Rosa's DopplerBench

Percentage Accurate: 72.6% → 97.9%

Time: 9.6s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% 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: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. associate-/l* 73.0%

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

   2. neg-mul-1 73.0%

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

   3. associate-*r/ 84.3%

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

   4. times-frac 96.4%

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

   5. div-inv 96.3%

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

   6. clear-num 97.7%

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

3. Applied egg-rr 97.7%

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

4. Final simplification 97.7%

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

Alternative 2: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -3 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -4.9 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (* (- v) (/ t1 (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -3e+98)
     t_1
     (if (<= t1 -4.9e-144)
       t_2
       (if (<= t1 6e-185)
         (/ (/ (- t1) (/ u v)) (+ t1 u))
         (if (<= t1 2.4e+144) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = -v * (t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -3e+98) {
		tmp = t_1;
	} else if (t1 <= -4.9e-144) {
		tmp = t_2;
	} else if (t1 <= 6e-185) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (t1 <= 2.4e+144) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    t_2 = -v * (t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-3d+98)) then
        tmp = t_1
    else if (t1 <= (-4.9d-144)) then
        tmp = t_2
    else if (t1 <= 6d-185) then
        tmp = (-t1 / (u / v)) / (t1 + u)
    else if (t1 <= 2.4d+144) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = -v * (t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -3e+98) {
		tmp = t_1;
	} else if (t1 <= -4.9e-144) {
		tmp = t_2;
	} else if (t1 <= 6e-185) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (t1 <= 2.4e+144) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	t_2 = -v * (t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -3e+98:
		tmp = t_1
	elif t1 <= -4.9e-144:
		tmp = t_2
	elif t1 <= 6e-185:
		tmp = (-t1 / (u / v)) / (t1 + u)
	elif t1 <= 2.4e+144:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	t_2 = Float64(Float64(-v) * Float64(t1 / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -3e+98)
		tmp = t_1;
	elseif (t1 <= -4.9e-144)
		tmp = t_2;
	elseif (t1 <= 6e-185)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	elseif (t1 <= 2.4e+144)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	t_2 = -v * (t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -3e+98)
		tmp = t_1;
	elseif (t1 <= -4.9e-144)
		tmp = t_2;
	elseif (t1 <= 6e-185)
		tmp = (-t1 / (u / v)) / (t1 + u);
	elseif (t1 <= 2.4e+144)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-v) * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3e+98], t$95$1, If[LessEqual[t1, -4.9e-144], t$95$2, If[LessEqual[t1, 6e-185], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.4e+144], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.0000000000000001e98 or 2.4000000000000001e144 < t1

   1. Initial program 45.6%

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

      1. associate-/r* 60.5%

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

      2. associate-/l* 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around inf 93.9%

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

      1. neg-mul-1 93.9%

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

   6. Simplified 93.9%

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

   if -3.0000000000000001e98 < t1 < -4.9000000000000001e-144 or 6.00000000000000061e-185 < t1 < 2.4000000000000001e144

   1. Initial program 86.9%

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

      1. associate-*l/ 92.6%

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

      2. *-commutative 92.6%

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

   3. Simplified 92.6%

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

   if -4.9000000000000001e-144 < t1 < 6.00000000000000061e-185

   1. Initial program 85.6%

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

      1. associate-/r* 93.4%

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

      2. associate-/l* 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in t1 around 0 91.8%

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

      1. mul-1-neg 91.8%

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

      2. associate-/l* 92.0%

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

      3. distribute-neg-frac 92.0%

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

   6. Simplified 92.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{+98}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -4.9 \cdot 10^{-144}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{+144}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 3: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t1 + u\right) \cdot \left(t1 + u\right)\\ t_2 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.85 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -4.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{t1 \cdot \left(-v\right)}{t_1}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-189}:\\ \;\;\;\;\frac{-v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (+ t1 u) (+ t1 u))) (t_2 (/ (- v) (+ t1 u))))
   (if (<= t1 -1.85e+96)
     t_2
     (if (<= t1 -4.2e-204)
       (/ (* t1 (- v)) t_1)
       (if (<= t1 1.1e-189)
         (/ (- v) (/ u (/ t1 u)))
         (if (<= t1 2.5e+142) (* (- v) (/ t1 t_1)) t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 + u) * (t1 + u);
	double t_2 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1.85e+96) {
		tmp = t_2;
	} else if (t1 <= -4.2e-204) {
		tmp = (t1 * -v) / t_1;
	} else if (t1 <= 1.1e-189) {
		tmp = -v / (u / (t1 / u));
	} else if (t1 <= 2.5e+142) {
		tmp = -v * (t1 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t1 + u) * (t1 + u)
    t_2 = -v / (t1 + u)
    if (t1 <= (-1.85d+96)) then
        tmp = t_2
    else if (t1 <= (-4.2d-204)) then
        tmp = (t1 * -v) / t_1
    else if (t1 <= 1.1d-189) then
        tmp = -v / (u / (t1 / u))
    else if (t1 <= 2.5d+142) then
        tmp = -v * (t1 / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 + u) * (t1 + u);
	double t_2 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1.85e+96) {
		tmp = t_2;
	} else if (t1 <= -4.2e-204) {
		tmp = (t1 * -v) / t_1;
	} else if (t1 <= 1.1e-189) {
		tmp = -v / (u / (t1 / u));
	} else if (t1 <= 2.5e+142) {
		tmp = -v * (t1 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 + u) * (t1 + u)
	t_2 = -v / (t1 + u)
	tmp = 0
	if t1 <= -1.85e+96:
		tmp = t_2
	elif t1 <= -4.2e-204:
		tmp = (t1 * -v) / t_1
	elif t1 <= 1.1e-189:
		tmp = -v / (u / (t1 / u))
	elif t1 <= 2.5e+142:
		tmp = -v * (t1 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 + u) * Float64(t1 + u))
	t_2 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1.85e+96)
		tmp = t_2;
	elseif (t1 <= -4.2e-204)
		tmp = Float64(Float64(t1 * Float64(-v)) / t_1);
	elseif (t1 <= 1.1e-189)
		tmp = Float64(Float64(-v) / Float64(u / Float64(t1 / u)));
	elseif (t1 <= 2.5e+142)
		tmp = Float64(Float64(-v) * Float64(t1 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 + u) * (t1 + u);
	t_2 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -1.85e+96)
		tmp = t_2;
	elseif (t1 <= -4.2e-204)
		tmp = (t1 * -v) / t_1;
	elseif (t1 <= 1.1e-189)
		tmp = -v / (u / (t1 / u));
	elseif (t1 <= 2.5e+142)
		tmp = -v * (t1 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.85e+96], t$95$2, If[LessEqual[t1, -4.2e-204], N[(N[(t1 * (-v)), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t1, 1.1e-189], N[((-v) / N[(u / N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.5e+142], N[((-v) * N[(t1 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.84999999999999996e96 or 2.5000000000000001e142 < t1

