Rosa's DopplerBench

Percentage Accurate: 73.8% → 97.7%

Time: 11.4s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. associate-/l* 74.5%

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

   2. distribute-lft-neg-out 74.5%

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

   3. distribute-rgt-neg-in 74.5%

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

   4. associate-/r* 86.2%

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

   5. distribute-neg-frac2 86.2%

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

3. Simplified 86.2%

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

   1. associate-*r/ 98.0%

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

   2. neg-mul-1 98.0%

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

   3. associate-/r* 98.0%

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

6. Applied egg-rr 98.0%

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

   1. clear-num 97.8%

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

   2. un-div-inv 98.0%

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

8. Applied egg-rr 98.0%

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

Alternative 2: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-u\right) - t1\\ \mathbf{if}\;t1 \leq -9.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{v}{t\_1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{+52}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- (- u) t1)))
   (if (<= t1 -9.4e+122)
     (/ v t_1)
     (if (<= t1 4.6e+52) (* t1 (/ (/ v (+ t1 u)) t_1)) (/ v (- u t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -9.4e+122) {
		tmp = v / t_1;
	} else if (t1 <= 4.6e+52) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -u - t1
    if (t1 <= (-9.4d+122)) then
        tmp = v / t_1
    else if (t1 <= 4.6d+52) then
        tmp = t1 * ((v / (t1 + u)) / t_1)
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -u - t1;
	double tmp;
	if (t1 <= -9.4e+122) {
		tmp = v / t_1;
	} else if (t1 <= 4.6e+52) {
		tmp = t1 * ((v / (t1 + u)) / t_1);
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -u - t1
	tmp = 0
	if t1 <= -9.4e+122:
		tmp = v / t_1
	elif t1 <= 4.6e+52:
		tmp = t1 * ((v / (t1 + u)) / t_1)
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-u) - t1)
	tmp = 0.0
	if (t1 <= -9.4e+122)
		tmp = Float64(v / t_1);
	elseif (t1 <= 4.6e+52)
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / t_1));
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -u - t1;
	tmp = 0.0;
	if (t1 <= -9.4e+122)
		tmp = v / t_1;
	elseif (t1 <= 4.6e+52)
		tmp = t1 * ((v / (t1 + u)) / t_1);
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-u) - t1), $MachinePrecision]}, If[LessEqual[t1, -9.4e+122], N[(v / t$95$1), $MachinePrecision], If[LessEqual[t1, 4.6e+52], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.40000000000000047e122

   1. Initial program 41.2%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 95.0%

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

   6. Taylor expanded in v around 0 95.0%

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

      1. neg-mul-1 95.0%

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

      2. distribute-neg-frac2 95.0%

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

      3. distribute-neg-in 95.0%

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

      4. sub-neg 95.0%

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

   8. Simplified 95.0%

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

   if -9.40000000000000047e122 < t1 < 4.6e52

   1. Initial program 87.4%

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

      1. associate-/l* 87.2%

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

      2. distribute-lft-neg-out 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. associate-/r* 93.3%

