Rosa's DopplerBench

Percentage Accurate: 72.5% → 97.8%

Time: 8.7s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. associate-/r* 83.5%

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

   2. associate-/l* 97.2%

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

3. Simplified 97.2%

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

4. Final simplification 97.2%

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

Alternative 2: 89.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t1 \leq -3.3 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))) (t_2 (* v (/ (- t1) (* (+ t1 u) (+ t1 u))))))
   (if (<= t1 -3.3e+80)
     t_1
     (if (<= t1 -3.2e-201)
       t_2
       (if (<= t1 2.2e-149)
         (/ (/ (- t1) (/ u v)) (+ t1 u))
         (if (<= t1 2.3e+86) t_2 t_1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -3.3e+80) {
		tmp = t_1;
	} else if (t1 <= -3.2e-201) {
		tmp = t_2;
	} else if (t1 <= 2.2e-149) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (t1 <= 2.3e+86) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    t_2 = v * (-t1 / ((t1 + u) * (t1 + u)))
    if (t1 <= (-3.3d+80)) then
        tmp = t_1
    else if (t1 <= (-3.2d-201)) then
        tmp = t_2
    else if (t1 <= 2.2d-149) then
        tmp = (-t1 / (u / v)) / (t1 + u)
    else if (t1 <= 2.3d+86) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	double tmp;
	if (t1 <= -3.3e+80) {
		tmp = t_1;
	} else if (t1 <= -3.2e-201) {
		tmp = t_2;
	} else if (t1 <= 2.2e-149) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if (t1 <= 2.3e+86) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	t_2 = v * (-t1 / ((t1 + u) * (t1 + u)))
	tmp = 0
	if t1 <= -3.3e+80:
		tmp = t_1
	elif t1 <= -3.2e-201:
		tmp = t_2
	elif t1 <= 2.2e-149:
		tmp = (-t1 / (u / v)) / (t1 + u)
	elif t1 <= 2.3e+86:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	t_2 = Float64(v * Float64(Float64(-t1) / Float64(Float64(t1 + u) * Float64(t1 + u))))
	tmp = 0.0
	if (t1 <= -3.3e+80)
		tmp = t_1;
	elseif (t1 <= -3.2e-201)
		tmp = t_2;
	elseif (t1 <= 2.2e-149)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	elseif (t1 <= 2.3e+86)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	t_2 = v * (-t1 / ((t1 + u) * (t1 + u)));
	tmp = 0.0;
	if (t1 <= -3.3e+80)
		tmp = t_1;
	elseif (t1 <= -3.2e-201)
		tmp = t_2;
	elseif (t1 <= 2.2e-149)
		tmp = (-t1 / (u / v)) / (t1 + u);
	elseif (t1 <= 2.3e+86)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(v * N[((-t1) / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.3e+80], t$95$1, If[LessEqual[t1, -3.2e-201], t$95$2, If[LessEqual[t1, 2.2e-149], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.3e+86], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.29999999999999991e80 or 2.2999999999999999e86 < t1

   1. Initial program 60.0%

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

      1. associate-/r* 77.1%

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

      2. associate-/l* 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t1 around inf 92.2%

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

      1. neg-mul-1 92.2%

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

   6. Simplified 92.2%

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

   if -3.29999999999999991e80 < t1 < -3.2000000000000001e-201 or 2.1999999999999998e-149 < t1 < 2.2999999999999999e86

   1. Initial program 89.2%

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

      1. associate-*l/ 94.5%

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

      2. *-commutative 94.5%

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

   3. Simplified 94.5%

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

   if -3.2000000000000001e-201 < t1 < 2.1999999999999998e-149

   1. Initial program 72.8%

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

      1. associate-/r* 79.6%

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

      2. associate-/l* 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in t1 around 0 78.1%

