 * N[(t$95$2 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.3e+154], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.00000000000000003e122

   1. Initial program 44.9%

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

      1. associate-/l* 39.9%

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

      2. distribute-lft-neg-out 39.9%

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

      3. distribute-rgt-neg-in 39.9%

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

      4. associate-/r* 68.4%

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

      5. distribute-neg-frac2 68.4%

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

   3. Simplified 68.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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 87.2%

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

      1. mul-1-neg 87.2%

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

   9. Simplified 87.2%

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

   if -2.00000000000000003e122 < t1 < 1.54999999999999998e-211

   1. Initial program 82.3%

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

      1. associate-/l* 85.0%

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

      2. distribute-lft-neg-out 85.0%

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

      3. distribute-rgt-neg-in 85.0%

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

      4. associate-/r* 92.4%

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

      5. distribute-neg-frac2 92.4%

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

   3. Simplified 92.4%

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

   if 1.54999999999999998e-211 < t1 < 4.2999999999999998e154

   1. Initial program 87.0%

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

      1. associate-*l/ 91.0%

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

      2. *-commutative 91.0%

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

   3. Simplified 91.0%

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

   if 4.2999999999999998e154 < t1

   1. Initial program 24.7%

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

      1. associate-/l* 26.3%

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

      2. distribute-lft-neg-out 26.3%

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

      3. distribute-rgt-neg-in 26.3%

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

      4. associate-/r* 61.7%

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

      5. distribute-neg-frac2 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in t1 around inf 90.6%

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

      1. associate-*r/ 90.6%

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

      2. neg-mul-1 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{+122}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.55 \cdot 10^{-211}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(-u\right) - t1}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 4.3 \cdot 10^{+154}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-u\right) - t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\left(-u\right) - t1}\\ \mathbf{if}\;t1 \leq -4.6 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.5 \cdot 10^{+154}:\\ \;\;\;\;t1 \cdot \frac{t\_1}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (- u) t1))))
   (if (<= t1 -4.6e+120)
     t_1
     (if (<= t1 5.5e+154) (* t1 (/ t_1 (+ t1 u))) (/ v (- t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = v / (-u - t1);
	double tmp;
	if (t1 <= -4.6e+120) {
		tmp = t_1;
	} else if (t1 <= 5.5e+154) {
		tmp = t1 * (t_1 / (t1 + u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (-u - t1)
    if (t1 <= (-4.6d+120)) then
        tmp = t_1
    else if (t1 <= 5.5d+154) then
        tmp = t1 * (t_1 / (t1 + u))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / (-u - t1);
	double tmp;
	if (t1 <= -4.6e+120) {
		tmp = t_1;
	} else if (t1 <= 5.5e+154) {
		tmp = t1 * (t_1 / (t1 + u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / (-u - t1)
	tmp = 0
	if t1 <= -4.6e+120:
		tmp = t_1
	elif t1 <= 5.5e+154:
		tmp = t1 * (t_1 / (t1 + u))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(-u) - t1))
	tmp = 0.0
	if (t1 <= -4.6e+120)
		tmp = t_1;
	elseif (t1 <= 5.5e+154)
		tmp = Float64(t1 * Float64(t_1 / Float64(t1 + u)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / (-u - t1);
	tmp = 0.0;
	if (t1 <= -4.6e+120)
		tmp = t_1;
	elseif (t1 <= 5.5e+154)
		tmp = t1 * (t_1 / (t1 + u));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -4.6e+120], t$95$1, If[LessEqual[t1, 5.5e+154], N[(t1 * N[(t$95$1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.59999999999999985e120

   1. Initial program 44.9%

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

      1. associate-/l* 39.9%

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

      2. distribute-lft-neg-out 39.9%

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

      3. distribute-rgt-neg-in 39.9%

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

      4. associate-/r* 68.4%

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

      5. distribute-neg-frac2 68.4%

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

   3. Simplified 68.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   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 87.2%

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

      1. mul-1-neg 87.2%

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

   9. Simplified 87.2%

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

   if -4.59999999999999985e120 < t1 < 5.5000000000000006e154

   1. Initial program 84.1%

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

      1. associate-/l* 83.1%

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

      2. distribute-lft-neg-out 83.1%

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

      3. distribute-rgt-neg-in 83.1%

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

      4. associate-/r* 89.5%

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

      5. distribute-neg-frac2 89.5%

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

   3. Simplified 89.5%

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

   if 5.5000000000000006e154 < t1

   1. Initial program 24.7%

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

      1. associate-/l* 26.3%

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

      2. distribute-lft-neg-out 26.3%

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

      3. distribute-rgt-neg-in 26.3%

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

      4. associate-/r* 61.7%

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

      5. distribute-neg-frac2 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in t1 around inf 90.6%

