Rosa's DopplerBench

Percentage Accurate: 72.9% → 97.5%

Time: 10.1s

Alternatives: 16

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: 72.9% 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: 97.5% 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 73.6%

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

   1. associate-/r* 83.2%

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

   2. associate-/l* 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 2: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -1.15 \cdot 10^{+126}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -1.28 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 9.2 \cdot 10^{-267}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -1.15e+126)
     (/ (- v) t1)
     (if (<= t1 -1.28e-213)
       t_1
       (if (<= t1 9.2e-267)
         (/ (/ (- t1) (/ u v)) (+ t1 u))
         (if (<= t1 5.2e+148) t_1 (/ (- v) (+ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.15e+126) {
		tmp = -v / t1;
	} else if (t1 <= -1.28e-213) {
		tmp = t_1;
	} else if (t1 <= 9.2e-267) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (t1 <= 5.2e+148) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-1.15d+126)) then
        tmp = -v / t1
    else if (t1 <= (-1.28d-213)) then
        tmp = t_1
    else if (t1 <= 9.2d-267) then
        tmp = (-t1 / (u / v)) / (t1 + u)
    else if (t1 <= 5.2d+148) then
        tmp = t_1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -1.15e+126) {
		tmp = -v / t1;
	} else if (t1 <= -1.28e-213) {
		tmp = t_1;
	} else if (t1 <= 9.2e-267) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (t1 <= 5.2e+148) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -1.15e+126:
		tmp = -v / t1
	elif t1 <= -1.28e-213:
		tmp = t_1
	elif t1 <= 9.2e-267:
		tmp = (-t1 / (u / v)) / (t1 + u)
	elif t1 <= 5.2e+148:
		tmp = t_1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -1.15e+126)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= -1.28e-213)
		tmp = t_1;
	elseif (t1 <= 9.2e-267)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	elseif (t1 <= 5.2e+148)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -1.15e+126)
		tmp = -v / t1;
	elseif (t1 <= -1.28e-213)
		tmp = t_1;
	elseif (t1 <= 9.2e-267)
		tmp = (-t1 / (u / v)) / (t1 + u);
	elseif (t1 <= 5.2e+148)
		tmp = t_1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.15e+126], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, -1.28e-213], t$95$1, If[LessEqual[t1, 9.2e-267], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 5.2e+148], t$95$1, N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.15e126

   1. Initial program 49.5%

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

      1. associate-*l/ 53.6%

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

      2. *-commutative 53.6%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in t1 around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if -1.15e126 < t1 < -1.28000000000000005e-213 or 9.2000000000000002e-267 < t1 < 5.2e148

   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-*l/ 92.1%

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

      2. *-commutative 92.1%

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

   3. Simplified 92.1%

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

   if -1.28000000000000005e-213 < t1 < 9.2000000000000002e-267

   1. Initial program 75.7%

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

      1. associate-/r* 82.8%

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

      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 0 82.8%

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

      1. mul-1-neg 82.8%

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

      2. associate-/l* 96.5%

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

      3. distribute-neg-frac 96.5%

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

   6. Simplified 96.5%

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

   if 5.2e148 < t1

   1. Initial program 41.8%

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

      1. associate-/r* 64.1%

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

      2. associate-/l* 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in t1 around inf 95.0%

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

      1. neg-mul-1 95.0%

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

   6. Simplified 95.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{+126}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq -1.28 \cdot 10^{-213}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 9.2 \cdot 10^{-267}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{+148}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 3: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{t1 - u}\\ \mathbf{elif}\;t1 \leq -2.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{v}{\frac{t1 + u}{\frac{u}{t1} + -1}}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.65e-17)
   (/ v (- u t1))
   (if (<= t1 -1.25e-94)
     (/ (/ (* t1 v) u) (- t1 u))
     (if (<= t1 -2.05e-119)
       (/ v (/ (+ t1 u) (+ (/ u t1) -1.0)))
       (if (<= t1 4.8e-25) (/ t1 (* u (/ (- u) v))) (/ (- v) (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.65e-17) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.25e-94) {
		tmp = ((t1 * v) / u) / (t1 - u);
	} else if (t1 <= -2.05e-119) {
		tmp = v / ((t1 + u) / ((u / t1) + -1.0));
	} else if (t1 <= 4.8e-25) {
		tmp = t1 / (u * (-u / v));
	} 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 (t1 <= (-2.65d-17)) then
        tmp = v / (u - t1)
    else if (t1 <= (-1.25d-94)) then
        tmp = ((t1 * v) / u) / (t1 - u)
    else if (t1 <= (-2.05d-119)) then
        tmp = v / ((t1 + u) / ((u / t1) + (-1.0d0)))
    else if (t1 <= 4.8d-25) then
        tmp = t1 / (u * (-u / v))
    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 (t1 <= -2.65e-17) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.25e-94) {
		tmp = ((t1 * v) / u) / (t1 - u);
	} else if (t1 <= -2.05e-119) {
		tmp = v / ((t1 + u) / ((u / t1) + -1.0));
	} else if (t1 <= 4.8e-25) {
		tmp = t1 / (u * (-u / v));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.65e-17:
		tmp = v / (u - t1)
	elif t1 <= -1.25e-94:
		tmp = ((t1 * v) / u) / (t1 - u)
	elif t1 <= -2.05e-119:
		tmp = v / ((t1 + u) / ((u / t1) + -1.0))
	elif t1 <= 4.8e-25:
		tmp = t1 / (u * (-u / v))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.65e-17)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -1.25e-94)
		tmp = Float64(Float64(Float64(t1 * v) / u) / Float64(t1 - u));
	elseif (t1 <= -2.05e-119)
		tmp = Float64(v / Float64(Float64(t1 + u) / Float64(Float64(u / t1) + -1.0)));
	elseif (t1 <= 4.8e-25)
		tmp = Float64(t1 / Float64(u * Float64(Float64(-u) / v)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.65e-17)
		tmp = v / (u - t1);
	elseif (t1 <= -1.25e-94)
		tmp = ((t1 * v) / u) / (t1 - u);
	elseif (t1 <= -2.05e-119)
		tmp = v / ((t1 + u) / ((u / t1) + -1.0));
	elseif (t1 <= 4.8e-25)
		tmp = t1 / (u * (-u / v));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.65e-17], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.25e-94], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -2.05e-119], N[(v / N[(N[(t1 + u), $MachinePrecision] / N[(N[(u / t1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 4.8e-25], N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t1 < -2.6499999999999999e-17