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

      1. mul-1-neg 78.1%

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

      2. associate-/l* 83.4%

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

      3. distribute-neg-frac 83.4%

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

   6. Simplified 83.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.3 \cdot 10^{+86}:\\ \;\;\;\;v \cdot \frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 3: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37} \lor \neg \left(t1 \leq 2.5 \cdot 10^{-86} \lor \neg \left(t1 \leq 1.3 \cdot 10^{-9}\right) \land t1 \leq 29.5\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.22e+37)
         (not (or (<= t1 2.5e-86) (and (not (<= t1 1.3e-9)) (<= t1 29.5)))))
   (/ (- v) (+ t1 u))
   (* v (/ (- t1) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.22e+37) || !((t1 <= 2.5e-86) || (!(t1 <= 1.3e-9) && (t1 <= 29.5)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.22d+37)) .or. (.not. (t1 <= 2.5d-86) .or. (.not. (t1 <= 1.3d-9)) .and. (t1 <= 29.5d0))) then
        tmp = -v / (t1 + u)
    else
        tmp = v * (-t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.22e+37) || !((t1 <= 2.5e-86) || (!(t1 <= 1.3e-9) && (t1 <= 29.5)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * (-t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.22e+37) or not ((t1 <= 2.5e-86) or (not (t1 <= 1.3e-9) and (t1 <= 29.5))):
		tmp = -v / (t1 + u)
	else:
		tmp = v * (-t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.22e+37) || !((t1 <= 2.5e-86) || (!(t1 <= 1.3e-9) && (t1 <= 29.5))))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(v * Float64(Float64(-t1) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.22e+37) || ~(((t1 <= 2.5e-86) || (~((t1 <= 1.3e-9)) && (t1 <= 29.5)))))
		tmp = -v / (t1 + u);
	else
		tmp = v * (-t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.22e+37], N[Not[Or[LessEqual[t1, 2.5e-86], And[N[Not[LessEqual[t1, 1.3e-9]], $MachinePrecision], LessEqual[t1, 29.5]]]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v * N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.22e37 or 2.4999999999999999e-86 < t1 < 1.3000000000000001e-9 or 29.5 < t1

   1. Initial program 67.5%

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

      1. associate-/r* 80.9%

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

      2. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 87.9%

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

      1. neg-mul-1 87.9%

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

   6. Simplified 87.9%

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

   if -1.22e37 < t1 < 2.4999999999999999e-86 or 1.3000000000000001e-9 < t1 < 29.5

   1. Initial program 80.7%

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

      1. associate-*l/ 82.2%

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

      2. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in t1 around 0 76.3%

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

      1. associate-*r/ 76.3%

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

      2. neg-mul-1 76.3%

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

      3. unpow2 76.3%

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

   6. Simplified 76.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37} \lor \neg \left(t1 \leq 2.5 \cdot 10^{-86} \lor \neg \left(t1 \leq 1.3 \cdot 10^{-9}\right) \land t1 \leq 29.5\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \end{array} \]

Alternative 4: 76.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-87} \lor \neg \left(t1 \leq 1.6 \cdot 10^{-8}\right) \land t1 \leq 7.2 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.22e+37)
         (not (or (<= t1 1.8e-87) (and (not (<= t1 1.6e-8)) (<= t1 7.2e+71)))))
   (/ (- v) (+ t1 u))
   (* (- v) (/ (/ t1 u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.22e+37) || !((t1 <= 1.8e-87) || (!(t1 <= 1.6e-8) && (t1 <= 7.2e+71)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.22d+37)) .or. (.not. (t1 <= 1.8d-87) .or. (.not. (t1 <= 1.6d-8)) .and. (t1 <= 7.2d+71))) then
        tmp = -v / (t1 + u)
    else
        tmp = -v * ((t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.22e+37) || !((t1 <= 1.8e-87) || (!(t1 <= 1.6e-8) && (t1 <= 7.2e+71)))) {
		tmp = -v / (t1 + u);
	} else {
		tmp = -v * ((t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.22e+37) or not ((t1 <= 1.8e-87) or (not (t1 <= 1.6e-8) and (t1 <= 7.2e+71))):
		tmp = -v / (t1 + u)
	else:
		tmp = -v * ((t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.22e+37) || !((t1 <= 1.8e-87) || (!(t1 <= 1.6e-8) && (t1 <= 7.2e+71))))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.22e+37) || ~(((t1 <= 1.8e-87) || (~((t1 <= 1.6e-8)) && (t1 <= 7.2e+71)))))
		tmp = -v / (t1 + u);
	else
		tmp = -v * ((t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.22e+37], N[Not[Or[LessEqual[t1, 1.8e-87], And[N[Not[LessEqual[t1, 1.6e-8]], $MachinePrecision], LessEqual[t1, 7.2e+71]]]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.22e37 or 1.79999999999999996e-87 < t1 < 1.6000000000000001e-8 or 7.1999999999999999e71 < t1