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

      1. associate-*r/ 90.6%

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

      2. neg-mul-1 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.6 \cdot 10^{+120}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 5.5 \cdot 10^{+154}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(-u\right) - t1}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{v}{u}\\ t_2 := \left(-u\right) - t1\\ \mathbf{if}\;u \leq -1.7 \cdot 10^{+42}:\\ \;\;\;\;\frac{t\_1}{t\_2}\\ \mathbf{elif}\;u \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{t\_2}}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{-1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ v u))) (t_2 (- (- u) t1)))
   (if (<= u -1.7e+42)
     (/ t_1 t_2)
     (if (<= u 2.7e+16) (/ (* v (/ t1 t_2)) t1) (* t_1 (/ -1.0 u))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (v / u);
	double t_2 = -u - t1;
	double tmp;
	if (u <= -1.7e+42) {
		tmp = t_1 / t_2;
	} else if (u <= 2.7e+16) {
		tmp = (v * (t1 / t_2)) / t1;
	} else {
		tmp = t_1 * (-1.0 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t1 * (v / u)
    t_2 = -u - t1
    if (u <= (-1.7d+42)) then
        tmp = t_1 / t_2
    else if (u <= 2.7d+16) then
        tmp = (v * (t1 / t_2)) / t1
    else
        tmp = t_1 * ((-1.0d0) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * (v / u);
	double t_2 = -u - t1;
	double tmp;
	if (u <= -1.7e+42) {
		tmp = t_1 / t_2;
	} else if (u <= 2.7e+16) {
		tmp = (v * (t1 / t_2)) / t1;
	} else {
		tmp = t_1 * (-1.0 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * (v / u)
	t_2 = -u - t1
	tmp = 0
	if u <= -1.7e+42:
		tmp = t_1 / t_2
	elif u <= 2.7e+16:
		tmp = (v * (t1 / t_2)) / t1
	else:
		tmp = t_1 * (-1.0 / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(v / u))
	t_2 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (u <= -1.7e+42)
		tmp = Float64(t_1 / t_2);
	elseif (u <= 2.7e+16)
		tmp = Float64(Float64(v * Float64(t1 / t_2)) / t1);
	else
		tmp = Float64(t_1 * Float64(-1.0 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * (v / u);
	t_2 = -u - t1;
	tmp = 0.0;
	if (u <= -1.7e+42)
		tmp = t_1 / t_2;
	elseif (u <= 2.7e+16)
		tmp = (v * (t1 / t_2)) / t1;
	else
		tmp = t_1 * (-1.0 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[u, -1.7e+42], N[(t$95$1 / t$95$2), $MachinePrecision], If[LessEqual[u, 2.7e+16], N[(N[(v * N[(t1 / t$95$2), $MachinePrecision]), $MachinePrecision] / t1), $MachinePrecision], N[(t$95$1 * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.69999999999999988e42

   1. Initial program 77.0%

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

      1. associate-/l* 74.2%

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

      2. distribute-lft-neg-out 74.2%

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

      3. distribute-rgt-neg-in 74.2%

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

      4. associate-/r* 87.4%

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

      5. distribute-neg-frac2 87.4%

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

   3. Simplified 87.4%

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

      1. associate-*r/ 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t1 around 0 86.0%

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

      1. mul-1-neg 86.0%

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

      2. associate-*r/ 90.9%

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

      3. distribute-rgt-neg-in 90.9%

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

      4. mul-1-neg 90.9%

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

      5. associate-*r/ 90.9%

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

      6. mul-1-neg 90.9%

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

   9. Simplified 90.9%

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

   if -1.69999999999999988e42 < u < 2.7e16

   1. Initial program 64.3%

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

      1. associate-/l* 63.2%

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

      2. distribute-lft-neg-out 63.2%

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

      3. distribute-rgt-neg-in 63.2%

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

      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. Step-by-step derivation

      1. distribute-frac-neg2 77.5%

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

      2. distribute-rgt-neg-out 77.5%

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

      3. associate-/r* 63.2%

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

      4. distribute-lft-neg-out 63.2%

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

      5. associate-/l* 64.3%

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

      6. times-frac 96.7%

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

      7. frac-2neg 96.7%

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

      8. associate-*r/ 98.3%

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

      9. add-sqr-sqrt 49.7%

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

      10. sqrt-unprod 32.9%

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

      11. sqr-neg 32.9%

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

      12. sqrt-unprod 6.8%

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

      13. add-sqr-sqrt 12.7%

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

      14. add-sqr-sqrt 7.2%

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

      15. sqrt-unprod 39.9%

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

      16. sqr-neg 39.9%

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

      17. sqrt-prod 47.1%

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

      18. add-sqr-sqrt 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in t1 around inf 78.9%

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

   if 2.7e16 < u

   1. Initial program 80.0%

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

      1. associate-/l* 78.9%

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

      2. distribute-lft-neg-out 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. associate-/r* 90.8%

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

      5. distribute-neg-frac2 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in t1 around 0 82.7%

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

   6. Taylor expanded in t1 around 0 82.2%

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

      1. associate-*r/ 84.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 61.4%

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

      4. sqr-neg 61.4%

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

      5. sqrt-unprod 59.7%

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

      6. add-sqr-sqrt 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-unprod 71.3%