   1. Initial program 45.1%

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

      1. associate-/r* 59.8%

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

      2. associate-/l* 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in t1 around inf 94.0%

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

      1. neg-mul-1 94.0%

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

   6. Simplified 94.0%

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

   if -1.84999999999999996e96 < t1 < -4.20000000000000018e-204

   1. Initial program 94.6%

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

   if -4.20000000000000018e-204 < t1 < 1.1000000000000001e-189

   1. Initial program 81.6%

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

      1. associate-/l* 81.1%

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

      2. neg-mul-1 81.1%

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

      3. associate-*r/ 85.1%

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

      4. times-frac 93.2%

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

      5. div-inv 93.2%

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

      6. clear-num 93.2%

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

   3. Applied egg-rr 93.2%

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

   4. Taylor expanded in t1 around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. *-commutative 81.6%

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

      3. unpow2 81.6%

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

      4. times-frac 91.0%

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

      5. associate-*l/ 88.8%

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

      6. associate-/l* 93.1%

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

   6. Simplified 93.1%

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

   if 1.1000000000000001e-189 < t1 < 2.5000000000000001e142

   1. Initial program 82.8%

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

      1. associate-*l/ 89.8%

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

      2. *-commutative 89.8%

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

   3. Simplified 89.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{+96}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -4.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-189}:\\ \;\;\;\;\frac{-v}{\frac{u}{\frac{t1}{u}}}\\ \mathbf{elif}\;t1 \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 4: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-49} \lor \neg \left(u \leq 2.8 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{t1}{\frac{t1 - u}{\frac{v}{t1 + u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.8e-49) (not (<= u 2.8e-79)))
   (/ t1 (/ (- t1 u) (/ v (+ t1 u))))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e-49) || !(u <= 2.8e-79)) {
		tmp = t1 / ((t1 - u) / (v / (t1 + u)));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.8d-49)) .or. (.not. (u <= 2.8d-79))) then
        tmp = t1 / ((t1 - u) / (v / (t1 + u)))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e-49) || !(u <= 2.8e-79)) {
		tmp = t1 / ((t1 - u) / (v / (t1 + u)));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.8e-49) or not (u <= 2.8e-79):
		tmp = t1 / ((t1 - u) / (v / (t1 + u)))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.8e-49) || !(u <= 2.8e-79))
		tmp = Float64(t1 / Float64(Float64(t1 - u) / Float64(v / Float64(t1 + u))));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.8e-49) || ~((u <= 2.8e-79)))
		tmp = t1 / ((t1 - u) / (v / (t1 + u)));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.8e-49], N[Not[LessEqual[u, 2.8e-79]], $MachinePrecision]], N[(t1 / N[(N[(t1 - u), $MachinePrecision] / N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.8e-49 or 2.80000000000000012e-79 < u