   1. Initial program 66.8%

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

      1. associate-/r* 80.6%

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

      2. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 91.3%

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

      1. neg-mul-1 91.3%

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

   6. Simplified 91.3%

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

   if -1.22e37 < t1 < 1.79999999999999996e-87 or 1.6000000000000001e-8 < t1 < 7.1999999999999999e71

   1. Initial program 80.2%

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

      1. associate-*l/ 82.3%

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

      2. *-commutative 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in t1 around 0 73.1%

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

      1. associate-*r/ 73.1%

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

      2. neg-mul-1 73.1%

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

      3. unpow2 73.1%

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

   6. Simplified 73.1%

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

      1. neg-mul-1 73.1%

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

      2. times-frac 79.5%

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

   8. Applied egg-rr 79.5%

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

      1. associate-*l/ 79.5%

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

      2. mul-1-neg 79.5%

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

   10. Simplified 79.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-87} \lor \neg \left(t1 \leq 1.6 \cdot 10^{-8}\right) \land t1 \leq 7.2 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 5: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3.4 \cdot 10^{-85}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -1.22e+37)
     t_1
     (if (<= t1 3.4e-85)
       (* (- v) (/ (/ t1 u) u))
       (if (or (<= t1 4.6e-9) (not (<= t1 4.2e+86)))
         t_1
         (/ t1 (/ (- u) (/ v u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1.22e+37) {
		tmp = t_1;
	} else if (t1 <= 3.4e-85) {
		tmp = -v * ((t1 / u) / u);
	} else if ((t1 <= 4.6e-9) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = t1 / (-u / (v / u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    if (t1 <= (-1.22d+37)) then
        tmp = t_1
    else if (t1 <= 3.4d-85) then
        tmp = -v * ((t1 / u) / u)
    else if ((t1 <= 4.6d-9) .or. (.not. (t1 <= 4.2d+86))) then
        tmp = t_1
    else
        tmp = t1 / (-u / (v / u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -1.22e+37) {
		tmp = t_1;
	} else if (t1 <= 3.4e-85) {
		tmp = -v * ((t1 / u) / u);
	} else if ((t1 <= 4.6e-9) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = t1 / (-u / (v / u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -1.22e+37:
		tmp = t_1
	elif t1 <= 3.4e-85:
		tmp = -v * ((t1 / u) / u)
	elif (t1 <= 4.6e-9) or not (t1 <= 4.2e+86):
		tmp = t_1
	else:
		tmp = t1 / (-u / (v / u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -1.22e+37)
		tmp = t_1;
	elseif (t1 <= 3.4e-85)
		tmp = Float64(Float64(-v) * Float64(Float64(t1 / u) / u));
	elseif ((t1 <= 4.6e-9) || !(t1 <= 4.2e+86))
		tmp = t_1;
	else
		tmp = Float64(t1 / Float64(Float64(-u) / Float64(v / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -1.22e+37)
		tmp = t_1;
	elseif (t1 <= 3.4e-85)
		tmp = -v * ((t1 / u) / u);
	elseif ((t1 <= 4.6e-9) || ~((t1 <= 4.2e+86)))
		tmp = t_1;
	else
		tmp = t1 / (-u / (v / u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.22e+37], t$95$1, If[LessEqual[t1, 3.4e-85], N[((-v) * N[(N[(t1 / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 4.6e-9], N[Not[LessEqual[t1, 4.2e+86]], $MachinePrecision]], t$95$1, N[(t1 / N[((-u) / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.22e37 or 3.4e-85 < t1 < 4.5999999999999998e-9 or 4.1999999999999998e86 < t1