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

      3. sqr-neg 71.3%

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

      4. distribute-frac-neg 71.3%

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

      5. distribute-frac-neg 71.3%

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

      6. sqrt-unprod 60.7%

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

      7. add-sqr-sqrt 84.7%

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

      8. distribute-frac-neg 84.7%

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

      9. distribute-rgt-neg-in 84.7%

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

      10. neg-mul-1 84.7%

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

      11. *-commutative 84.7%

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

      12. associate-/l* 84.8%

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

   10. Applied egg-rr 84.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \cdot 10^{+42}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{\left(-u\right) - t1}\\ \mathbf{elif}\;u \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{\left(-u\right) - t1}}{t1}\\ \mathbf{else}:\\ \;\;\;\;\left(t1 \cdot \frac{v}{u}\right) \cdot \frac{-1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\frac{t1}{\frac{u}{v}}}{-1}}{u}\\ \mathbf{elif}\;u \leq 8.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\left(t1 \cdot \frac{v}{u}\right) \cdot \frac{-1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5e-79)
   (/ (/ (/ t1 (/ u v)) -1.0) u)
   (if (<= u 8.8e+14) (/ v (- t1)) (* (* t1 (/ v u)) (/ -1.0 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e-79) {
		tmp = ((t1 / (u / v)) / -1.0) / u;
	} else if (u <= 8.8e+14) {
		tmp = v / -t1;
	} else {
		tmp = (t1 * (v / u)) * (-1.0 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5d-79)) then
        tmp = ((t1 / (u / v)) / (-1.0d0)) / u
    else if (u <= 8.8d+14) then
        tmp = v / -t1
    else
        tmp = (t1 * (v / u)) * ((-1.0d0) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e-79) {
		tmp = ((t1 / (u / v)) / -1.0) / u;
	} else if (u <= 8.8e+14) {
		tmp = v / -t1;
	} else {
		tmp = (t1 * (v / u)) * (-1.0 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5e-79:
		tmp = ((t1 / (u / v)) / -1.0) / u
	elif u <= 8.8e+14:
		tmp = v / -t1
	else:
		tmp = (t1 * (v / u)) * (-1.0 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5e-79)
		tmp = Float64(Float64(Float64(t1 / Float64(u / v)) / -1.0) / u);
	elseif (u <= 8.8e+14)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) * Float64(-1.0 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5e-79)
		tmp = ((t1 / (u / v)) / -1.0) / u;
	elseif (u <= 8.8e+14)
		tmp = v / -t1;
	else
		tmp = (t1 * (v / u)) * (-1.0 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5e-79], N[(N[(N[(t1 / N[(u / v), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 8.8e+14], N[(v / (-t1)), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.99999999999999999e-79

   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* 72.3%

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

      2. distribute-lft-neg-out 72.3%

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

      3. distribute-rgt-neg-in 72.3%

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

      4. associate-/r* 86.0%

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

      5. distribute-neg-frac2 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in t1 around 0 75.0%

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

   6. Taylor expanded in t1 around 0 72.1%

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

      1. associate-*r/ 76.9%

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

      2. neg-mul-1 76.9%

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

      3. associate-/r* 76.9%

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

   8. Applied egg-rr 76.9%

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

   9. Taylor expanded in t1 around 0 71.3%

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

       1. *-rgt-identity 71.3%

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

       2. times-frac 70.0%

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

       3. /-rgt-identity 70.0%

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

       4. associate-/r/ 78.0%

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

   11. Simplified 78.0%

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

   if -4.99999999999999999e-79 < u < 8.8e14

   1. Initial program 63.4%

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

      1. associate-/l* 62.0%

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

      2. distribute-lft-neg-out 62.0%

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

      3. distribute-rgt-neg-in 62.0%

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

      4. associate-/r* 76.3%

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

      5. distribute-neg-frac2 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in t1 around inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. neg-mul-1 82.6%

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

   7. Simplified 82.6%

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

   if 8.8e14 < u

   1. Initial program 80.0%

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

      1. associate-/l* 78.9%

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

      2. distribute-lft-neg-out 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. associate-/r* 90.8%

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

      5. distribute-neg-frac2 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in t1 around 0 82.7%

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

   6. Taylor expanded in t1 around 0 82.2%

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

      1. associate-*r/ 84.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 61.4%

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

      4. sqr-neg 61.4%

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

      5. sqrt-unprod 59.7%

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

      6. add-sqr-sqrt 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-unprod 71.3%