   1. Initial program 78.6%

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

      1. associate-*l/ 79.0%

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

      2. *-commutative 79.0%

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

   3. Simplified 79.0%

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

      1. associate-/r* 91.7%

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

      2. associate-*r/ 97.2%

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

      3. *-commutative 97.2%

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

      4. associate-/r/ 98.8%

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

      5. frac-2neg 98.8%

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

      6. distribute-frac-neg 98.8%

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

      7. remove-double-neg 98.8%

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

      8. div-inv 98.8%

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

      9. clear-num 98.9%

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

      10. distribute-neg-in 98.9%

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

      11. add-sqr-sqrt 52.5%

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

      12. sqrt-unprod 87.1%

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

      13. sqr-neg 87.1%

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

      14. sqrt-unprod 40.8%

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

      15. add-sqr-sqrt 84.1%

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

      16. sub-neg 84.1%

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

   5. Applied egg-rr 84.1%

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

      1. associate-/l* 82.8%

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

   7. Simplified 82.8%

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

   if -5.8e-49 < u < 2.80000000000000012e-79

   1. Initial program 65.9%

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

      1. associate-*l/ 72.8%

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

      2. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   6. Simplified 85.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-49} \lor \neg \left(u \leq 2.8 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{t1}{\frac{t1 - u}{\frac{v}{t1 + u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 5: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{t1}{\frac{t1 - u}{\frac{v}{t1 + u}}}\\ \mathbf{elif}\;u \leq 1.9 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.8e-49)
   (/ t1 (/ (- t1 u) (/ v (+ t1 u))))
   (if (<= u 1.9e-79) (/ (- v) t1) (/ (/ t1 (+ t1 u)) (/ (- t1 u) v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e-49) {
		tmp = t1 / ((t1 - u) / (v / (t1 + u)));
	} else if (u <= 1.9e-79) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.8d-49)) then
        tmp = t1 / ((t1 - u) / (v / (t1 + u)))
    else if (u <= 1.9d-79) then
        tmp = -v / t1
    else
        tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.8e-49) {
		tmp = t1 / ((t1 - u) / (v / (t1 + u)));
	} else if (u <= 1.9e-79) {
		tmp = -v / t1;
	} else {
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.8e-49:
		tmp = t1 / ((t1 - u) / (v / (t1 + u)))
	elif u <= 1.9e-79:
		tmp = -v / t1
	else:
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.8e-49)
		tmp = Float64(t1 / Float64(Float64(t1 - u) / Float64(v / Float64(t1 + u))));
	elseif (u <= 1.9e-79)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 / Float64(t1 + u)) / Float64(Float64(t1 - u) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.8e-49)
		tmp = t1 / ((t1 - u) / (v / (t1 + u)));
	elseif (u <= 1.9e-79)
		tmp = -v / t1;
	else
		tmp = (t1 / (t1 + u)) / ((t1 - u) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.8e-49], N[(t1 / N[(N[(t1 - u), $MachinePrecision] / N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.9e-79], N[((-v) / t1), $MachinePrecision], N[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.79999999999999985e-49

   1. Initial program 78.4%

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

      1. associate-*l/ 78.4%

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

      2. *-commutative 78.4%

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

   3. Simplified 78.4%

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

      1. associate-/r* 92.1%

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

      2. associate-*r/ 96.4%

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

      3. *-commutative 96.4%

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

      4. associate-/r/ 98.6%

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

      5. frac-2neg 98.6%

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

      6. distribute-frac-neg 98.6%

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

      7. remove-double-neg 98.6%

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

      8. div-inv 98.6%

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

      9. clear-num 98.7%

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

      10. distribute-neg-in 98.7%

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

      11. add-sqr-sqrt 48.0%

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

      12. sqrt-unprod 87.5%

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

      13. sqr-neg 87.5%

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

      14. sqrt-unprod 41.9%

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

      15. add-sqr-sqrt 83.6%

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

      16. sub-neg 83.6%

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

   5. Applied egg-rr 83.6%

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

      1. associate-/l* 84.0%

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

   7. Simplified 84.0%

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

   if -4.79999999999999985e-49 < u < 1.9000000000000001e-79

   1. Initial program 65.9%

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

      1. associate-*l/ 72.8%

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

      2. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   6. Simplified 85.4%

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

   if 1.9000000000000001e-79 < 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/ 79.5%

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

      2. *-commutative 79.5%

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

   3. Simplified 79.5%

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

      1. associate-/r* 91.3%

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

      2. associate-*r/ 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-/r/ 99.0%

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

      5. div-inv 99.0%

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

      6. clear-num 98.7%

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

      7. associate-/r/ 98.9%

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

      8. clear-num 98.9%

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

      9. *-commutative 98.9%

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

      10. clear-num 98.9%

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

      11. div-inv 99.0%

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

      12. frac-2neg 99.0%

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

      13. remove-double-neg 99.0%

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

      14. associate-*l/ 98.6%

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

   5. Applied egg-rr 84.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{t1}{\frac{t1 - u}{\frac{v}{t1 + u}}}\\ \mathbf{elif}\;u \leq 1.9 \cdot 10^{-79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{t1 + u}}{\frac{t1 - u}{v}}\\ \end{array} \]

Alternative 6: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.8e-45)
   (/ (- t1) (/ u (/ v u)))
   (if (<= u 3.1e+19) (/ (- v) t1) (/ (/ (- t1) (/ u v)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e-45) {
		tmp = -t1 / (u / (v / u));
	} else if (u <= 3.1e+19) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (u / v)) / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.8d-45)) then
        tmp = -t1 / (u / (v / u))
    else if (u <= 3.1d+19) then
        tmp = -v / t1
    else
        tmp = (-t1 / (u / v)) / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.8e-45) {
		tmp = -t1 / (u / (v / u));
	} else if (u <= 3.1e+19) {
		tmp = -v / t1;
	} else {
		tmp = (-t1 / (u / v)) / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.8e-45:
		tmp = -t1 / (u / (v / u))
	elif u <= 3.1e+19:
		tmp = -v / t1
	else:
		tmp = (-t1 / (u / v)) / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.8e-45)
		tmp = Float64(Float64(-t1) / Float64(u / Float64(v / u)));
	elseif (u <= 3.1e+19)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.8e-45)
		tmp = -t1 / (u / (v / u));
	elseif (u <= 3.1e+19)
		tmp = -v / t1;
	else
		tmp = (-t1 / (u / v)) / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.8e-45], N[((-t1) / N[(u / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.1e+19], N[((-v) / t1), $MachinePrecision], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.79999999999999997e-45

   1. Initial program 78.9%

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

      1. associate-*l/ 77.7%

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

      2. *-commutative 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in t1 around 0 69.1%

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

      1. mul-1-neg 69.1%

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

      2. associate-/l* 73.1%

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

      3. distribute-neg-frac 73.1%

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

      4. unpow2 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in u around 0 73.1%