   1. Initial program 68.4%

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

      1. associate-*l/ 74.2%

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

      2. *-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. neg-mul-1 74.2%

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

      2. times-frac 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.8%

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

      2. mul-1-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in t1 around inf 80.6%

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

      1. frac-2neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. metadata-eval 80.6%

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

      4. un-div-inv 80.9%

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

      5. +-commutative 80.9%

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

      6. distribute-neg-in 80.9%

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

      7. add-sqr-sqrt 44.9%

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

      8. sqrt-unprod 80.4%

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

      9. sqr-neg 80.4%

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

      10. sqrt-unprod 36.5%

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

      11. add-sqr-sqrt 81.0%

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

   10. Applied egg-rr 81.0%

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

       1. sub-neg 81.0%

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

   12. Simplified 81.0%

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

   if -2.6499999999999999e-17 < t1 < -1.2499999999999999e-94

   1. Initial program 99.5%

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

      1. associate-/r* 100.0%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 92.4%

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

      1. mul-1-neg 92.4%

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

      2. associate-/l* 92.1%

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

      3. distribute-neg-frac 92.1%

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

   6. Simplified 92.1%

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

      1. expm1-log1p-u 84.5%

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

      2. expm1-udef 54.6%

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

   8. Applied egg-rr 55.2%

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

      1. expm1-def 85.2%

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

      2. expm1-log1p 92.9%

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

      3. associate-*l/ 92.9%

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

   10. Simplified 92.9%

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

   if -1.2499999999999999e-94 < t1 < -2.0500000000000001e-119

   1. Initial program 86.0%

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

      1. associate-/r* 86.2%

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

      2. associate-/l* 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in t1 around inf 79.6%

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

      1. neg-mul-1 79.6%

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

      2. +-commutative 79.6%

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

      3. unsub-neg 79.6%

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

      4. associate-/l* 79.6%

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

   6. Simplified 79.6%

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

   7. Taylor expanded in v around 0 79.6%

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

      1. associate-/l* 79.6%

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

      2. sub-neg 79.6%

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

      3. metadata-eval 79.6%

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

   9. Simplified 79.6%

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

   if -2.0500000000000001e-119 < t1 < 4.80000000000000018e-25

   1. Initial program 83.8%

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

      1. associate-/r* 89.1%

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

      2. associate-/l* 94.7%

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

   3. Simplified 94.7%

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

      1. clear-num 94.6%

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

      2. associate-/r/ 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in t1 around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. *-commutative 78.6%