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

      3. sqr-neg 71.3%

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

      4. distribute-frac-neg 71.3%

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

      5. distribute-frac-neg 71.3%

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

      6. sqrt-unprod 60.7%

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

      7. add-sqr-sqrt 84.7%

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

      8. distribute-frac-neg 84.7%

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

      9. distribute-rgt-neg-in 84.7%

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

      10. neg-mul-1 84.7%

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

      11. *-commutative 84.7%

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

      12. associate-/l* 84.8%

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

   10. Applied egg-rr 84.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\frac{t1}{\frac{u}{v}}}{-1}}{u}\\ \mathbf{elif}\;u \leq 8.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\left(t1 \cdot \frac{v}{u}\right) \cdot \frac{-1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -4.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{t\_1}{-u}\\ \mathbf{elif}\;u \leq 9.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{-1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ v u))))
   (if (<= u -4.1e-55)
     (/ t_1 (- u))
     (if (<= u 9.4e+14) (/ v (- t1)) (* t_1 (/ -1.0 u))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (v / u);
	double tmp;
	if (u <= -4.1e-55) {
		tmp = t_1 / -u;
	} else if (u <= 9.4e+14) {
		tmp = v / -t1;
	} else {
		tmp = t_1 * (-1.0 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t1 * (v / u)
    if (u <= (-4.1d-55)) then
        tmp = t_1 / -u
    else if (u <= 9.4d+14) then
        tmp = v / -t1
    else
        tmp = t_1 * ((-1.0d0) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * (v / u);
	double tmp;
	if (u <= -4.1e-55) {
		tmp = t_1 / -u;
	} else if (u <= 9.4e+14) {
		tmp = v / -t1;
	} else {
		tmp = t_1 * (-1.0 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * (v / u)
	tmp = 0
	if u <= -4.1e-55:
		tmp = t_1 / -u
	elif u <= 9.4e+14:
		tmp = v / -t1
	else:
		tmp = t_1 * (-1.0 / u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(v / u))
	tmp = 0.0
	if (u <= -4.1e-55)
		tmp = Float64(t_1 / Float64(-u));
	elseif (u <= 9.4e+14)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(t_1 * Float64(-1.0 / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * (v / u);
	tmp = 0.0;
	if (u <= -4.1e-55)
		tmp = t_1 / -u;
	elseif (u <= 9.4e+14)
		tmp = v / -t1;
	else
		tmp = t_1 * (-1.0 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.1e-55], N[(t$95$1 / (-u)), $MachinePrecision], If[LessEqual[u, 9.4e+14], N[(v / (-t1)), $MachinePrecision], N[(t$95$1 * N[(-1.0 / u), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.0999999999999998e-55

   1. Initial program 74.3%

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

      1. associate-/l* 74.0%

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

      2. distribute-lft-neg-out 74.0%

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

      3. distribute-rgt-neg-in 74.0%

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

      4. associate-/r* 86.8%

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

      5. distribute-neg-frac2 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in t1 around 0 76.8%

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

   6. Taylor expanded in t1 around 0 73.7%

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

      1. associate-*r/ 78.8%

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

      2. neg-mul-1 78.8%

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

      3. associate-/r* 78.8%

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

   8. Applied egg-rr 78.8%

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

      1. frac-2neg 78.8%

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

      2. metadata-eval 78.8%

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

      3. /-rgt-identity 78.8%

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

      4. *-commutative 78.8%

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

      5. distribute-rgt-neg-in 78.8%

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

   10. Applied egg-rr 78.8%

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

   if -4.0999999999999998e-55 < u < 9.4e14

   1. Initial program 63.5%

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

      1. associate-/l* 61.3%

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

      2. distribute-lft-neg-out 61.3%

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

      3. distribute-rgt-neg-in 61.3%

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

      4. associate-/r* 76.1%

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

      5. distribute-neg-frac2 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in t1 around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   7. Simplified 81.5%

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

   if 9.4e14 < u

   1. Initial program 80.0%

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

      1. associate-/l* 78.9%

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

      2. distribute-lft-neg-out 78.9%

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

      3. distribute-rgt-neg-in 78.9%

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

      4. associate-/r* 90.8%

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

      5. distribute-neg-frac2 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in t1 around 0 82.7%

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

   6. Taylor expanded in t1 around 0 82.2%

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

      1. associate-*r/ 84.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 61.4%

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

      4. sqr-neg 61.4%

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

      5. sqrt-unprod 59.7%

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

      6. add-sqr-sqrt 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. add-sqr-sqrt 38.2%

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

      2. sqrt-unprod 71.3%

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

      3. sqr-neg 71.3%

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

      4. distribute-frac-neg 71.3%

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

      5. distribute-frac-neg 71.3%

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

      6. sqrt-unprod 60.7%

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

      7. add-sqr-sqrt 84.7%

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

      8. distribute-frac-neg 84.7%

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

      9. distribute-rgt-neg-in 84.7%

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

      10. neg-mul-1 84.7%

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

      11. *-commutative 84.7%

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

      12. associate-/l* 84.8%

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

   10. Applied egg-rr 84.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{elif}\;u \leq 9.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\left(t1 \cdot \frac{v}{u}\right) \cdot \frac{-1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.4 \cdot 10^{-55} \lor \neg \left(u \leq 8.8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.4e-55) (not (<= u 8.8e+14)))
   (/ (* t1 (/ v u)) (- u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.4e-55) || !(u <= 8.8e+14)) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.4d-55)) .or. (.not. (u <= 8.8d+14))) then
        tmp = (t1 * (v / u)) / -u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.4e-55) || !(u <= 8.8e+14)) {
		tmp = (t1 * (v / u)) / -u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.4e-55) or not (u <= 8.8e+14):
		tmp = (t1 * (v / u)) / -u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.4e-55) || !(u <= 8.8e+14))
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.4e-55) || ~((u <= 8.8e+14)))
		tmp = (t1 * (v / u)) / -u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.4e-55], N[Not[LessEqual[u, 8.8e+14]], $MachinePrecision]], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.3999999999999999e-55 or 8.8e14 < u