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

      1. unpow2 73.1%

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

      2. associate-/l* 80.8%

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

   9. Simplified 80.8%

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

   if -3.79999999999999997e-45 < u < 3.1e19

   1. Initial program 67.0%

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

      1. associate-*l/ 75.8%

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

      2. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in t1 around inf 79.5%

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

      1. associate-*r/ 79.5%

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

      2. neg-mul-1 79.5%

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

   6. Simplified 79.5%

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

   if 3.1e19 < u

   1. Initial program 80.3%

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

      1. associate-/r* 87.3%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 83.9%

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

      1. mul-1-neg 83.9%

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

      2. associate-/l* 89.0%

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

      3. distribute-neg-frac 89.0%

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

   6. Simplified 89.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \end{array} \]

Alternative 7: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{-83} \lor \neg \left(t1 \leq 175\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\frac{u}{\frac{t1}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -6e-83) (not (<= t1 175.0)))
   (/ (- v) (+ t1 u))
   (/ (- v) (/ u (/ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6e-83) || !(t1 <= 175.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -v / (u / (t1 / u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-6d-83)) .or. (.not. (t1 <= 175.0d0))) then
        tmp = -v / (t1 + u)
    else
        tmp = -v / (u / (t1 / u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6e-83) || !(t1 <= 175.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -v / (u / (t1 / u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -6e-83) or not (t1 <= 175.0):
		tmp = -v / (t1 + u)
	else:
		tmp = -v / (u / (t1 / u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -6e-83) || !(t1 <= 175.0))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) / Float64(u / Float64(t1 / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -6e-83) || ~((t1 <= 175.0)))
		tmp = -v / (t1 + u);
	else
		tmp = -v / (u / (t1 / u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -6e-83], N[Not[LessEqual[t1, 175.0]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(u / N[(t1 / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.00000000000000021e-83 or 175 < t1

   1. Initial program 61.7%

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

      1. associate-/r* 74.1%

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

      2. associate-/l* 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 82.8%

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

      1. neg-mul-1 82.8%

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

   6. Simplified 82.8%

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

   if -6.00000000000000021e-83 < t1 < 175

   1. Initial program 88.3%

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

      1. associate-/l* 85.0%

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

      2. neg-mul-1 85.0%

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

      3. associate-*r/ 88.7%

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

      4. times-frac 95.2%

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

      5. div-inv 95.1%

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

      6. clear-num 95.2%

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

   3. Applied egg-rr 95.2%

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

   4. Taylor expanded in t1 around 0 76.6%

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

      1. mul-1-neg 76.6%

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

      2. *-commutative 76.6%

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

      3. unpow2 76.6%

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

      4. times-frac 81.9%

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

      5. associate-*l/ 81.9%

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

      6. associate-/l* 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{-83} \lor \neg \left(t1 \leq 175\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\frac{u}{\frac{t1}{u}}}\\ \end{array} \]

Alternative 8: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{-83} \lor \neg \left(t1 \leq 400\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -6e-83) (not (<= t1 400.0)))
   (/ (- v) (+ t1 u))
   (* (/ v u) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6e-83) || !(t1 <= 400.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-6d-83)) .or. (.not. (t1 <= 400.0d0))) then
        tmp = -v / (t1 + u)
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6e-83) || !(t1 <= 400.0)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -6e-83) or not (t1 <= 400.0):
		tmp = -v / (t1 + u)
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -6e-83) || !(t1 <= 400.0))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -6e-83) || ~((t1 <= 400.0)))
		tmp = -v / (t1 + u);
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -6e-83], N[Not[LessEqual[t1, 400.0]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.00000000000000021e-83 or 400 < t1

   1. Initial program 61.7%

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

      1. associate-/r* 74.1%

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

      2. associate-/l* 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around inf 82.8%

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

      1. neg-mul-1 82.8%

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

   6. Simplified 82.8%

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

   if -6.00000000000000021e-83 < t1 < 400

   1. Initial program 88.3%

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

      1. associate-*l/ 88.2%

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

      2. *-commutative 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in t1 around 0 76.6%

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

      1. mul-1-neg 76.6%

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

      2. associate-/l* 74.9%

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

      3. distribute-neg-frac 74.9%

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

      4. unpow2 74.9%

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

   6. Simplified 74.9%

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

      1. frac-2neg 74.9%

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

      2. associate-/r/ 75.6%

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

      3. add-sqr-sqrt 40.4%

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

      4. sqrt-unprod 52.0%

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

      5. sqr-neg 52.0%

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

      6. sqrt-unprod 18.0%

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

      7. add-sqr-sqrt 44.2%

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

      8. frac-2neg 44.2%

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

      9. add-sqr-sqrt 18.0%

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

      10. sqrt-unprod 52.0%

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

      11. sqr-neg 52.0%

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

      12. sqrt-unprod 40.4%

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

      13. add-sqr-sqrt 75.6%

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

   8. Applied egg-rr 75.6%

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

   9. Taylor expanded in t1 around 0 76.6%

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

       1. mul-1-neg 76.6%

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

       2. unpow2 76.6%

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

       3. associate-*l/ 75.6%

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

       4. distribute-lft-neg-in 75.6%

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

       5. associate-/r* 80.0%

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

       6. *-rgt-identity 80.0%

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

       7. associate-*r/ 80.0%

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

       8. *-lft-identity 80.0%

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

       9. associate-*l* 80.0%

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

       10. metadata-eval 80.0%

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

       11. associate-*r/ 80.0%

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

       12. *-rgt-identity 80.0%

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

       13. associate-/r* 75.6%

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

       14. times-frac 75.6%

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

       15. neg-mul-1 75.6%

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

       16. neg-mul-1 75.6%

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

       17. distribute-neg-frac 75.6%

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

       18. remove-double-neg 75.6%

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

       19. associate-/r/ 74.9%

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

       20. associate-/l* 76.6%

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

       21. *-commutative 76.6%

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

       22. distribute-rgt-neg-in 76.6%

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

       23. times-frac 81.9%

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

   11. Simplified 81.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{-83} \lor \neg \left(t1 \leq 400\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]