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

      5. distribute-neg-frac2 93.3%

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

   3. Simplified 93.3%

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

   if 4.6e52 < t1

   1. Initial program 52.9%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 87.1%

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

      1. clear-num 87.1%

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

      2. clear-num 87.0%

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

      3. frac-times 87.0%

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

      4. metadata-eval 87.0%

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

      5. add-sqr-sqrt 42.3%

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

      6. sqrt-unprod 85.0%

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

      7. sqr-neg 85.0%

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

      8. sqrt-unprod 44.6%

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

      9. add-sqr-sqrt 87.3%

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

   7. Applied egg-rr 87.3%

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

      1. *-commutative 87.3%

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

      2. associate-*l/ 87.3%

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

      3. associate-*r/ 57.6%

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

      4. associate-/r* 52.9%

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

      5. times-frac 87.3%

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

      6. *-inverses 87.3%

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

   9. Simplified 87.3%

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

   10. Taylor expanded in v around 0 87.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.6 \cdot 10^{+52}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.9:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t1 \cdot \left(-v\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-1 + \frac{u}{t1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.9)
   (/ v (- (- u) t1))
   (if (<= t1 1.25e-31)
     (/ (/ (* t1 (- v)) u) u)
     (* (/ v (+ t1 u)) (+ -1.0 (/ u t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.9) {
		tmp = v / (-u - t1);
	} else if (t1 <= 1.25e-31) {
		tmp = ((t1 * -v) / u) / u;
	} else {
		tmp = (v / (t1 + u)) * (-1.0 + (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.9d0)) then
        tmp = v / (-u - t1)
    else if (t1 <= 1.25d-31) then
        tmp = ((t1 * -v) / u) / u
    else
        tmp = (v / (t1 + u)) * ((-1.0d0) + (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.9) {
		tmp = v / (-u - t1);
	} else if (t1 <= 1.25e-31) {
		tmp = ((t1 * -v) / u) / u;
	} else {
		tmp = (v / (t1 + u)) * (-1.0 + (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.9:
		tmp = v / (-u - t1)
	elif t1 <= 1.25e-31:
		tmp = ((t1 * -v) / u) / u
	else:
		tmp = (v / (t1 + u)) * (-1.0 + (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.9)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= 1.25e-31)
		tmp = Float64(Float64(Float64(t1 * Float64(-v)) / u) / u);
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(-1.0 + Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.9)
		tmp = v / (-u - t1);
	elseif (t1 <= 1.25e-31)
		tmp = ((t1 * -v) / u) / u;
	else
		tmp = (v / (t1 + u)) * (-1.0 + (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.9], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.25e-31], N[(N[(N[(t1 * (-v)), $MachinePrecision] / u), $MachinePrecision] / u), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.8999999999999999

   1. Initial program 63.1%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.4%

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

   6. Taylor expanded in v around 0 88.4%

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

      1. neg-mul-1 88.4%

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

      2. distribute-neg-frac2 88.4%

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

      3. distribute-neg-in 88.4%

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

      4. sub-neg 88.4%

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

   8. Simplified 88.4%

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

   if -1.8999999999999999 < t1 < 1.25e-31

   1. Initial program 85.0%

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

      1. associate-/l* 84.0%

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

      2. distribute-lft-neg-out 84.0%

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

      3. distribute-rgt-neg-in 84.0%

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

      4. associate-/r* 90.8%

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

      5. distribute-neg-frac2 90.8%

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

   3. Simplified 90.8%

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

      1. associate-*r/ 96.0%

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

      2. neg-mul-1 96.0%

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

      3. associate-/r* 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in t1 around 0 79.9%

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

      1. associate-*r/ 79.9%

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

      2. neg-mul-1 79.9%

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

      3. distribute-rgt-neg-in 79.9%

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

   9. Simplified 79.9%

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

   10. Taylor expanded in t1 around 0 80.9%

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

   if 1.25e-31 < t1

   1. Initial program 67.7%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 85.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.9:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{t1 \cdot \left(-v\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \left(-1 + \frac{u}{t1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{t1 \cdot \left(-v\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1e-14)
   (/ v (- (- u) t1))
   (if (<= t1 3e-81) (/ (/ (* t1 (- v)) u) u) (/ v (- u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1e-14) {
		tmp = v / (-u - t1);
	} else if (t1 <= 3e-81) {
		tmp = ((t1 * -v) / u) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1d-14)) then
        tmp = v / (-u - t1)
    else if (t1 <= 3d-81) then
        tmp = ((t1 * -v) / u) / u
    else
        tmp = v / (u - t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1e-14) {
		tmp = v / (-u - t1);
	} else if (t1 <= 3e-81) {
		tmp = ((t1 * -v) / u) / u;
	} else {
		tmp = v / (u - t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1e-14:
		tmp = v / (-u - t1)
	elif t1 <= 3e-81:
		tmp = ((t1 * -v) / u) / u
	else:
		tmp = v / (u - t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1e-14)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	elseif (t1 <= 3e-81)
		tmp = Float64(Float64(Float64(t1 * Float64(-v)) / u) / u);
	else
		tmp = Float64(v / Float64(u - t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1e-14)
		tmp = v / (-u - t1);
	elseif (t1 <= 3e-81)
		tmp = ((t1 * -v) / u) / u;
	else
		tmp = v / (u - t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1e-14], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3e-81], N[(N[(N[(t1 * (-v)), $MachinePrecision] / u), $MachinePrecision] / u), $MachinePrecision], N[(v / N[(u - t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -9.99999999999999999e-15