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

      3. unpow2 78.6%

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

      4. times-frac 84.3%

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

      5. distribute-rgt-neg-in 84.3%

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

      6. distribute-neg-frac 84.3%

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

   8. Simplified 84.3%

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

      1. clear-num 84.3%

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

      2. frac-2neg 84.3%

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

      3. frac-times 86.0%

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

      4. *-un-lft-identity 86.0%

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

      5. remove-double-neg 86.0%

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

   10. Applied egg-rr 86.0%

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

   if 4.80000000000000018e-25 < t1

   1. Initial program 60.8%

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

      1. associate-/r* 75.6%

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

      2. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.4%

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

      1. neg-mul-1 85.4%

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

   6. Simplified 85.4%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{t1 - u}\\ \mathbf{elif}\;t1 \leq -2.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{v}{\frac{t1 + u}{\frac{u}{t1} + -1}}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 4: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{if}\;t1 \leq -1.62 \cdot 10^{-17}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -5.4 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* v (/ (- t1) (* u u)))))
   (if (<= t1 -1.62e-17)
     (/ v (- u t1))
     (if (<= t1 -5.4e-91)
       t_1
       (if (<= t1 -1.65e-125)
         (/ (- v) t1)
         (if (<= t1 7.2e-27) t_1 (/ (- v) (+ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / (u * u));
	double tmp;
	if (t1 <= -1.62e-17) {
		tmp = v / (u - t1);
	} else if (t1 <= -5.4e-91) {
		tmp = t_1;
	} else if (t1 <= -1.65e-125) {
		tmp = -v / t1;
	} else if (t1 <= 7.2e-27) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = v * (-t1 / (u * u))
    if (t1 <= (-1.62d-17)) then
        tmp = v / (u - t1)
    else if (t1 <= (-5.4d-91)) then
        tmp = t_1
    else if (t1 <= (-1.65d-125)) then
        tmp = -v / t1
    else if (t1 <= 7.2d-27) then
        tmp = t_1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v * (-t1 / (u * u));
	double tmp;
	if (t1 <= -1.62e-17) {
		tmp = v / (u - t1);
	} else if (t1 <= -5.4e-91) {
		tmp = t_1;
	} else if (t1 <= -1.65e-125) {
		tmp = -v / t1;
	} else if (t1 <= 7.2e-27) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v * (-t1 / (u * u))
	tmp = 0
	if t1 <= -1.62e-17:
		tmp = v / (u - t1)
	elif t1 <= -5.4e-91:
		tmp = t_1
	elif t1 <= -1.65e-125:
		tmp = -v / t1
	elif t1 <= 7.2e-27:
		tmp = t_1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v * Float64(Float64(-t1) / Float64(u * u)))
	tmp = 0.0
	if (t1 <= -1.62e-17)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -5.4e-91)
		tmp = t_1;
	elseif (t1 <= -1.65e-125)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 7.2e-27)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v * (-t1 / (u * u));
	tmp = 0.0;
	if (t1 <= -1.62e-17)
		tmp = v / (u - t1);
	elseif (t1 <= -5.4e-91)
		tmp = t_1;
	elseif (t1 <= -1.65e-125)
		tmp = -v / t1;
	elseif (t1 <= 7.2e-27)
		tmp = t_1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.62e-17], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -5.4e-91], t$95$1, If[LessEqual[t1, -1.65e-125], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 7.2e-27], t$95$1, N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.62000000000000001e-17

   1. Initial program 68.4%

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

      1. associate-*l/ 74.2%

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

      2. *-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. neg-mul-1 74.2%

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

      2. times-frac 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.8%

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

      2. mul-1-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in t1 around inf 80.6%

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

      1. frac-2neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. metadata-eval 80.6%

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

      4. un-div-inv 80.9%

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

      5. +-commutative 80.9%

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

      6. distribute-neg-in 80.9%

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

      7. add-sqr-sqrt 44.9%

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

      8. sqrt-unprod 80.4%

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

      9. sqr-neg 80.4%

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

      10. sqrt-unprod 36.5%

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

      11. add-sqr-sqrt 81.0%

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

   10. Applied egg-rr 81.0%

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

       1. sub-neg 81.0%

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

   12. Simplified 81.0%

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

   if -1.62000000000000001e-17 < t1 < -5.3999999999999995e-91 or -1.65e-125 < t1 < 7.1999999999999997e-27

   1. Initial program 86.4%

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

      1. associate-*l/ 86.6%

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

      2. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in t1 around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. neg-mul-1 80.3%

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

      3. unpow2 80.3%

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

   6. Simplified 80.3%

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

   if -5.3999999999999995e-91 < t1 < -1.65e-125

   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/ 99.5%

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

      2. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t1 around inf 72.0%

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

      1. associate-*r/ 72.0%

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

      2. neg-mul-1 72.0%

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

   6. Simplified 72.0%

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

   if 7.1999999999999997e-27 < t1

   1. Initial program 60.8%

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

      1. associate-/r* 75.6%

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

      2. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.4%

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

      1. neg-mul-1 85.4%

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

   6. Simplified 85.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.62 \cdot 10^{-17}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -5.4 \cdot 10^{-91}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{-27}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 5: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ t1 u) (/ (- v) u))))
   (if (<= t1 -1.4e-18)
     (/ v (- u t1))
     (if (<= t1 -1.25e-94)
       t_1
       (if (<= t1 -1.65e-125)
         (/ (- v) t1)
         (if (<= t1 3.8e-27) t_1 (/ (- v) (+ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (t1 / u) * (-v / u);
	double tmp;
	if (t1 <= -1.4e-18) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.25e-94) {
		tmp = t_1;
	} else if (t1 <= -1.65e-125) {
		tmp = -v / t1;
	} else if (t1 <= 3.8e-27) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (t1 / u) * (-v / u)
    if (t1 <= (-1.4d-18)) then
        tmp = v / (u - t1)
    else if (t1 <= (-1.25d-94)) then
        tmp = t_1
    else if (t1 <= (-1.65d-125)) then
        tmp = -v / t1
    else if (t1 <= 3.8d-27) then
        tmp = t_1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (t1 / u) * (-v / u);
	double tmp;
	if (t1 <= -1.4e-18) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.25e-94) {
		tmp = t_1;
	} else if (t1 <= -1.65e-125) {
		tmp = -v / t1;
	} else if (t1 <= 3.8e-27) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (t1 / u) * (-v / u)
	tmp = 0
	if t1 <= -1.4e-18:
		tmp = v / (u - t1)
	elif t1 <= -1.25e-94:
		tmp = t_1
	elif t1 <= -1.65e-125:
		tmp = -v / t1
	elif t1 <= 3.8e-27:
		tmp = t_1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(t1 / u) * Float64(Float64(-v) / u))
	tmp = 0.0
	if (t1 <= -1.4e-18)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -1.25e-94)
		tmp = t_1;
	elseif (t1 <= -1.65e-125)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 3.8e-27)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (t1 / u) * (-v / u);
	tmp = 0.0;
	if (t1 <= -1.4e-18)
		tmp = v / (u - t1);
	elseif (t1 <= -1.25e-94)
		tmp = t_1;
	elseif (t1 <= -1.65e-125)
		tmp = -v / t1;
	elseif (t1 <= 3.8e-27)
		tmp = t_1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.4e-18], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.25e-94], t$95$1, If[LessEqual[t1, -1.65e-125], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 3.8e-27], t$95$1, N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -1.40000000000000006e-18