   1. Initial program 67.3%

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

      1. associate-/r* 81.2%

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

      2. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 92.0%

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

      1. neg-mul-1 92.0%

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

   6. Simplified 92.0%

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

   if -1.22e37 < t1 < 3.4e-85

   1. Initial program 79.8%

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

      1. associate-*l/ 81.4%

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

      2. *-commutative 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in t1 around 0 75.2%

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

      1. associate-*r/ 75.2%

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

      2. neg-mul-1 75.2%

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

      3. unpow2 75.2%

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

   6. Simplified 75.2%

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

      1. neg-mul-1 75.2%

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

      2. times-frac 81.7%

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

   8. Applied egg-rr 81.7%

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

      1. associate-*l/ 81.7%

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

      2. mul-1-neg 81.7%

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

   10. Simplified 81.7%

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

   if 4.5999999999999998e-9 < t1 < 4.1999999999999998e86

   1. Initial program 77.9%

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

      1. associate-*l/ 83.7%

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

      2. *-commutative 83.7%

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

   3. Simplified 83.7%

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

      1. associate-/r* 89.1%

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

      2. associate-*r/ 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r/ 99.3%

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

      5. div-inv 99.5%

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

      6. frac-2neg 99.5%

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

      7. frac-times 99.3%

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

      8. remove-double-neg 99.3%

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

      9. *-commutative 99.3%

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

      10. *-un-lft-identity 99.3%

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

      11. distribute-neg-frac 99.3%

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

      12. distribute-neg-in 99.3%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 72.1%

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

      15. sqr-neg 72.1%

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

      16. sqrt-unprod 72.1%

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

      17. add-sqr-sqrt 72.1%

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

      18. sub-neg 72.1%

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

   5. Applied egg-rr 72.1%

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

   6. Taylor expanded in t1 around 0 56.5%

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

      1. unpow2 56.5%

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

      2. neg-mul-1 56.5%

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

      3. associate-/l* 72.4%

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

      4. distribute-neg-frac 72.4%

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

   8. Simplified 72.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+37}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.4 \cdot 10^{-85}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 6: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.8 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{v}{\frac{u}{t1}}}{-u}\\ \mathbf{elif}\;t1 \leq 3.5 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -7.8e+39)
     t_1
     (if (<= t1 5.2e-88)
       (/ (/ v (/ u t1)) (- u))
       (if (or (<= t1 3.5e-9) (not (<= t1 4.2e+86)))
         t_1
         (/ t1 (/ (- u) (/ v u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7.8e+39) {
		tmp = t_1;
	} else if (t1 <= 5.2e-88) {
		tmp = (v / (u / t1)) / -u;
	} else if ((t1 <= 3.5e-9) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = t1 / (-u / (v / u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    if (t1 <= (-7.8d+39)) then
        tmp = t_1
    else if (t1 <= 5.2d-88) then
        tmp = (v / (u / t1)) / -u
    else if ((t1 <= 3.5d-9) .or. (.not. (t1 <= 4.2d+86))) then
        tmp = t_1
    else
        tmp = t1 / (-u / (v / u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -7.8e+39) {
		tmp = t_1;
	} else if (t1 <= 5.2e-88) {
		tmp = (v / (u / t1)) / -u;
	} else if ((t1 <= 3.5e-9) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = t1 / (-u / (v / u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -7.8e+39:
		tmp = t_1
	elif t1 <= 5.2e-88:
		tmp = (v / (u / t1)) / -u
	elif (t1 <= 3.5e-9) or not (t1 <= 4.2e+86):
		tmp = t_1
	else:
		tmp = t1 / (-u / (v / u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -7.8e+39)
		tmp = t_1;
	elseif (t1 <= 5.2e-88)
		tmp = Float64(Float64(v / Float64(u / t1)) / Float64(-u));
	elseif ((t1 <= 3.5e-9) || !(t1 <= 4.2e+86))
		tmp = t_1;
	else
		tmp = Float64(t1 / Float64(Float64(-u) / Float64(v / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -7.8e+39)
		tmp = t_1;
	elseif (t1 <= 5.2e-88)
		tmp = (v / (u / t1)) / -u;
	elseif ((t1 <= 3.5e-9) || ~((t1 <= 4.2e+86)))
		tmp = t_1;
	else
		tmp = t1 / (-u / (v / u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -7.8e+39], t$95$1, If[LessEqual[t1, 5.2e-88], N[(N[(v / N[(u / t1), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], If[Or[LessEqual[t1, 3.5e-9], N[Not[LessEqual[t1, 4.2e+86]], $MachinePrecision]], t$95$1, N[(t1 / N[((-u) / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -7.8000000000000002e39 or 5.20000000000000027e-88 < t1 < 3.4999999999999999e-9 or 4.1999999999999998e86 < t1