   1. Initial program 63.1%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.4%

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

   6. Taylor expanded in v around 0 88.4%

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

      1. neg-mul-1 88.4%

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

      2. distribute-neg-frac2 88.4%

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

      3. distribute-neg-in 88.4%

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

      4. sub-neg 88.4%

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

   8. Simplified 88.4%

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

   if -9.99999999999999999e-15 < t1 < 2.9999999999999999e-81

   1. Initial program 86.2%

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

      1. associate-/l* 84.4%

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

      2. distribute-lft-neg-out 84.4%

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

      3. distribute-rgt-neg-in 84.4%

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

      4. associate-/r* 90.1%

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

      5. distribute-neg-frac2 90.1%

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

   3. Simplified 90.1%

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

      1. associate-*r/ 95.7%

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

      2. neg-mul-1 95.7%

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

      3. associate-/r* 95.7%

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

   6. Applied egg-rr 95.7%

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

   7. Taylor expanded in t1 around 0 81.6%

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

      1. associate-*r/ 81.6%

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

      2. neg-mul-1 81.6%

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

      3. distribute-rgt-neg-in 81.6%

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

   9. Simplified 81.6%

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

   10. Taylor expanded in t1 around 0 82.6%

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

   if 2.9999999999999999e-81 < t1

   1. Initial program 67.8%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around inf 80.6%

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

      1. clear-num 80.6%

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

      2. clear-num 80.4%

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

      3. frac-times 80.4%

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

      4. metadata-eval 80.4%

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

      5. add-sqr-sqrt 35.3%

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

      6. sqrt-unprod 82.8%

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

      7. sqr-neg 82.8%

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

      8. sqrt-unprod 44.9%

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

      9. add-sqr-sqrt 80.5%

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

   7. Applied egg-rr 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*l/ 80.5%

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

      3. associate-*r/ 62.7%

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

      4. associate-/r* 58.6%

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

      5. times-frac 80.5%

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

      6. *-inverses 80.5%

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

   9. Simplified 80.5%

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

   10. Taylor expanded in v around 0 80.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{t1 \cdot \left(-v\right)}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-149} \lor \neg \left(t1 \leq 1.5 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9e-149) (not (<= t1 1.5e-168)))
   (/ v (- (- u) t1))
   (/ (/ (* t1 v) u) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9e-149) || !(t1 <= 1.5e-168)) {
		tmp = v / (-u - t1);
	} else {
		tmp = ((t1 * v) / u) / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-9d-149)) .or. (.not. (t1 <= 1.5d-168))) then
        tmp = v / (-u - t1)
    else
        tmp = ((t1 * v) / u) / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -9e-149) || !(t1 <= 1.5e-168)) {
		tmp = v / (-u - t1);
	} else {
		tmp = ((t1 * v) / u) / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -9e-149) or not (t1 <= 1.5e-168):
		tmp = v / (-u - t1)
	else:
		tmp = ((t1 * v) / u) / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -9e-149) || !(t1 <= 1.5e-168))
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(Float64(t1 * v) / u) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -9e-149) || ~((t1 <= 1.5e-168)))
		tmp = v / (-u - t1);
	else
		tmp = ((t1 * v) / u) / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -9e-149], N[Not[LessEqual[t1, 1.5e-168]], $MachinePrecision]], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8.9999999999999996e-149 or 1.49999999999999996e-168 < t1

   1. Initial program 70.4%

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

      1. times-frac 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-neg-frac2 98.8%

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

      4. +-commutative 98.8%

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

      5. distribute-neg-in 98.8%

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

      6. unsub-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in t1 around inf 75.9%

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

   6. Taylor expanded in v around 0 75.5%

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

      1. neg-mul-1 75.5%

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

      2. distribute-neg-frac2 75.5%

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

      3. distribute-neg-in 75.5%

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

      4. sub-neg 75.5%

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

   8. Simplified 75.5%

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

   if -8.9999999999999996e-149 < t1 < 1.49999999999999996e-168

   1. Initial program 86.3%

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

      1. times-frac 92.0%

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

      2. distribute-frac-neg 92.0%

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

      3. distribute-neg-frac2 92.0%

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

      4. +-commutative 92.0%

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

      5. distribute-neg-in 92.0%

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

      6. unsub-neg 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in t1 around inf 31.4%