Alternative 9: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{-45} \lor \neg \left(u \leq 1.02 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.85e-45) (not (<= u 1.02e+14)))
   (/ (- t1) (/ u (/ v u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.85e-45) || !(u <= 1.02e+14)) {
		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 <= (-1.85d-45)) .or. (.not. (u <= 1.02d+14))) 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 <= -1.85e-45) || !(u <= 1.02e+14)) {
		tmp = -t1 / (u / (v / u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.85e-45) or not (u <= 1.02e+14):
		tmp = -t1 / (u / (v / u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.85e-45) || !(u <= 1.02e+14))
		tmp = Float64(Float64(-t1) / Float64(u / Float64(v / u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.85e-45) || ~((u <= 1.02e+14)))
		tmp = -t1 / (u / (v / u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.85e-45], N[Not[LessEqual[u, 1.02e+14]], $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 < -1.85e-45 or 1.02e14 < u

   1. Initial program 79.1%

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

      1. associate-*l/ 77.5%

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

      2. *-commutative 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in t1 around 0 73.4%

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

      1. mul-1-neg 73.4%

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

      2. associate-/l* 75.8%

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

      3. distribute-neg-frac 75.8%

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

      4. unpow2 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in u around 0 75.8%

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

      1. unpow2 75.8%

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

      2. associate-/l* 84.1%

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

   9. Simplified 84.1%

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

   if -1.85e-45 < u < 1.02e14

   1. Initial program 67.3%

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

      1. associate-*l/ 75.4%

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

      2. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in t1 around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. neg-mul-1 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.85 \cdot 10^{-45} \lor \neg \left(u \leq 1.02 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. associate-/r* 83.0%

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

   2. associate-/l* 96.5%

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

3. Simplified 96.5%

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

4. Final simplification 96.5%

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

Alternative 11: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+29} \lor \neg \left(u \leq 7.2 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.6e+29) (not (<= u 7.2e+94)))
   (* (/ t1 u) (/ v u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.6e+29) || !(u <= 7.2e+94)) {
		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 <= (-3.6d+29)) .or. (.not. (u <= 7.2d+94))) 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 <= -3.6e+29) || !(u <= 7.2e+94)) {
		tmp = (t1 / u) * (v / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.6e+29) or not (u <= 7.2e+94):
		tmp = (t1 / u) * (v / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.6e+29) || !(u <= 7.2e+94))
		tmp = Float64(Float64(t1 / u) * Float64(v / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.6e+29) || ~((u <= 7.2e+94)))
		tmp = (t1 / u) * (v / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.6e+29], N[Not[LessEqual[u, 7.2e+94]], $MachinePrecision]], N[(N[(t1 / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.59999999999999976e29 or 7.19999999999999985e94 < u

   1. Initial program 79.2%

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

      1. associate-*l/ 75.5%

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

      2. *-commutative 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in t1 around 0 76.0%

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

      1. mul-1-neg 76.0%

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

      2. associate-/l* 77.2%

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

      3. distribute-neg-frac 77.2%

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

      4. unpow2 77.2%

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

   6. Simplified 77.2%

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

      1. add-sqr-sqrt 39.3%

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

      2. sqrt-unprod 60.0%

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

      3. sqr-neg 60.0%

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

      4. sqrt-unprod 36.1%

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

      5. add-sqr-sqrt 68.6%

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

      6. associate-/l* 68.4%

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

      7. associate-/r/ 65.8%

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

   8. Applied egg-rr 65.8%

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

   if -3.59999999999999976e29 < u < 7.19999999999999985e94

   1. Initial program 69.1%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in t1 around inf 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

   6. Simplified 72.6%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+29} \lor \neg \left(u \leq 7.2 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.6e+29)
   (* (/ t1 u) (/ v u))
   (if (<= u 1.1e+94) (/ (- v) t1) (* v (/ t1 (* u u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e+29) {
		tmp = (t1 / u) * (v / u);
	} else if (u <= 1.1e+94) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.6d+29)) then
        tmp = (t1 / u) * (v / u)
    else if (u <= 1.1d+94) then
        tmp = -v / t1
    else
        tmp = v * (t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e+29) {
		tmp = (t1 / u) * (v / u);
	} else if (u <= 1.1e+94) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.6e+29:
		tmp = (t1 / u) * (v / u)
	elif u <= 1.1e+94:
		tmp = -v / t1
	else:
		tmp = v * (t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.6e+29)
		tmp = Float64(Float64(t1 / u) * Float64(v / u));
	elseif (u <= 1.1e+94)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.6e+29)
		tmp = (t1 / u) * (v / u);
	elseif (u <= 1.1e+94)
		tmp = -v / t1;
	else
		tmp = v * (t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.6e+29], N[(N[(t1 / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.1e+94], N[((-v) / t1), $MachinePrecision], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.59999999999999976e29

   1. Initial program 78.2%

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

      1. associate-*l/ 75.3%

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

      2. *-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in t1 around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. associate-/l* 75.3%