   1. Initial program 68.4%

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

      1. associate-*l/ 74.2%

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

      2. *-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. neg-mul-1 74.2%

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

      2. times-frac 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.8%

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

      2. mul-1-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in t1 around inf 80.6%

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

      1. frac-2neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. metadata-eval 80.6%

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

      4. un-div-inv 80.9%

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

      5. +-commutative 80.9%

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

      6. distribute-neg-in 80.9%

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

      7. add-sqr-sqrt 44.9%

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

      8. sqrt-unprod 80.4%

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

      9. sqr-neg 80.4%

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

      10. sqrt-unprod 36.5%

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

      11. add-sqr-sqrt 81.0%

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

   10. Applied egg-rr 81.0%

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

       1. sub-neg 81.0%

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

   12. Simplified 81.0%

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

   if -1.40000000000000006e-18 < t1 < -1.2499999999999999e-94 or -1.65e-125 < t1 < 3.8e-27

   1. Initial program 86.4%

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

      1. associate-/r* 91.2%

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

      2. associate-/l* 95.3%

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

   3. Simplified 95.3%

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

      1. clear-num 95.2%

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

      2. associate-/r/ 95.2%

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

   5. Applied egg-rr 95.2%

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

   6. Taylor expanded in t1 around 0 80.9%

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

      1. mul-1-neg 80.9%

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

      2. *-commutative 80.9%

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

      3. unpow2 80.9%

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

      4. times-frac 86.0%

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

      5. distribute-rgt-neg-in 86.0%

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

      6. distribute-neg-frac 86.0%

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

   8. Simplified 86.0%

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

   if -1.2499999999999999e-94 < t1 < -1.65e-125

   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/ 99.5%

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

      2. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t1 around inf 72.0%

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

      1. associate-*r/ 72.0%

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

      2. neg-mul-1 72.0%

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

   6. Simplified 72.0%

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

   if 3.8e-27 < t1

   1. Initial program 60.8%

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

      1. associate-/r* 75.6%

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

      2. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.4%

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

      1. neg-mul-1 85.4%

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

   6. Simplified 85.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 6: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{if}\;t1 \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.3 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ t1 (* u (/ (- u) v)))))
   (if (<= t1 -2.5e-17)
     (/ v (- u t1))
     (if (<= t1 -1.3e-94)
       t_1
       (if (<= t1 -1.65e-125)
         (/ (- v) t1)
         (if (<= t1 7.2e-27) t_1 (/ (- v) (+ t1 u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 / (u * (-u / v));
	double tmp;
	if (t1 <= -2.5e-17) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.3e-94) {
		tmp = t_1;
	} else if (t1 <= -1.65e-125) {
		tmp = -v / t1;
	} else if (t1 <= 7.2e-27) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = t1 / (u * (-u / v))
    if (t1 <= (-2.5d-17)) then
        tmp = v / (u - t1)
    else if (t1 <= (-1.3d-94)) then
        tmp = t_1
    else if (t1 <= (-1.65d-125)) then
        tmp = -v / t1
    else if (t1 <= 7.2d-27) then
        tmp = t_1
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 / (u * (-u / v));
	double tmp;
	if (t1 <= -2.5e-17) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.3e-94) {
		tmp = t_1;
	} else if (t1 <= -1.65e-125) {
		tmp = -v / t1;
	} else if (t1 <= 7.2e-27) {
		tmp = t_1;
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 / (u * (-u / v))
	tmp = 0
	if t1 <= -2.5e-17:
		tmp = v / (u - t1)
	elif t1 <= -1.3e-94:
		tmp = t_1
	elif t1 <= -1.65e-125:
		tmp = -v / t1
	elif t1 <= 7.2e-27:
		tmp = t_1
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 / Float64(u * Float64(Float64(-u) / v)))
	tmp = 0.0
	if (t1 <= -2.5e-17)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -1.3e-94)
		tmp = t_1;
	elseif (t1 <= -1.65e-125)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 7.2e-27)
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 / (u * (-u / v));
	tmp = 0.0;
	if (t1 <= -2.5e-17)
		tmp = v / (u - t1);
	elseif (t1 <= -1.3e-94)
		tmp = t_1;
	elseif (t1 <= -1.65e-125)
		tmp = -v / t1;
	elseif (t1 <= 7.2e-27)
		tmp = t_1;
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.5e-17], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.3e-94], t$95$1, If[LessEqual[t1, -1.65e-125], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 7.2e-27], t$95$1, N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t1 < -2.4999999999999999e-17

   1. Initial program 68.4%

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

      1. associate-*l/ 74.2%

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

      2. *-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. neg-mul-1 74.2%

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

      2. times-frac 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.8%

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

      2. mul-1-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in t1 around inf 80.6%

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

      1. frac-2neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. metadata-eval 80.6%

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

      4. un-div-inv 80.9%

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

      5. +-commutative 80.9%

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

      6. distribute-neg-in 80.9%

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

      7. add-sqr-sqrt 44.9%

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

      8. sqrt-unprod 80.4%

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

      9. sqr-neg 80.4%

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

      10. sqrt-unprod 36.5%

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

      11. add-sqr-sqrt 81.0%

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

   10. Applied egg-rr 81.0%

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

       1. sub-neg 81.0%

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

   12. Simplified 81.0%

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

   if -2.4999999999999999e-17 < t1 < -1.29999999999999997e-94 or -1.65e-125 < t1 < 7.1999999999999997e-27