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

      1. associate-*r/ 49.9%

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

      2. associate-*l/ 40.5%

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

      3. *-commutative 40.5%

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

      4. add-sqr-sqrt 20.8%

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

      5. sqrt-unprod 35.1%

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

      6. sqr-neg 35.1%

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

      7. sqrt-unprod 15.8%

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

      8. add-sqr-sqrt 35.5%

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

      9. associate-*l/ 35.4%

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

      10. sub-neg 35.4%

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

      11. distribute-neg-in 35.4%

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

      12. +-commutative 35.4%

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

      13. clear-num 35.4%

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

      14. frac-2neg 35.4%

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

      15. associate-/r/ 35.4%

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

      16. *-commutative 35.4%

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

      17. associate-*l/ 35.4%

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

      18. *-un-lft-identity 35.4%

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

   7. Applied egg-rr 35.4%

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

   8. Taylor expanded in t1 around 0 35.6%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-149} \lor \neg \left(t1 \leq 1.5 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+115}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.3e+115) (/ v u) (if (<= u 5e+62) (/ v (- t1)) (/ 1.0 (/ u v)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e+115) {
		tmp = v / u;
	} else if (u <= 5e+62) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.3d+115)) then
        tmp = v / u
    else if (u <= 5d+62) then
        tmp = v / -t1
    else
        tmp = 1.0d0 / (u / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.3e+115) {
		tmp = v / u;
	} else if (u <= 5e+62) {
		tmp = v / -t1;
	} else {
		tmp = 1.0 / (u / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.3e+115:
		tmp = v / u
	elif u <= 5e+62:
		tmp = v / -t1
	else:
		tmp = 1.0 / (u / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.3e+115)
		tmp = Float64(v / u);
	elseif (u <= 5e+62)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(1.0 / Float64(u / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.3e+115)
		tmp = v / u;
	elseif (u <= 5e+62)
		tmp = v / -t1;
	else
		tmp = 1.0 / (u / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.3e+115], N[(v / u), $MachinePrecision], If[LessEqual[u, 5e+62], N[(v / (-t1)), $MachinePrecision], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.3e115

   1. Initial program 75.1%

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

      1. times-frac 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-neg-frac2 97.5%

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

      4. +-commutative 97.5%

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

      5. distribute-neg-in 97.5%

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

      6. unsub-neg 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around inf 53.2%

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

      1. clear-num 53.2%

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

      2. clear-num 53.2%

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

      3. frac-times 55.8%

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

      4. metadata-eval 55.8%

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

      5. add-sqr-sqrt 55.8%

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

      6. sqrt-unprod 62.5%

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

      7. sqr-neg 62.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l/ 64.0%

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

      3. associate-*r/ 43.9%

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

      4. associate-/r* 43.6%

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

      5. times-frac 46.6%

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

      6. *-inverses 46.6%

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

   9. Simplified 46.6%

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

   10. Taylor expanded in u around inf 42.0%

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

   if -1.3e115 < u < 5.00000000000000029e62

   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-/l* 70.8%

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

      2. distribute-lft-neg-out 70.8%

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

      3. distribute-rgt-neg-in 70.8%

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

      4. associate-/r* 82.2%

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

      5. distribute-neg-frac2 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in t1 around inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   7. Simplified 67.4%

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

   if 5.00000000000000029e62 < u

   1. Initial program 87.7%

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

      1. times-frac 93.9%

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

      2. distribute-frac-neg 93.9%

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

      3. distribute-neg-frac2 93.9%

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

      4. +-commutative 93.9%

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

      5. distribute-neg-in 93.9%

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

      6. unsub-neg 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in t1 around inf 52.5%