   1. Initial program 77.2%

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

      1. associate-/l* 76.5%

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

      2. distribute-lft-neg-out 76.5%

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

      3. distribute-rgt-neg-in 76.5%

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

      4. associate-/r* 88.8%

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

      5. distribute-neg-frac2 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in t1 around 0 79.8%

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

   6. Taylor expanded in t1 around 0 78.1%

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

      1. associate-*r/ 81.8%

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

      2. neg-mul-1 81.8%

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

      3. associate-/r* 81.8%

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

   8. Applied egg-rr 81.8%

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

      1. frac-2neg 81.8%

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

      2. metadata-eval 81.8%

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

      3. /-rgt-identity 81.8%

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

      4. *-commutative 81.8%

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

      5. distribute-rgt-neg-in 81.8%

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

   10. Applied egg-rr 81.8%

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

   if -4.3999999999999999e-55 < u < 8.8e14

   1. Initial program 63.5%

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

      1. associate-/l* 61.3%

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

      2. distribute-lft-neg-out 61.3%

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

      3. distribute-rgt-neg-in 61.3%

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

      4. associate-/r* 76.1%

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

      5. distribute-neg-frac2 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in t1 around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.4 \cdot 10^{-55} \lor \neg \left(u \leq 8.8 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{-55} \lor \neg \left(u \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.3e-55) (not (<= u 2.2e+15)))
   (* t1 (/ (/ v (- u)) u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e-55) || !(u <= 2.2e+15)) {
		tmp = t1 * ((v / -u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.3d-55)) .or. (.not. (u <= 2.2d+15))) then
        tmp = t1 * ((v / -u) / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e-55) || !(u <= 2.2e+15)) {
		tmp = t1 * ((v / -u) / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.3e-55) or not (u <= 2.2e+15):
		tmp = t1 * ((v / -u) / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.3e-55) || !(u <= 2.2e+15))
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / u));
	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-55) || ~((u <= 2.2e+15)))
		tmp = t1 * ((v / -u) / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.3e-55], N[Not[LessEqual[u, 2.2e+15]], $MachinePrecision]], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.30000000000000011e-55 or 2.2e15 < u

   1. Initial program 77.2%

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

      1. associate-/l* 76.5%

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

      2. distribute-lft-neg-out 76.5%

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

      3. distribute-rgt-neg-in 76.5%

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

      4. associate-/r* 88.8%

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

      5. distribute-neg-frac2 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in t1 around 0 79.8%

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

   6. Taylor expanded in t1 around 0 78.1%

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

   if -2.30000000000000011e-55 < u < 2.2e15

   1. Initial program 63.5%

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

      1. associate-/l* 61.3%

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

      2. distribute-lft-neg-out 61.3%

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

      3. distribute-rgt-neg-in 61.3%

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

      4. associate-/r* 76.1%

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

      5. distribute-neg-frac2 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in t1 around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{-55} \lor \neg \left(u \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+44} \lor \neg \left(u \leq 3.9 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5e+44) (not (<= u 3.9e+16)))
   (/ t1 (/ u (/ v u)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e+44) || !(u <= 3.9e+16)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5d+44)) .or. (.not. (u <= 3.9d+16))) then
        tmp = t1 / (u / (v / u))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5e+44) || !(u <= 3.9e+16)) {
		tmp = t1 / (u / (v / u));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5e+44) or not (u <= 3.9e+16):
		tmp = t1 / (u / (v / u))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5e+44) || !(u <= 3.9e+16))
		tmp = Float64(t1 / Float64(u / 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 <= -5e+44) || ~((u <= 3.9e+16)))
		tmp = t1 / (u / (v / u));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5e+44], N[Not[LessEqual[u, 3.9e+16]], $MachinePrecision]], N[(t1 / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.9999999999999996e44 or 3.9e16 < u

   1. Initial program 78.8%

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

      1. associate-/l* 77.1%

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

      2. distribute-lft-neg-out 77.1%

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

      3. distribute-rgt-neg-in 77.1%

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

      4. associate-/r* 89.5%

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

      5. distribute-neg-frac2 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in t1 around 0 82.8%