   1. Initial program 67.3%

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

      1. associate-/r* 81.2%

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

      2. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 92.0%

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

      1. neg-mul-1 92.0%

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

   6. Simplified 92.0%

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

   if -7.8000000000000002e39 < t1 < 5.20000000000000027e-88

   1. Initial program 79.8%

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

      1. associate-*l/ 81.4%

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

      2. *-commutative 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in t1 around 0 75.2%

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

      1. associate-*r/ 75.2%

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

      2. neg-mul-1 75.2%

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

      3. unpow2 75.2%

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

   6. Simplified 75.2%

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

      1. associate-*r/ 75.4%

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

      2. *-commutative 75.4%

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

      3. associate-/r* 80.1%

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

      4. frac-2neg 80.1%

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

   8. Applied egg-rr 83.1%

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

   if 3.4999999999999999e-9 < t1 < 4.1999999999999998e86

   1. Initial program 77.9%

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

      1. associate-*l/ 83.7%

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

      2. *-commutative 83.7%

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

   3. Simplified 83.7%

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

      1. associate-/r* 89.1%

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

      2. associate-*r/ 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r/ 99.3%

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

      5. div-inv 99.5%

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

      6. frac-2neg 99.5%

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

      7. frac-times 99.3%

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

      8. remove-double-neg 99.3%

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

      9. *-commutative 99.3%

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

      10. *-un-lft-identity 99.3%

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

      11. distribute-neg-frac 99.3%

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

      12. distribute-neg-in 99.3%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 72.1%