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

      3. distribute-neg-frac 75.3%

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

      4. unpow2 75.3%

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

   6. Simplified 75.3%

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

      1. add-sqr-sqrt 39.7%

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

      2. sqrt-unprod 60.3%

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

      3. sqr-neg 60.3%

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

      4. sqrt-unprod 32.3%

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

      5. add-sqr-sqrt 67.2%

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

      6. associate-/l* 67.0%

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

      7. associate-/r/ 65.4%

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

   8. Applied egg-rr 65.4%

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

   if -3.59999999999999976e29 < u < 1.10000000000000006e94

   1. Initial program 69.1%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in t1 around inf 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

   6. Simplified 72.6%

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

   if 1.10000000000000006e94 < u

   1. Initial program 80.4%

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

      1. associate-*l/ 75.7%

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

      2. *-commutative 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in t1 around 0 80.6%

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

      1. mul-1-neg 80.6%

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

      2. associate-/l* 79.3%

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

      3. distribute-neg-frac 79.3%

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

      4. unpow2 79.3%

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

   6. Simplified 79.3%

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

      1. associate-/r/ 75.7%

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

      2. add-sqr-sqrt 33.7%

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

      3. sqrt-unprod 57.2%

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

      4. sqr-neg 57.2%

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

      5. sqrt-unprod 40.4%

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

      6. add-sqr-sqrt 70.2%

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

   8. Applied egg-rr 70.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 1.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 13: 67.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+29}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.6e+29)
   (* t1 (/ v (* u u)))
   (if (<= u 1.8e+92) (/ (- v) t1) (* v (/ t1 (* u u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e+29) {
		tmp = t1 * (v / (u * u));
	} else if (u <= 1.8e+92) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.6d+29)) then
        tmp = t1 * (v / (u * u))
    else if (u <= 1.8d+92) then
        tmp = -v / t1
    else
        tmp = v * (t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.6e+29) {
		tmp = t1 * (v / (u * u));
	} else if (u <= 1.8e+92) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.6e+29:
		tmp = t1 * (v / (u * u))
	elif u <= 1.8e+92:
		tmp = -v / t1
	else:
		tmp = v * (t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.6e+29)
		tmp = Float64(t1 * Float64(v / Float64(u * u)));
	elseif (u <= 1.8e+92)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.6e+29)
		tmp = t1 * (v / (u * u));
	elseif (u <= 1.8e+92)
		tmp = -v / t1;
	else
		tmp = v * (t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.6e+29], N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.8e+92], N[((-v) / t1), $MachinePrecision], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.59999999999999976e29

   1. Initial program 78.2%

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

      1. associate-*l/ 75.3%

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

      2. *-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in t1 around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. associate-/l* 75.3%

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

      3. distribute-neg-frac 75.3%

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

      4. unpow2 75.3%

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

   6. Simplified 75.3%

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

      1. clear-num 75.3%

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

      2. associate-/r/ 75.3%

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

      3. clear-num 75.3%

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

      4. add-sqr-sqrt 39.7%

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

      5. sqrt-unprod 60.3%

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

      6. sqr-neg 60.3%

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

      7. sqrt-unprod 32.3%

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

      8. add-sqr-sqrt 67.2%

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

   8. Applied egg-rr 67.2%

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

   if -3.59999999999999976e29 < u < 1.8e92

   1. Initial program 69.1%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in t1 around inf 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

   6. Simplified 72.6%

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

   if 1.8e92 < u

   1. Initial program 80.4%

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

      1. associate-*l/ 75.7%

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

      2. *-commutative 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in t1 around 0 80.6%

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

      1. mul-1-neg 80.6%

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

      2. associate-/l* 79.3%

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

      3. distribute-neg-frac 79.3%

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

      4. unpow2 79.3%

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

   6. Simplified 79.3%

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

      1. associate-/r/ 75.7%

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

      2. add-sqr-sqrt 33.7%

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

      3. sqrt-unprod 57.2%

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

      4. sqr-neg 57.2%

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

      5. sqrt-unprod 40.4%

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

      6. add-sqr-sqrt 70.2%

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

   8. Applied egg-rr 70.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+29}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 14: 57.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+189}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4e+189)
   (/ (- v) u)
   (if (<= u 1.8e+87) (/ (- v) t1) (/ 1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+189) {
		tmp = -v / u;
	} else if (u <= 1.8e+87) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4d+189)) then
        tmp = -v / u
    else if (u <= 1.8d+87) then
        tmp = -v / t1
    else
        tmp = 1.0d0 / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4e+189) {
		tmp = -v / u;
	} else if (u <= 1.8e+87) {
		tmp = -v / t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4e+189:
		tmp = -v / u
	elif u <= 1.8e+87:
		tmp = -v / t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4e+189)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 1.8e+87)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4e+189)
		tmp = -v / u;
	elseif (u <= 1.8e+87)
		tmp = -v / t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4e+189], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 1.8e+87], N[((-v) / t1), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.0000000000000001e189

   1. Initial program 85.2%

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

      1. associate-/r* 96.6%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 99.9%

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

      3. distribute-neg-frac 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in t1 around inf 44.2%

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

      1. neg-mul-1 44.2%

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

      2. distribute-neg-frac 44.2%

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

   9. Simplified 44.2%

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

   if -4.0000000000000001e189 < u < 1.79999999999999997e87

   1. Initial program 69.4%

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

      1. associate-*l/ 75.1%

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

      2. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in t1 around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   6. Simplified 66.2%

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

   if 1.79999999999999997e87 < u

   1. Initial program 80.8%

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

      1. associate-/l* 79.7%

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

      2. neg-mul-1 79.7%

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

      3. associate-*r/ 92.3%

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

      4. times-frac 99.9%

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

      5. div-inv 99.8%

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

      6. clear-num 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in t1 around inf 47.9%