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

      1. clear-num 54.6%

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

      2. clear-num 54.6%

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

      3. frac-times 56.1%

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

      4. metadata-eval 56.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 64.1%

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

      7. sqr-neg 64.1%

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

      8. sqrt-unprod 56.0%

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

      9. add-sqr-sqrt 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l/ 65.2%

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

      3. associate-*r/ 36.1%

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

      4. associate-/r* 39.9%

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

      5. times-frac 36.1%

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

      6. *-inverses 36.1%

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

   9. Simplified 36.1%

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

   10. Taylor expanded in u around inf 28.6%

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

       1. clear-num 30.0%

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

       2. inv-pow 30.0%

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

   12. Applied egg-rr 30.0%

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

       1. unpow-1 30.0%

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

   14. Simplified 30.0%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+115}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+114} \lor \neg \left(u \leq 6.8 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.1e+114) (not (<= u 6.8e+86))) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.1e+114) || !(u <= 6.8e+86)) {
		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 <= (-4.1d+114)) .or. (.not. (u <= 6.8d+86))) 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 <= -4.1e+114) || !(u <= 6.8e+86)) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.1e+114) or not (u <= 6.8e+86):
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.1e+114) || !(u <= 6.8e+86))
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.1e+114) || ~((u <= 6.8e+86)))
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.1e+114], N[Not[LessEqual[u, 6.8e+86]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.1000000000000001e114 or 6.7999999999999995e86 < 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. times-frac 95.3%

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

      2. distribute-frac-neg 95.3%

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

      3. distribute-neg-frac2 95.3%

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

      4. +-commutative 95.3%

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

      5. distribute-neg-in 95.3%

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

      6. unsub-neg 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in t1 around inf 53.1%

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

      1. clear-num 54.3%

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

      2. clear-num 54.3%

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

      3. frac-times 55.5%

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

      4. metadata-eval 55.5%

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

      5. add-sqr-sqrt 27.2%

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

      6. sqrt-unprod 63.5%

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

      7. sqr-neg 63.5%

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

      8. sqrt-unprod 28.3%

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

      9. add-sqr-sqrt 55.6%

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

   7. Applied egg-rr 55.6%

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

      1. *-commutative 55.6%

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

      2. associate-*l/ 64.8%

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

      3. associate-*r/ 39.4%

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

      4. associate-/r* 40.3%

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

      5. times-frac 40.7%

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

      6. *-inverses 40.7%

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

   9. Simplified 40.7%

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

   10. Taylor expanded in u around inf 36.1%

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

   if -4.1000000000000001e114 < u < 6.7999999999999995e86

   1. Initial program 72.4%

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

      1. associate-/l* 71.6%

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

      2. distribute-lft-neg-out 71.6%

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

      3. distribute-rgt-neg-in 71.6%

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

      4. associate-/r* 82.7%

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

      5. distribute-neg-frac2 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in t1 around inf 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

   7. Simplified 66.1%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+114} \lor \neg \left(u \leq 6.8 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+109}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.88 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.1e+109) (/ v u) (if (<= u 1.88e+84) (/ v (- t1)) (/ v (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+109) {
		tmp = v / u;
	} else if (u <= 1.88e+84) {
		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 <= (-2.1d+109)) then
        tmp = v / u
    else if (u <= 1.88d+84) 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 <= -2.1e+109) {
		tmp = v / u;
	} else if (u <= 1.88e+84) {
		tmp = v / -t1;
	} else {
		tmp = v / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.1e+109:
		tmp = v / u
	elif u <= 1.88e+84:
		tmp = v / -t1
	else:
		tmp = v / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.1e+109)
		tmp = Float64(v / u);
	elseif (u <= 1.88e+84)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(v / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.1e+109)
		tmp = v / u;
	elseif (u <= 1.88e+84)
		tmp = v / -t1;
	else
		tmp = v / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.1e+109], N[(v / u), $MachinePrecision], If[LessEqual[u, 1.88e+84], N[(v / (-t1)), $MachinePrecision], N[(v / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.1000000000000001e109

   1. Initial program 75.1%

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

      1. times-frac 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-neg-frac2 97.5%

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

      4. +-commutative 97.5%

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

      5. distribute-neg-in 97.5%

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

      6. unsub-neg 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around inf 53.2%

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

      1. clear-num 53.2%

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

      2. clear-num 53.2%

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

      3. frac-times 55.8%

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

      4. metadata-eval 55.8%

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

      5. add-sqr-sqrt 55.8%

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

      6. sqrt-unprod 62.5%

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

      7. sqr-neg 62.5%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l/ 64.0%

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

      3. associate-*r/ 43.9%

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

      4. associate-/r* 43.6%

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

      5. times-frac 46.6%

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

      6. *-inverses 46.6%

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

   9. Simplified 46.6%

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

   10. Taylor expanded in u around inf 42.0%

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

   if -2.1000000000000001e109 < u < 1.8799999999999999e84

   1. Initial program 72.4%

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

      1. associate-/l* 71.6%

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

      2. distribute-lft-neg-out 71.6%

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

      3. distribute-rgt-neg-in 71.6%

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

      4. associate-/r* 82.7%

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

      5. distribute-neg-frac2 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in t1 around inf 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