   1. Initial program 86.4%

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

      1. associate-/r* 91.2%

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

      2. associate-/l* 95.3%

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

   3. Simplified 95.3%

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

      1. clear-num 95.2%

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

      2. associate-/r/ 95.2%

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

   5. Applied egg-rr 95.2%

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

   6. Taylor expanded in t1 around 0 80.9%

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

      1. mul-1-neg 80.9%

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

      2. *-commutative 80.9%

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

      3. unpow2 80.9%

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

      4. times-frac 86.0%

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

      5. distribute-rgt-neg-in 86.0%

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

      6. distribute-neg-frac 86.0%

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

   8. Simplified 86.0%

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

      1. clear-num 86.0%

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

      2. frac-2neg 86.0%

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

      3. frac-times 87.5%

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

      4. *-un-lft-identity 87.5%

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

      5. remove-double-neg 87.5%

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

   10. Applied egg-rr 87.5%

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

   if -1.29999999999999997e-94 < t1 < -1.65e-125

   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/ 99.5%

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

      2. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t1 around inf 72.0%

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

      1. associate-*r/ 72.0%

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

      2. neg-mul-1 72.0%

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

   6. Simplified 72.0%

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

   if 7.1999999999999997e-27 < t1

   1. Initial program 60.8%

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

      1. associate-/r* 75.6%

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

      2. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.4%

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

      1. neg-mul-1 85.4%

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

   6. Simplified 85.4%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.3 \cdot 10^{-94}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.62 \cdot 10^{-18}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq -1.4 \cdot 10^{-126}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.62e-18)
   (/ v (- u t1))
   (if (<= t1 -1.25e-94)
     (/ (* v (/ t1 u)) (- u))
     (if (<= t1 -1.4e-126)
       (/ (- v) t1)
       (if (<= t1 9.5e-26) (/ t1 (* u (/ (- u) v))) (/ (- v) (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.62e-18) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.25e-94) {
		tmp = (v * (t1 / u)) / -u;
	} else if (t1 <= -1.4e-126) {
		tmp = -v / t1;
	} else if (t1 <= 9.5e-26) {
		tmp = t1 / (u * (-u / v));
	} 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 (t1 <= (-1.62d-18)) then
        tmp = v / (u - t1)
    else if (t1 <= (-1.25d-94)) then
        tmp = (v * (t1 / u)) / -u
    else if (t1 <= (-1.4d-126)) then
        tmp = -v / t1
    else if (t1 <= 9.5d-26) then
        tmp = t1 / (u * (-u / v))
    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 (t1 <= -1.62e-18) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.25e-94) {
		tmp = (v * (t1 / u)) / -u;
	} else if (t1 <= -1.4e-126) {
		tmp = -v / t1;
	} else if (t1 <= 9.5e-26) {
		tmp = t1 / (u * (-u / v));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.62e-18:
		tmp = v / (u - t1)
	elif t1 <= -1.25e-94:
		tmp = (v * (t1 / u)) / -u
	elif t1 <= -1.4e-126:
		tmp = -v / t1
	elif t1 <= 9.5e-26:
		tmp = t1 / (u * (-u / v))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.62e-18)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -1.25e-94)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	elseif (t1 <= -1.4e-126)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 9.5e-26)
		tmp = Float64(t1 / Float64(u * Float64(Float64(-u) / v)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.62e-18)
		tmp = v / (u - t1);
	elseif (t1 <= -1.25e-94)
		tmp = (v * (t1 / u)) / -u;
	elseif (t1 <= -1.4e-126)
		tmp = -v / t1;
	elseif (t1 <= 9.5e-26)
		tmp = t1 / (u * (-u / v));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.62e-18], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.25e-94], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[LessEqual[t1, -1.4e-126], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 9.5e-26], N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t1 < -1.62000000000000005e-18

   1. Initial program 68.4%

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

      1. associate-*l/ 74.2%

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

      2. *-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. neg-mul-1 74.2%

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

      2. times-frac 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.8%

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

      2. mul-1-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in t1 around inf 80.6%

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

      1. frac-2neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. metadata-eval 80.6%

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

      4. un-div-inv 80.9%

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

      5. +-commutative 80.9%

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

      6. distribute-neg-in 80.9%

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

      7. add-sqr-sqrt 44.9%

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

      8. sqrt-unprod 80.4%

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

      9. sqr-neg 80.4%

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

      10. sqrt-unprod 36.5%

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

      11. add-sqr-sqrt 81.0%

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

   10. Applied egg-rr 81.0%

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

       1. sub-neg 81.0%

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

   12. Simplified 81.0%

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

   if -1.62000000000000005e-18 < t1 < -1.2499999999999999e-94

   1. Initial program 99.5%

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

      1. associate-/r* 100.0%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in t1 around 0 92.4%