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

   6. Taylor expanded in t1 around 0 82.5%

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

      1. clear-num 82.0%

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

      2. un-div-inv 82.1%

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

      3. add-sqr-sqrt 31.7%

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

      4. sqrt-unprod 65.9%

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

      5. sqr-neg 65.9%

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

      6. sqrt-unprod 37.5%

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

      7. add-sqr-sqrt 60.5%

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

   8. Applied egg-rr 60.5%

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

   if -4.9999999999999996e44 < u < 3.9e16

   1. Initial program 64.3%

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

      1. associate-/l* 63.2%

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

      2. distribute-lft-neg-out 63.2%

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

      3. distribute-rgt-neg-in 63.2%

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

      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 76.9%

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

      1. associate-*r/ 76.9%

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

      2. neg-mul-1 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{+44} \lor \neg \left(u \leq 3.9 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+43} \lor \neg \left(u \leq 2.6 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.2e+43) (not (<= u 2.6e+16)))
   (* (/ v u) (/ t1 u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+43) || !(u <= 2.6e+16)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.2d+43)) .or. (.not. (u <= 2.6d+16))) then
        tmp = (v / u) * (t1 / u)
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+43) || !(u <= 2.6e+16)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.2e+43) or not (u <= 2.6e+16):
		tmp = (v / u) * (t1 / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.2e+43) || !(u <= 2.6e+16))
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.2e+43) || ~((u <= 2.6e+16)))
		tmp = (v / u) * (t1 / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.2e+43], N[Not[LessEqual[u, 2.6e+16]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.20000000000000012e43 or 2.6e16 < u

   1. Initial program 78.8%

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

      1. associate-/l* 77.1%

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

      2. distribute-lft-neg-out 77.1%

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

      3. distribute-rgt-neg-in 77.1%

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

      4. associate-/r* 89.5%

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

      5. distribute-neg-frac2 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in t1 around 0 82.8%

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

   6. Taylor expanded in t1 around 0 82.5%

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

      1. associate-*r/ 87.0%

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

      2. add-sqr-sqrt 35.0%

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

      3. sqrt-unprod 67.4%

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

      4. sqr-neg 67.4%

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

      5. sqrt-unprod 36.6%

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

      6. add-sqr-sqrt 59.5%

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

   8. Applied egg-rr 59.5%

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

      1. *-commutative 59.5%

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

      2. associate-/l* 59.6%

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

   10. Applied egg-rr 59.6%

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

   if -1.20000000000000012e43 < u < 2.6e16

   1. Initial program 64.3%

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

      1. associate-/l* 63.2%

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

      2. distribute-lft-neg-out 63.2%

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

      3. distribute-rgt-neg-in 63.2%

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

      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 76.9%

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

      1. associate-*r/ 76.9%

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

      2. neg-mul-1 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+43} \lor \neg \left(u \leq 2.6 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;v \leq 3.4 \cdot 10^{+205}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t\_1}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= v 3.4e+205)
     (* (/ v (+ t1 u)) (/ t1 t_1))
     (* t1 (/ (/ v t_1) (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (v <= 3.4e+205) {
		tmp = (v / (t1 + u)) * (t1 / t_1);
	} else {
		tmp = t1 * ((v / t_1) / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -u - t1
    if (v <= 3.4d+205) then
        tmp = (v / (t1 + u)) * (t1 / t_1)
    else
        tmp = t1 * ((v / t_1) / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (v <= 3.4e+205) {
		tmp = (v / (t1 + u)) * (t1 / t_1);
	} else {
		tmp = t1 * ((v / t_1) / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if v <= 3.4e+205:
		tmp = (v / (t1 + u)) * (t1 / t_1)
	else:
		tmp = t1 * ((v / t_1) / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (v <= 3.4e+205)
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / t_1));
	else
		tmp = Float64(t1 * Float64(Float64(v / t_1) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (v <= 3.4e+205)
		tmp = (v / (t1 + u)) * (t1 / t_1);
	else
		tmp = t1 * ((v / t_1) / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[v, 3.4e+205], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / t$95$1), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 3.4e205

   1. Initial program 72.2%

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

      1. times-frac 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-neg-frac2 97.5%

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

      4. +-commutative 97.5%

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

      5. distribute-neg-in 97.5%

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

      6. unsub-neg 97.5%

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

   3. Simplified 97.5%

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

   if 3.4e205 < v

   1. Initial program 51.7%

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

      1. associate-/l* 48.0%

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

      2. distribute-lft-neg-out 48.0%

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

      3. distribute-rgt-neg-in 48.0%

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

      4. associate-/r* 85.9%

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

      5. distribute-neg-frac2 85.9%

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

   3. Simplified 85.9%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 3.4 \cdot 10^{+205}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{\left(-u\right) - t1}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9.5e+112)
   (/ 1.0 (/ u v))
   (if (<= u 1.6e+136) (/ v (- t1)) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9.5e+112) {
		tmp = 1.0 / (u / v);
	} else if (u <= 1.6e+136) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-9.5d+112)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 1.6d+136) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9.5e+112) {
		tmp = 1.0 / (u / v);
	} else if (u <= 1.6e+136) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9.5e+112:
		tmp = 1.0 / (u / v)
	elif u <= 1.6e+136:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9.5e+112)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 1.6e+136)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -9.5e+112)
		tmp = 1.0 / (u / v);
	elseif (u <= 1.6e+136)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9.5e+112], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.6e+136], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -9.5000000000000008e112