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

      15. sqr-neg 72.1%

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

      16. sqrt-unprod 72.1%

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

      17. add-sqr-sqrt 72.1%

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

      18. sub-neg 72.1%

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

   5. Applied egg-rr 72.1%

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

   6. Taylor expanded in t1 around 0 56.5%

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

      1. unpow2 56.5%

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

      2. neg-mul-1 56.5%

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

      3. associate-/l* 72.4%

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

      4. distribute-neg-frac 72.4%

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

   8. Simplified 72.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{v}{\frac{u}{t1}}}{-u}\\ \mathbf{elif}\;t1 \leq 3.5 \cdot 10^{-9} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 7: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.15 \cdot 10^{+38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{-8} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{u}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (+ t1 u))))
   (if (<= t1 -3.15e+38)
     t_1
     (if (<= t1 6e-85)
       (/ (/ (- t1) (/ u v)) (+ t1 u))
       (if (or (<= t1 1.5e-8) (not (<= t1 4.2e+86)))
         t_1
         (/ t1 (/ (- u) (/ v u))))))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -3.15e+38) {
		tmp = t_1;
	} else if (t1 <= 6e-85) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if ((t1 <= 1.5e-8) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = t1 / (-u / (v / u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -v / (t1 + u)
    if (t1 <= (-3.15d+38)) then
        tmp = t_1
    else if (t1 <= 6d-85) then
        tmp = (-t1 / (u / v)) / (t1 + u)
    else if ((t1 <= 1.5d-8) .or. (.not. (t1 <= 4.2d+86))) then
        tmp = t_1
    else
        tmp = t1 / (-u / (v / u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / (t1 + u);
	double tmp;
	if (t1 <= -3.15e+38) {
		tmp = t_1;
	} else if (t1 <= 6e-85) {
		tmp = (-t1 / (u / v)) / (t1 + u);
	} else if ((t1 <= 1.5e-8) || !(t1 <= 4.2e+86)) {
		tmp = t_1;
	} else {
		tmp = t1 / (-u / (v / u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / (t1 + u)
	tmp = 0
	if t1 <= -3.15e+38:
		tmp = t_1
	elif t1 <= 6e-85:
		tmp = (-t1 / (u / v)) / (t1 + u)
	elif (t1 <= 1.5e-8) or not (t1 <= 4.2e+86):
		tmp = t_1
	else:
		tmp = t1 / (-u / (v / u))
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -3.15e+38)
		tmp = t_1;
	elseif (t1 <= 6e-85)
		tmp = Float64(Float64(Float64(-t1) / Float64(u / v)) / Float64(t1 + u));
	elseif ((t1 <= 1.5e-8) || !(t1 <= 4.2e+86))
		tmp = t_1;
	else
		tmp = Float64(t1 / Float64(Float64(-u) / Float64(v / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -3.15e+38)
		tmp = t_1;
	elseif (t1 <= 6e-85)
		tmp = (-t1 / (u / v)) / (t1 + u);
	elseif ((t1 <= 1.5e-8) || ~((t1 <= 4.2e+86)))
		tmp = t_1;
	else
		tmp = t1 / (-u / (v / u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.15e+38], t$95$1, If[LessEqual[t1, 6e-85], N[(N[((-t1) / N[(u / v), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t1, 1.5e-8], N[Not[LessEqual[t1, 4.2e+86]], $MachinePrecision]], t$95$1, N[(t1 / N[((-u) / N[(v / u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -3.15000000000000001e38 or 6.00000000000000044e-85 < t1 < 1.49999999999999987e-8 or 4.1999999999999998e86 < t1

   1. Initial program 67.3%

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

      1. associate-/r* 81.2%

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

      2. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t1 around inf 92.0%

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

      1. neg-mul-1 92.0%

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

   6. Simplified 92.0%

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

   if -3.15000000000000001e38 < t1 < 6.00000000000000044e-85

   1. Initial program 79.8%

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

      1. associate-/r* 86.2%

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

      2. associate-/l* 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in t1 around 0 80.1%

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

      1. mul-1-neg 80.1%

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

      2. associate-/l* 83.9%

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

      3. distribute-neg-frac 83.9%

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

   6. Simplified 83.9%

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

   if 1.49999999999999987e-8 < t1 < 4.1999999999999998e86

   1. Initial program 77.9%

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

      1. associate-*l/ 83.7%

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

      2. *-commutative 83.7%

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

   3. Simplified 83.7%

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

      1. associate-/r* 89.1%

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

      2. associate-*r/ 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r/ 99.3%

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

      5. div-inv 99.5%

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

      6. frac-2neg 99.5%

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

      7. frac-times 99.3%

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

      8. remove-double-neg 99.3%

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

      9. *-commutative 99.3%

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

      10. *-un-lft-identity 99.3%

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

      11. distribute-neg-frac 99.3%

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

      12. distribute-neg-in 99.3%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 72.1%

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

      15. sqr-neg 72.1%

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

      16. sqrt-unprod 72.1%

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

      17. add-sqr-sqrt 72.1%

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

      18. sub-neg 72.1%

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

   5. Applied egg-rr 72.1%

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

   6. Taylor expanded in t1 around 0 56.5%

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

      1. unpow2 56.5%

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

      2. neg-mul-1 56.5%

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

      3. associate-/l* 72.4%

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

      4. distribute-neg-frac 72.4%

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

   8. Simplified 72.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.15 \cdot 10^{+38}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 6 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{-8} \lor \neg \left(t1 \leq 4.2 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{-u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 8: 95.2% 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 73.3%