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

      1. associate-*l/ 47.9%

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

      2. neg-mul-1 47.9%

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

      3. clear-num 48.1%

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

      4. add-sqr-sqrt 23.8%

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

      5. sqrt-unprod 43.9%

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

      6. sqr-neg 43.9%

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

      7. sqrt-unprod 20.7%

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

      8. add-sqr-sqrt 42.7%

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

   6. Applied egg-rr 42.7%

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

   7. Taylor expanded in t1 around 0 39.4%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+189}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]

Alternative 15: 57.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+189}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2 \cdot 10^{+89}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.1e+189)
   (/ (- v) u)
   (if (<= u 2e+89) (/ (- v) t1) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.1e+189) {
		tmp = -v / u;
	} else if (u <= 2e+89) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.1d+189)) then
        tmp = -v / u
    else if (u <= 2d+89) then
        tmp = -v / t1
    else
        tmp = v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.1e+189) {
		tmp = -v / u;
	} else if (u <= 2e+89) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.1e+189:
		tmp = -v / u
	elif u <= 2e+89:
		tmp = -v / t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.1e+189)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 2e+89)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.1e+189)
		tmp = -v / u;
	elseif (u <= 2e+89)
		tmp = -v / t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.1e+189], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 2e+89], N[((-v) / t1), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.0999999999999999e189

   1. Initial program 85.2%

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

      1. associate-/r* 96.6%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 99.9%

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

      3. distribute-neg-frac 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in t1 around inf 44.2%

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

      1. neg-mul-1 44.2%

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

      2. distribute-neg-frac 44.2%

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

   9. Simplified 44.2%

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

   if -3.0999999999999999e189 < u < 1.99999999999999999e89

   1. Initial program 69.4%

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

      1. associate-*l/ 75.1%

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

      2. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in t1 around inf 66.2%

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

      1. associate-*r/ 66.2%

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

      2. neg-mul-1 66.2%

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

   6. Simplified 66.2%

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

   if 1.99999999999999999e89 < u

   1. Initial program 80.8%

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

      1. associate-/l* 79.7%

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

      2. neg-mul-1 79.7%

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

      3. associate-*r/ 92.3%

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

      4. times-frac 99.9%

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

      5. div-inv 99.8%

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

      6. clear-num 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in t1 around inf 47.9%

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

      1. expm1-log1p-u 47.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1}{t1 + u} \cdot v\right)\right)} \]

      2. expm1-udef 70.1%

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

      3. associate-*l/ 70.1%

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

      4. neg-mul-1 70.1%

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

      5. add-sqr-sqrt 33.8%

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

      6. sqrt-unprod 66.6%

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

      7. sqr-neg 66.6%

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

      8. sqrt-unprod 34.9%

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

      9. add-sqr-sqrt 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. expm1-def 42.4%

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

      2. expm1-log1p 42.5%

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

      3. +-commutative 42.5%

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

   8. Simplified 42.5%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+189}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2 \cdot 10^{+89}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]

Alternative 16: 57.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.9 \cdot 10^{+189}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.16 \cdot 10^{+95}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.9e+189) (/ v u) (if (<= u 1.16e+95) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.9e+189) {
		tmp = v / u;
	} else if (u <= 1.16e+95) {
		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 <= (-3.9d+189)) then
        tmp = v / u
    else if (u <= 1.16d+95) 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 <= -3.9e+189) {
		tmp = v / u;
	} else if (u <= 1.16e+95) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.9e+189:
		tmp = v / u
	elif u <= 1.16e+95:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.9e+189)
		tmp = Float64(v / u);
	elseif (u <= 1.16e+95)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.9e+189)
		tmp = v / u;
	elseif (u <= 1.16e+95)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.9e+189], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.16e+95], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.9e189 or 1.1599999999999999e95 < u

   1. Initial program 82.1%

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

      1. associate-/l* 81.5%

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

      2. neg-mul-1 81.5%

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

      3. associate-*r/ 94.0%

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

      4. times-frac 99.9%

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

      5. div-inv 99.9%

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

      6. clear-num 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in t1 around inf 46.0%

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

      1. associate-*l/ 46.0%

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

      2. neg-mul-1 46.0%

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

      3. clear-num 46.1%

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

      4. add-sqr-sqrt 24.9%

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

      5. sqrt-unprod 43.1%

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

      6. sqr-neg 43.1%

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

      7. sqrt-unprod 18.8%

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

      8. add-sqr-sqrt 42.5%

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

   6. Applied egg-rr 42.5%

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

   7. Taylor expanded in t1 around 0 40.2%

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

   if -3.9e189 < u < 1.1599999999999999e95

   1. Initial program 69.5%

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

      1. associate-*l/ 75.3%

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

      2. *-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in t1 around inf 66.4%

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

      1. associate-*r/ 66.4%

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

      2. neg-mul-1 66.4%

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

   6. Simplified 66.4%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.9 \cdot 10^{+189}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.16 \cdot 10^{+95}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 17: 57.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+95}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.2e+189) (/ (- v) u) (if (<= u 2.05e+95) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.2e+189) {
		tmp = -v / u;
	} else if (u <= 2.05e+95) {
		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 <= (-3.2d+189)) then
        tmp = -v / u
    else if (u <= 2.05d+95) 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 <= -3.2e+189) {
		tmp = -v / u;
	} else if (u <= 2.05e+95) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.2e+189:
		tmp = -v / u
	elif u <= 2.05e+95:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.2e+189)
		tmp = Float64(Float64(-v) / u);
	elseif (u <= 2.05e+95)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.2e+189)
		tmp = -v / u;
	elseif (u <= 2.05e+95)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.2e+189], N[((-v) / u), $MachinePrecision], If[LessEqual[u, 2.05e+95], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.2000000000000001e189