   1. Initial program 76.8%

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

      1. associate-/l* 77.2%

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

      2. distribute-lft-neg-out 77.2%

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

      3. distribute-rgt-neg-in 77.2%

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

      4. associate-/r* 92.9%

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

      5. distribute-neg-frac2 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in t1 around 0 90.0%

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

   6. Taylor expanded in t1 around inf 25.0%

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

      1. associate-*r/ 25.0%

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

      2. mul-1-neg 25.0%

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

   8. Simplified 25.0%

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

      1. add-sqr-sqrt 20.8%

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

      2. sqrt-unprod 51.4%

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

      3. distribute-frac-neg 51.4%

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

      4. distribute-frac-neg 51.4%

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

      5. sqr-neg 51.4%

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

      6. sqrt-unprod 20.9%

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

      7. add-sqr-sqrt 25.5%

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

      8. clear-num 27.9%

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

      9. inv-pow 27.9%

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

   10. Applied egg-rr 27.9%

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

       1. unpow-1 27.9%

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

   12. Simplified 27.9%

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

   if -9.5000000000000008e112 < u < 1.59999999999999994e136

   1. Initial program 68.0%

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

      1. associate-/l* 65.9%

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

      2. distribute-lft-neg-out 65.9%

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

      3. distribute-rgt-neg-in 65.9%

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

      4. associate-/r* 79.2%

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

      5. distribute-neg-frac2 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in t1 around inf 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   7. Simplified 65.8%

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

   if 1.59999999999999994e136 < 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* 78.4%

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

      2. distribute-lft-neg-out 78.4%

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

      3. distribute-rgt-neg-in 78.4%

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

      4. associate-/r* 90.9%

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

      5. distribute-neg-frac2 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in t1 around 0 84.5%

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

   6. Taylor expanded in t1 around inf 41.7%

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

      1. associate-*r/ 41.7%

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

      2. mul-1-neg 41.7%

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

   8. Simplified 41.7%

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

      1. add-sqr-sqrt 40.8%

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

      2. sqrt-unprod 53.5%

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

      3. distribute-frac-neg 53.5%

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

      4. distribute-frac-neg 53.5%

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

      5. sqr-neg 53.5%

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

      6. sqrt-unprod 40.2%

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

      7. add-sqr-sqrt 41.8%

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

      8. div-inv 41.8%

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

   10. Applied egg-rr 41.8%

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

       1. associate-*r/ 41.8%

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

       2. *-rgt-identity 41.8%

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

   12. Simplified 41.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 21.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.15 \cdot 10^{+232} \lor \neg \left(t1 \leq 2.8 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.15e+232) (not (<= t1 2.8e+185))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.15e+232) || !(t1 <= 2.8e+185)) {
		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 <= (-2.15d+232)) .or. (.not. (t1 <= 2.8d+185))) 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 <= -2.15e+232) || !(t1 <= 2.8e+185)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.15e+232) or not (t1 <= 2.8e+185):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.15e+232) || !(t1 <= 2.8e+185))
		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 <= -2.15e+232) || ~((t1 <= 2.8e+185)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.15e+232], N[Not[LessEqual[t1, 2.8e+185]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.1500000000000001e232 or 2.79999999999999982e185 < t1

   1. Initial program 44.8%

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

      1. associate-/l* 46.4%

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

      2. distribute-lft-neg-out 46.4%

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

      3. distribute-rgt-neg-in 46.4%

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

      4. associate-/r* 64.4%

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

      5. distribute-neg-frac2 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in t1 around inf 91.5%

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

      1. associate-*r/ 91.5%

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

      2. neg-mul-1 91.5%

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

   7. Simplified 91.5%

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

      1. *-un-lft-identity 91.5%

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

      2. *-commutative 91.5%

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

      3. add-sqr-sqrt 50.2%

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

      4. sqrt-unprod 63.5%

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

      5. sqr-neg 63.5%

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

      6. sqrt-unprod 13.6%

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

      7. add-sqr-sqrt 45.4%

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

   9. Applied egg-rr 45.4%

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

       1. *-rgt-identity 45.4%

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

   11. Simplified 45.4%

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

   if -2.1500000000000001e232 < t1 < 2.79999999999999982e185

   1. Initial program 74.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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* 85.3%

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

      5. distribute-neg-frac2 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in t1 around 0 54.4%

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

   6. Taylor expanded in t1 around inf 16.2%

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

      1. associate-*r/ 16.2%

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

      2. mul-1-neg 16.2%

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

   8. Simplified 16.2%

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

      1. add-sqr-sqrt 13.9%

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

      2. sqrt-unprod 26.4%

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

      3. distribute-frac-neg 26.4%

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

      4. distribute-frac-neg 26.4%

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

      5. sqr-neg 26.4%

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

      6. sqrt-unprod 11.9%

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

      7. add-sqr-sqrt 14.5%

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

      8. div-inv 14.5%

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

   10. Applied egg-rr 14.5%

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

       1. associate-*r/ 14.5%

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

       2. *-rgt-identity 14.5%

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

   12. Simplified 14.5%

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

4. Final simplification 18.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.15 \cdot 10^{+232} \lor \neg \left(t1 \leq 2.8 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. associate-/l* 69.2%