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

   1. associate-*l/ 76.3%

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

   2. *-commutative 76.3%

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

3. Simplified 76.3%

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

   1. neg-mul-1 76.3%

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

   2. times-frac 95.0%

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

5. Applied egg-rr 95.0%

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

   1. associate-*l/ 95.0%

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

   2. mul-1-neg 95.0%

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

7. Simplified 95.0%

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

8. Final simplification 95.0%

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

Alternative 9: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{+127} \lor \neg \left(u \leq 7.2 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.02e+127) (not (<= u 7.2e+170)))
   (/ v (/ (* u u) t1))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.02e+127) || !(u <= 7.2e+170)) {
		tmp = v / ((u * u) / t1);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.02d+127)) .or. (.not. (u <= 7.2d+170))) then
        tmp = v / ((u * u) / t1)
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.02e+127) || !(u <= 7.2e+170)) {
		tmp = v / ((u * u) / t1);
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.02e+127) or not (u <= 7.2e+170):
		tmp = v / ((u * u) / t1)
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.02e+127) || !(u <= 7.2e+170))
		tmp = Float64(v / Float64(Float64(u * u) / t1));
	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 <= -1.02e+127) || ~((u <= 7.2e+170)))
		tmp = v / ((u * u) / t1);
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.02e+127], N[Not[LessEqual[u, 7.2e+170]], $MachinePrecision]], N[(v / N[(N[(u * u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.02e127 or 7.1999999999999999e170 < u

   1. Initial program 78.2%

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

      1. associate-*l/ 74.5%

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

      2. *-commutative 74.5%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in t1 around 0 74.5%

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

      3. unpow2 74.5%

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

   6. Simplified 74.5%

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

      1. clear-num 74.5%

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

      2. un-div-inv 74.5%

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

      3. add-sqr-sqrt 45.5%

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

      4. sqrt-unprod 56.1%

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

      5. sqr-neg 56.1%

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

      6. sqrt-unprod 29.0%

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

      7. add-sqr-sqrt 74.5%

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

   8. Applied egg-rr 74.5%

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

   if -1.02e127 < u < 7.1999999999999999e170

   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* 82.2%

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

      2. associate-/l* 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in t1 around inf 65.9%

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

      1. neg-mul-1 65.9%

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

   6. Simplified 65.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{+127} \lor \neg \left(u \leq 7.2 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 10: 57.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+83} \lor \neg \left(u \leq 3.25 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.5e+83) (not (<= u 3.25e+84))) (/ v (+ t1 u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.5e+83) || !(u <= 3.25e+84)) {
		tmp = v / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-6.5d+83)) .or. (.not. (u <= 3.25d+84))) then
        tmp = v / (t1 + u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.5e+83) || !(u <= 3.25e+84)) {
		tmp = v / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.5e+83) or not (u <= 3.25e+84):
		tmp = v / (t1 + u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.5e+83) || !(u <= 3.25e+84))
		tmp = Float64(v / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.5e+83) || ~((u <= 3.25e+84)))
		tmp = v / (t1 + u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.5e+83], N[Not[LessEqual[u, 3.25e+84]], $MachinePrecision]], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.5000000000000003e83 or 3.25000000000000013e84 < u

   1. Initial program 79.5%

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

      1. associate-*l/ 74.0%

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

      2. *-commutative 74.0%

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

   3. Simplified 74.0%

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

      1. neg-mul-1 74.0%

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

      2. times-frac 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-*l/ 87.5%

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

      2. mul-1-neg 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in t1 around inf 57.3%