   1. Initial program 85.2%

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

      1. associate-/r* 96.6%

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

      2. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. associate-/l* 99.9%

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

      3. distribute-neg-frac 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in t1 around inf 44.2%

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

      1. neg-mul-1 44.2%

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

      2. distribute-neg-frac 44.2%

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

   9. Simplified 44.2%

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

   if -3.2000000000000001e189 < u < 2.04999999999999993e95

   1. Initial program 69.5%

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

      1. associate-*l/ 75.3%

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

      2. *-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in t1 around inf 66.4%

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

      1. associate-*r/ 66.4%

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

      2. neg-mul-1 66.4%

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

   6. Simplified 66.4%

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

   if 2.04999999999999993e95 < u

   1. Initial program 80.4%

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

      1. associate-/l* 79.3%

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

      2. neg-mul-1 79.3%

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

      3. associate-*r/ 92.1%

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

      4. times-frac 99.9%

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

      5. div-inv 99.8%

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

      6. clear-num 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in t1 around inf 46.9%

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

      1. associate-*l/ 46.9%

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

      2. neg-mul-1 46.9%

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

      3. clear-num 47.1%

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

      4. add-sqr-sqrt 22.4%

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

      5. sqrt-unprod 42.8%

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

      6. sqr-neg 42.8%

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

      7. sqrt-unprod 21.1%

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

      8. add-sqr-sqrt 41.6%

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

   6. Applied egg-rr 41.6%

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

   7. Taylor expanded in t1 around 0 38.1%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+95}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 18: 23.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+24}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.7e+24) (/ v t1) (if (<= t1 1.1e+22) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.7e+24) {
		tmp = v / t1;
	} else if (t1 <= 1.1e+22) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-2.7d+24)) then
        tmp = v / t1
    else if (t1 <= 1.1d+22) then
        tmp = v / u
    else
        tmp = v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.7e+24) {
		tmp = v / t1;
	} else if (t1 <= 1.1e+22) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.7e+24:
		tmp = v / t1
	elif t1 <= 1.1e+22:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.7e+24)
		tmp = Float64(v / t1);
	elseif (t1 <= 1.1e+22)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.7e+24)
		tmp = v / t1;
	elseif (t1 <= 1.1e+22)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.7e+24], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 1.1e+22], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.7e24 or 1.1e22 < t1

   1. Initial program 55.5%

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

      1. associate-/l* 56.6%

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

      2. neg-mul-1 56.6%

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

      3. associate-*r/ 77.3%

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

      4. times-frac 96.9%

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

      5. div-inv 96.8%

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

      6. clear-num 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in t1 around inf 84.1%

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

      1. associate-*l/ 84.3%

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

      2. neg-mul-1 84.3%

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

      3. clear-num 81.9%

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

      4. add-sqr-sqrt 50.1%

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

      5. sqrt-unprod 47.2%

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

      6. sqr-neg 47.2%

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

      7. sqrt-unprod 12.4%

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

      8. add-sqr-sqrt 33.5%

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

   6. Applied egg-rr 33.5%

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

   7. Taylor expanded in t1 around inf 29.2%

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

   if -2.7e24 < t1 < 1.1e22

   1. Initial program 89.5%

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

      1. associate-/l* 87.4%

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

      2. neg-mul-1 87.4%

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

      3. associate-*r/ 90.5%

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

      4. times-frac 96.0%

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

      5. div-inv 95.9%

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

      6. clear-num 95.9%

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

   3. Applied egg-rr 95.9%

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

   4. Taylor expanded in t1 around inf 38.9%

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

      1. associate-*l/ 39.0%

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

      2. neg-mul-1 39.0%

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

      3. clear-num 38.9%

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

      4. add-sqr-sqrt 22.2%

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

      5. sqrt-unprod 29.0%

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

      6. sqr-neg 29.0%

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

      7. sqrt-unprod 8.0%

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

      8. add-sqr-sqrt 19.7%

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

   6. Applied egg-rr 19.7%

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

   7. Taylor expanded in t1 around 0 21.0%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+24}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 19: 61.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. associate-/r* 83.0%

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

   2. associate-/l* 96.5%

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

3. Simplified 96.5%

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

4. Taylor expanded in t1 around inf 60.2%

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

   1. neg-mul-1 60.2%

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

6. Simplified 60.2%

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

7. Final simplification 60.2%

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

Alternative 20: 14.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. associate-/l* 73.0%

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

   2. neg-mul-1 73.0%

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

   3. associate-*r/ 84.3%

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

   4. times-frac 96.4%

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

   5. div-inv 96.3%

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

   6. clear-num 97.7%

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

3. Applied egg-rr 97.7%

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

4. Taylor expanded in t1 around inf 60.1%

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

   1. associate-*l/ 60.2%

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

   2. neg-mul-1 60.2%

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

   3. clear-num 59.1%

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

   4. add-sqr-sqrt 35.3%

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

   5. sqrt-unprod 37.5%

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

   6. sqr-neg 37.5%

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

   7. sqrt-unprod 10.1%

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

   8. add-sqr-sqrt 26.2%

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

6. Applied egg-rr 26.2%

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

7. Taylor expanded in t1 around inf 14.8%

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

8. Final simplification 14.8%

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

Reproduce

herbie shell --seed 2023272 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))