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

      1. mul-1-neg 92.4%

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

      2. *-commutative 92.4%

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

      3. unpow2 92.4%

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

      4. times-frac 92.5%

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

      5. distribute-rgt-neg-in 92.5%

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

      6. distribute-neg-frac 92.5%

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

   8. Simplified 92.5%

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

      1. frac-2neg 92.5%

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

      2. associate-*l/ 92.8%

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

      3. add-sqr-sqrt 69.1%

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

      4. sqrt-unprod 70.0%

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

      5. sqr-neg 70.0%

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

      6. sqrt-unprod 15.7%

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

      7. add-sqr-sqrt 47.2%

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

      8. add-sqr-sqrt 47.2%

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

      9. sqrt-unprod 47.2%

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

      10. sqr-neg 47.2%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 92.8%

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

   10. Applied egg-rr 92.8%

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

   if -1.2499999999999999e-94 < t1 < -1.39999999999999996e-126

   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/ 99.5%

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

      2. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t1 around inf 72.0%

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

      1. associate-*r/ 72.0%

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

      2. neg-mul-1 72.0%

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

   6. Simplified 72.0%

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

   if -1.39999999999999996e-126 < t1 < 9.4999999999999995e-26

   1. Initial program 84.5%

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

      1. associate-/r* 89.9%

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

      2. associate-/l* 94.6%

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

   3. Simplified 94.6%

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

      1. clear-num 94.5%

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

      2. associate-/r/ 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in t1 around 0 79.2%

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

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. unpow2 79.2%

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

      4. times-frac 85.1%

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

      5. distribute-rgt-neg-in 85.1%

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

      6. distribute-neg-frac 85.1%

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

   8. Simplified 85.1%

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

      1. clear-num 85.1%

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

      2. frac-2neg 85.1%

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

      3. frac-times 86.8%

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

      4. *-un-lft-identity 86.8%

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

      5. remove-double-neg 86.8%

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

   10. Applied egg-rr 86.8%

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

   if 9.4999999999999995e-26 < t1

   1. Initial program 60.8%

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

      1. associate-/r* 75.6%

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

      2. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.4%

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

      1. neg-mul-1 85.4%

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

   6. Simplified 85.4%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.62 \cdot 10^{-18}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{elif}\;t1 \leq -1.4 \cdot 10^{-126}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 8: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{t1 - u}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 7.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.35e-18)
   (/ v (- u t1))
   (if (<= t1 -1.5e-94)
     (/ (/ (* t1 v) u) (- t1 u))
     (if (<= t1 -1.65e-125)
       (/ (- v) t1)
       (if (<= t1 7.8e-25) (/ t1 (* u (/ (- u) v))) (/ (- v) (+ t1 u)))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.35e-18) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.5e-94) {
		tmp = ((t1 * v) / u) / (t1 - u);
	} else if (t1 <= -1.65e-125) {
		tmp = -v / t1;
	} else if (t1 <= 7.8e-25) {
		tmp = t1 / (u * (-u / v));
	} 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 (t1 <= (-1.35d-18)) then
        tmp = v / (u - t1)
    else if (t1 <= (-1.5d-94)) then
        tmp = ((t1 * v) / u) / (t1 - u)
    else if (t1 <= (-1.65d-125)) then
        tmp = -v / t1
    else if (t1 <= 7.8d-25) then
        tmp = t1 / (u * (-u / v))
    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 (t1 <= -1.35e-18) {
		tmp = v / (u - t1);
	} else if (t1 <= -1.5e-94) {
		tmp = ((t1 * v) / u) / (t1 - u);
	} else if (t1 <= -1.65e-125) {
		tmp = -v / t1;
	} else if (t1 <= 7.8e-25) {
		tmp = t1 / (u * (-u / v));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.35e-18:
		tmp = v / (u - t1)
	elif t1 <= -1.5e-94:
		tmp = ((t1 * v) / u) / (t1 - u)
	elif t1 <= -1.65e-125:
		tmp = -v / t1
	elif t1 <= 7.8e-25:
		tmp = t1 / (u * (-u / v))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.35e-18)
		tmp = Float64(v / Float64(u - t1));
	elseif (t1 <= -1.5e-94)
		tmp = Float64(Float64(Float64(t1 * v) / u) / Float64(t1 - u));
	elseif (t1 <= -1.65e-125)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 7.8e-25)
		tmp = Float64(t1 / Float64(u * Float64(Float64(-u) / v)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.35e-18)
		tmp = v / (u - t1);
	elseif (t1 <= -1.5e-94)
		tmp = ((t1 * v) / u) / (t1 - u);
	elseif (t1 <= -1.65e-125)
		tmp = -v / t1;
	elseif (t1 <= 7.8e-25)
		tmp = t1 / (u * (-u / v));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.35e-18], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.5e-94], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, -1.65e-125], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 7.8e-25], N[(t1 / N[(u * N[((-u) / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t1 < -1.34999999999999994e-18

   1. Initial program 68.4%

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

      1. associate-*l/ 74.2%

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

      2. *-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. neg-mul-1 74.2%

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

      2. times-frac 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.8%

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

      2. mul-1-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in t1 around inf 80.6%

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

      1. frac-2neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. metadata-eval 80.6%

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

      4. un-div-inv 80.9%

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

      5. +-commutative 80.9%

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

      6. distribute-neg-in 80.9%

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

      7. add-sqr-sqrt 44.9%

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

      8. sqrt-unprod 80.4%

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

      9. sqr-neg 80.4%

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

      10. sqrt-unprod 36.5%

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

      11. add-sqr-sqrt 81.0%

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

   10. Applied egg-rr 81.0%

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

       1. sub-neg 81.0%

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

   12. Simplified 81.0%

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

   if -1.34999999999999994e-18 < t1 < -1.5000000000000001e-94

   1. Initial program 99.5%

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

      1. associate-/r* 100.0%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 92.4%