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

   2. distribute-lft-neg-out 69.2%

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

   3. distribute-rgt-neg-in 69.2%

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

   4. associate-/r* 82.7%

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

   5. distribute-neg-frac2 82.7%

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

3. Simplified 82.7%

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

   1. distribute-frac-neg2 82.7%

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

   2. distribute-rgt-neg-out 82.7%

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

   3. associate-/r* 69.2%

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

   4. distribute-lft-neg-out 69.2%

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

   5. associate-/l* 70.6%

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

   6. times-frac 95.5%

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

   7. frac-2neg 95.5%

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

   8. associate-*r/ 96.4%

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

   9. add-sqr-sqrt 46.7%

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

   10. sqrt-unprod 44.8%

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

   11. sqr-neg 44.8%

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

   12. sqrt-unprod 18.5%

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

   13. add-sqr-sqrt 33.6%

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

   14. add-sqr-sqrt 14.2%

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

   15. sqrt-unprod 53.4%

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

   16. sqr-neg 53.4%

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

   17. sqrt-prod 50.1%

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

   18. add-sqr-sqrt 96.4%

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

6. Applied egg-rr 96.4%

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

7. Final simplification 96.4%

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

Alternative 15: 56.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 8.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (if (<= u 8.2e+138) (/ v (- t1)) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 8.2e+138) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 8.2d+138) then
        tmp = v / -t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 8.2e+138) {
		tmp = v / -t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 8.2e+138:
		tmp = v / -t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 8.2e+138)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 8.2e+138)
		tmp = v / -t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 8.2e+138], N[(v / (-t1)), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 8.19999999999999961e138

   1. Initial program 69.2%

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

      1. associate-/l* 67.5%

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

      2. distribute-lft-neg-out 67.5%

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

      3. distribute-rgt-neg-in 67.5%

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

      4. associate-/r* 81.1%

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

      5. distribute-neg-frac2 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in t1 around inf 58.5%

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

      1. associate-*r/ 58.5%

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

      2. neg-mul-1 58.5%

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

   7. Simplified 58.5%

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

   if 8.19999999999999961e138 < 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* 78.4%

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

      2. distribute-lft-neg-out 78.4%

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

      3. distribute-rgt-neg-in 78.4%

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

      4. associate-/r* 90.9%

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

      5. distribute-neg-frac2 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in t1 around 0 84.5%

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

   6. Taylor expanded in t1 around inf 41.7%

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

      1. associate-*r/ 41.7%

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

      2. mul-1-neg 41.7%

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

   8. Simplified 41.7%

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

      1. add-sqr-sqrt 40.8%

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

      2. sqrt-unprod 53.5%

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

      3. distribute-frac-neg 53.5%

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

      4. distribute-frac-neg 53.5%

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

      5. sqr-neg 53.5%

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

      6. sqrt-unprod 40.2%

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

      7. add-sqr-sqrt 41.8%

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

      8. div-inv 41.8%

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

   10. Applied egg-rr 41.8%

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

       1. associate-*r/ 41.8%

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

       2. *-rgt-identity 41.8%

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

   12. Simplified 41.8%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 8.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.9% accurate, 2.0× speedup

Math

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

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(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 70.6%

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

   1. associate-/l* 69.2%

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

   2. distribute-lft-neg-out 69.2%

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

   3. distribute-rgt-neg-in 69.2%

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

   4. associate-/r* 82.7%

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

   5. distribute-neg-frac2 82.7%

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

3. Simplified 82.7%

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

   1. associate-*r/ 97.9%

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

   2. neg-mul-1 97.9%

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

   3. associate-/r* 97.9%

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

6. Applied egg-rr 97.9%

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

7. Taylor expanded in t1 around inf 59.6%

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

   1. mul-1-neg 59.6%

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

9. Simplified 59.6%

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

10. Final simplification 59.6%

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

Alternative 17: 14.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. associate-/l* 69.2%

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

   2. distribute-lft-neg-out 69.2%

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

   3. distribute-rgt-neg-in 69.2%

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

   4. associate-/r* 82.7%

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

   5. distribute-neg-frac2 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in t1 around inf 52.7%

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

   1. associate-*r/ 52.7%

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

   2. neg-mul-1 52.7%

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

7. Simplified 52.7%

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

   1. *-un-lft-identity 52.7%

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

   2. *-commutative 52.7%

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

   3. add-sqr-sqrt 24.1%

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

   4. sqrt-unprod 31.6%

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

   5. sqr-neg 31.6%

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

   6. sqrt-unprod 3.9%

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

   7. add-sqr-sqrt 10.3%

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

9. Applied egg-rr 10.3%

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

    1. *-rgt-identity 10.3%

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

11. Simplified 10.3%

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

Reproduce

herbie shell --seed 2024133 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))