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

      1. expm1-log1p-u 56.8%

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

      2. expm1-udef 73.0%

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

   10. Applied egg-rr 69.1%

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

       1. expm1-def 47.5%

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

       2. expm1-log1p 47.8%

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

   12. Simplified 47.8%

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

   if -6.5000000000000003e83 < u < 3.25000000000000013e84

   1. Initial program 69.9%

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

      1. associate-*l/ 77.5%

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

      2. *-commutative 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in t1 around inf 68.3%

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

      1. associate-*r/ 68.3%

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

      2. neg-mul-1 68.3%

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

   6. Simplified 68.3%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+83} \lor \neg \left(u \leq 3.25 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 58.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+158} \lor \neg \left(u \leq 5.8 \cdot 10^{+221}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.2e+158) (not (<= u 5.8e+221))) (/ (- v) u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+158) || !(u <= 5.8e+221)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.2d+158)) .or. (.not. (u <= 5.8d+221))) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.2e+158) || !(u <= 5.8e+221)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.2e+158) or not (u <= 5.8e+221):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.2e+158) || !(u <= 5.8e+221))
		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 <= -1.2e+158) || ~((u <= 5.8e+221)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.2e+158], N[Not[LessEqual[u, 5.8e+221]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.20000000000000004e158 or 5.7999999999999996e221 < u

   1. Initial program 83.0%

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

      1. associate-*l/ 83.4%

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

      2. *-commutative 83.4%

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

   3. Simplified 83.4%

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

      1. neg-mul-1 83.4%

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

      2. times-frac 91.7%

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

   5. Applied egg-rr 91.7%

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

      1. associate-*l/ 91.7%

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

      2. mul-1-neg 91.7%

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

   7. Simplified 91.7%

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

   8. Taylor expanded in t1 around inf 57.7%

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

   9. Taylor expanded in t1 around 0 51.6%

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

       1. neg-mul-1 51.6%

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

       2. distribute-neg-frac 51.6%

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

   11. Simplified 51.6%

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

   if -1.20000000000000004e158 < u < 5.7999999999999996e221

   1. Initial program 71.2%

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

      1. associate-*l/ 74.7%

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

      2. *-commutative 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in t1 around inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. neg-mul-1 62.0%

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

   6. Simplified 62.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+158} \lor \neg \left(u \leq 5.8 \cdot 10^{+221}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12: 61.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. associate-/r* 83.5%

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

   2. associate-/l* 97.2%

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

3. Simplified 97.2%

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

4. Taylor expanded in t1 around inf 63.1%

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

   1. neg-mul-1 63.1%

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

6. Simplified 63.1%

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

7. Final simplification 63.1%

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

Alternative 13: 54.1% accurate, 3.0× 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(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 73.3%

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

   1. associate-*l/ 76.3%

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

   2. *-commutative 76.3%

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

3. Simplified 76.3%

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

4. Taylor expanded in t1 around inf 55.4%

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

   1. associate-*r/ 55.4%

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

   2. neg-mul-1 55.4%

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

6. Simplified 55.4%

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

7. Final simplification 55.4%

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

Alternative 14: 14.3% 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 73.3%

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

   1. associate-*l/ 76.3%

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

   2. *-commutative 76.3%

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

3. Simplified 76.3%

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

   1. associate-/r* 95.0%

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

   2. associate-*r/ 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-/r/ 97.2%

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

   5. div-inv 97.1%

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

   6. frac-2neg 97.1%

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

   7. frac-times 80.8%

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

   8. remove-double-neg 80.8%

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

   9. *-commutative 80.8%

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

   10. *-un-lft-identity 80.8%

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

   11. distribute-neg-frac 80.8%

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

   12. distribute-neg-in 80.8%

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

   13. add-sqr-sqrt 42.8%

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

   14. sqrt-unprod 66.3%

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

   15. sqr-neg 66.3%

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

   16. sqrt-unprod 25.6%

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

   17. add-sqr-sqrt 60.0%

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

   18. sub-neg 60.0%

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

5. Applied egg-rr 60.0%

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

6. Taylor expanded in t1 around inf 19.6%

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

7. Final simplification 19.6%

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

Reproduce

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