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

      1. mul-1-neg 92.4%

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

      2. associate-/l* 92.1%

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

      3. distribute-neg-frac 92.1%

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

   6. Simplified 92.1%

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

      1. expm1-log1p-u 84.5%

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

      2. expm1-udef 54.6%

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

   8. Applied egg-rr 55.2%

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

      1. expm1-def 85.2%

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

      2. expm1-log1p 92.9%

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

      3. associate-*l/ 92.9%

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

   10. Simplified 92.9%

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

   if -1.5000000000000001e-94 < t1 < -1.65e-125

   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/ 99.5%

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

      2. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t1 around inf 72.0%

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

      1. associate-*r/ 72.0%

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

      2. neg-mul-1 72.0%

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

   6. Simplified 72.0%

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

   if -1.65e-125 < t1 < 7.8e-25

   1. Initial program 84.5%

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

      1. associate-/r* 89.9%

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

      2. associate-/l* 94.6%

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

   3. Simplified 94.6%

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

      1. clear-num 94.5%

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

      2. associate-/r/ 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in t1 around 0 79.2%

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

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. unpow2 79.2%

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

      4. times-frac 85.1%

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

      5. distribute-rgt-neg-in 85.1%

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

      6. distribute-neg-frac 85.1%

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

   8. Simplified 85.1%

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

      1. clear-num 85.1%

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

      2. frac-2neg 85.1%

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

      3. frac-times 86.8%

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

      4. *-un-lft-identity 86.8%

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

      5. remove-double-neg 86.8%

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

   10. Applied egg-rr 86.8%

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

   if 7.8e-25 < t1

   1. Initial program 60.8%

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

      1. associate-/r* 75.6%

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

      2. associate-/l* 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in t1 around inf 85.4%

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

      1. neg-mul-1 85.4%

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

   6. Simplified 85.4%

      \[\leadsto \frac{\color{blue}{-v}}{t1 + u} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq -1.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{t1 - u}\\ \mathbf{elif}\;t1 \leq -1.65 \cdot 10^{-125}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 7.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 9: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. neg-mul-1 77.7%

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

   2. times-frac 95.3%

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

5. Applied egg-rr 95.3%

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

   1. associate-*l/ 95.3%

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

   2. mul-1-neg 95.3%

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

7. Simplified 95.3%

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

8. Final simplification 95.3%

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

Alternative 10: 68.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+68} \lor \neg \left(u \leq 2.9 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.2e+68) (not (<= u 2.9e+137)))
   (/ t1 (* u (/ u v)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.2e+68) || !(u <= 2.9e+137)) {
		tmp = t1 / (u * (u / v));
	} 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 <= (-2.2d+68)) .or. (.not. (u <= 2.9d+137))) then
        tmp = t1 / (u * (u / v))
    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 <= -2.2e+68) || !(u <= 2.9e+137)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.2e+68) or not (u <= 2.9e+137):
		tmp = t1 / (u * (u / v))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.2e+68) || !(u <= 2.9e+137))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.2e+68) || ~((u <= 2.9e+137)))
		tmp = t1 / (u * (u / v));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.2e+68], N[Not[LessEqual[u, 2.9e+137]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.19999999999999987e68 or 2.89999999999999985e137 < u

   1. Initial program 78.0%

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

      1. associate-/r* 88.0%

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

      2. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in t1 around 0 74.6%

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

      1. mul-1-neg 74.6%

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

      2. *-commutative 74.6%

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

      3. unpow2 74.6%

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

      4. times-frac 87.3%

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

      5. distribute-rgt-neg-in 87.3%

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

      6. distribute-neg-frac 87.3%

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

   8. Simplified 87.3%

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

      1. clear-num 88.5%

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

      2. frac-times 84.5%

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

      3. *-un-lft-identity 84.5%

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

      4. add-sqr-sqrt 41.0%

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

      5. sqrt-unprod 66.0%

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

      6. sqr-neg 66.0%

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

      7. sqrt-unprod 35.1%

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

      8. add-sqr-sqrt 66.5%

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

   10. Applied egg-rr 66.5%

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

   if -2.19999999999999987e68 < u < 2.89999999999999985e137

   1. Initial program 71.6%

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

      1. associate-/r* 81.0%

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

      2. associate-/l* 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in t1 around inf 69.3%

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

      1. neg-mul-1 69.3%

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

   6. Simplified 69.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+68} \lor \neg \left(u \leq 2.9 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 11: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+68} \lor \neg \left(u \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.3e+68) (not (<= u 2e+200))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e+68) || !(u <= 2e+200)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.3d+68)) .or. (.not. (u <= 2d+200))) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.3e+68) || !(u <= 2e+200)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.3e+68) or not (u <= 2e+200):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.3e+68) || !(u <= 2e+200))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.3e+68) || ~((u <= 2e+200)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.3e+68], N[Not[LessEqual[u, 2e+200]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.3e68 or 1.9999999999999999e200 < 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-/r* 89.8%

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

      2. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. associate-/l* 94.7%

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

      3. distribute-neg-frac 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in t1 around inf 42.4%

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

      1. neg-mul-1 42.4%

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

      2. distribute-neg-frac 42.4%

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

   9. Simplified 42.4%

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

   if -2.3e68 < u < 1.9999999999999999e200

   1. Initial program 71.5%

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

      1. associate-*l/ 77.2%

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

      2. *-commutative 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in t1 around inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. neg-mul-1 64.2%

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

   6. Simplified 64.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+68} \lor \neg \left(u \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 57.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+68}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+199}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.3e+68) (/ v u) (if (<= u 1.55e+199) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.3e+68) {
		tmp = v / u;
	} else if (u <= 1.55e+199) {
		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 <= (-2.3d+68)) then
        tmp = v / u
    else if (u <= 1.55d+199) 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 <= -2.3e+68) {
		tmp = v / u;
	} else if (u <= 1.55e+199) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.3e+68:
		tmp = v / u
	elif u <= 1.55e+199:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.3e+68)
		tmp = Float64(v / u);
	elseif (u <= 1.55e+199)
		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 <= -2.3e+68)
		tmp = v / u;
	elseif (u <= 1.55e+199)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.3e+68], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.55e+199], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.3e68 or 1.54999999999999993e199 < 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-/r* 89.8%

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

      2. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t1 around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. associate-/l* 94.7%

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

      3. distribute-neg-frac 94.7%

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

   6. Simplified 94.7%

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

      1. expm1-log1p-u 92.8%

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

      2. expm1-udef 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. expm1-def 91.0%

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

      2. expm1-log1p 93.0%

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

      3. associate-/l* 93.4%

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

   10. Simplified 93.4%

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

   11. Taylor expanded in t1 around inf 42.3%

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

   if -2.3e68 < u < 1.54999999999999993e199

   1. Initial program 71.5%

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

      1. associate-*l/ 77.2%

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

      2. *-commutative 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in t1 around inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. neg-mul-1 64.2%

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

   6. Simplified 64.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+68}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+199}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 13: 23.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+120}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{+90}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.6e+120) (/ v t1) (if (<= t1 1.75e+90) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.6e+120) {
		tmp = v / t1;
	} else if (t1 <= 1.75e+90) {
		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 <= (-1.6d+120)) then
        tmp = v / t1
    else if (t1 <= 1.75d+90) 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 <= -1.6e+120) {
		tmp = v / t1;
	} else if (t1 <= 1.75e+90) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.6e+120:
		tmp = v / t1
	elif t1 <= 1.75e+90:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.6e+120)
		tmp = Float64(v / t1);
	elseif (t1 <= 1.75e+90)
		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 <= -1.6e+120)
		tmp = v / t1;
	elseif (t1 <= 1.75e+90)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.6e+120], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 1.75e+90], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.59999999999999991e120 or 1.7499999999999999e90 < t1

   1. Initial program 46.6%

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

      1. associate-/r* 66.8%

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

      2. associate-/l* 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t1 around inf 83.6%

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

      1. neg-mul-1 83.6%

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

      2. +-commutative 83.6%

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

      3. unsub-neg 83.6%

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

      4. associate-/l* 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in u around inf 35.2%

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

   if -1.59999999999999991e120 < t1 < 1.7499999999999999e90

   1. Initial program 87.1%

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

      1. associate-/r* 91.4%

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

      2. associate-/l* 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in t1 around 0 63.9%

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

      1. mul-1-neg 63.9%

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

      2. associate-/l* 67.5%

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

      3. distribute-neg-frac 67.5%

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

   6. Simplified 67.5%

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

      1. expm1-log1p-u 59.5%

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

      2. expm1-udef 38.8%

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

   8. Applied egg-rr 38.1%

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

      1. expm1-def 56.6%

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

      2. expm1-log1p 64.6%

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

      3. associate-/l* 64.4%

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

   10. Simplified 64.4%

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

   11. Taylor expanded in t1 around inf 15.6%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+120}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.75 \cdot 10^{+90}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 14: 61.8% 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 73.6%

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

   1. associate-/r* 83.2%

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

   2. associate-/l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in t1 around inf 61.8%

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

   1. neg-mul-1 61.8%

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

6. Simplified 61.8%

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

7. Final simplification 61.8%

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

Alternative 15: 61.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. neg-mul-1 77.7%

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

   2. times-frac 95.3%

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

5. Applied egg-rr 95.3%

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

   1. associate-*l/ 95.3%

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

   2. mul-1-neg 95.3%

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

7. Simplified 95.3%

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

8. Taylor expanded in t1 around inf 61.7%

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

   1. frac-2neg 61.7%

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

   2. metadata-eval 61.7%

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

   3. metadata-eval 61.7%

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

   4. un-div-inv 61.8%

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

   5. +-commutative 61.8%

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

   6. distribute-neg-in 61.8%

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

   7. add-sqr-sqrt 29.2%

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

   8. sqrt-unprod 66.2%

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

   9. sqr-neg 66.2%

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

   10. sqrt-unprod 32.0%

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

   11. add-sqr-sqrt 61.0%

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

10. Applied egg-rr 61.0%

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

    1. sub-neg 61.0%

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

12. Simplified 61.0%

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

13. Final simplification 61.0%

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

Alternative 16: 14.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

   1. associate-/r* 83.2%

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

   2. associate-/l* 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in t1 around inf 51.0%

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

   1. neg-mul-1 51.0%

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

   2. +-commutative 51.0%

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

   3. unsub-neg 51.0%

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

   4. associate-/l* 51.4%

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

6. Simplified 51.4%

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

7. Taylor expanded in u around inf 13.9%

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

8. Final simplification 13.9%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))