   7. Simplified 66.1%

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

   if 1.8799999999999999e84 < u

   1. Initial program 86.2%

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

      1. times-frac 93.2%

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

      2. distribute-frac-neg 93.2%

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

      3. distribute-neg-frac2 93.2%

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

      4. +-commutative 93.2%

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

      5. distribute-neg-in 93.2%

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

      6. unsub-neg 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in t1 around inf 53.0%

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

   6. Taylor expanded in t1 around 0 30.6%

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

      1. associate-*r/ 30.6%

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

      2. mul-1-neg 30.6%

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

   8. Simplified 30.6%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+109}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.88 \cdot 10^{+84}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 24.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+98} \lor \neg \left(t1 \leq 3.1 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.8e+98) (not (<= t1 3.1e+118))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e+98) || !(t1 <= 3.1e+118)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-2.8d+98)) .or. (.not. (t1 <= 3.1d+118))) then
        tmp = v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.8e+98) || !(t1 <= 3.1e+118)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.8e+98) or not (t1 <= 3.1e+118):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.8e+98) || !(t1 <= 3.1e+118))
		tmp = Float64(v / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.8e+98) || ~((t1 <= 3.1e+118)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.8e+98], N[Not[LessEqual[t1, 3.1e+118]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.8000000000000001e98 or 3.09999999999999986e118 < t1

   1. Initial program 45.4%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 90.7%

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

   6. Taylor expanded in u around inf 33.7%

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

   if -2.8000000000000001e98 < t1 < 3.09999999999999986e118

   1. Initial program 86.9%

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

      1. times-frac 95.6%

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

      2. distribute-frac-neg 95.6%

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

      3. distribute-neg-frac2 95.6%

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

      4. +-commutative 95.6%

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

      5. distribute-neg-in 95.6%

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

      6. unsub-neg 95.6%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in t1 around inf 51.6%

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

      1. clear-num 52.1%

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

      2. clear-num 52.0%

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

      3. frac-times 52.9%

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

      4. metadata-eval 52.9%

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

      5. add-sqr-sqrt 22.3%

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

      6. sqrt-unprod 56.4%

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

      7. sqr-neg 56.4%

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

      8. sqrt-unprod 29.4%

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

      9. add-sqr-sqrt 51.0%

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

   7. Applied egg-rr 51.0%

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

      1. *-commutative 51.0%

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

      2. associate-*l/ 55.1%

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

      3. associate-*r/ 42.9%

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

      4. associate-/r* 43.9%

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

      5. times-frac 44.1%

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

      6. *-inverses 44.1%

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

   9. Simplified 44.1%

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

   10. Taylor expanded in u around inf 17.9%

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

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+98} \lor \neg \left(t1 \leq 3.1 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. associate-/l* 74.5%

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

   2. distribute-lft-neg-out 74.5%

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

   3. distribute-rgt-neg-in 74.5%

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

   4. associate-/r* 86.2%

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

   5. distribute-neg-frac2 86.2%

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

3. Simplified 86.2%

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

   1. associate-*r/ 98.0%

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

   2. neg-mul-1 98.0%

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

   3. associate-/r* 98.0%

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

6. Applied egg-rr 98.0%

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

Alternative 11: 97.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. associate-/l* 74.5%

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

   2. distribute-lft-neg-out 74.5%

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

   3. distribute-rgt-neg-in 74.5%

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

   4. associate-/r* 86.2%

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

   5. distribute-neg-frac2 86.2%

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

3. Simplified 86.2%

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

   1. associate-*r/ 98.0%

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

   2. neg-mul-1 98.0%

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

   3. associate-/r* 98.0%

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

6. Applied egg-rr 98.0%

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

   1. associate-*r/ 85.9%

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

   2. associate-/l/ 85.9%

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

   3. neg-mul-1 85.9%

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

   4. +-commutative 85.9%

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

   5. distribute-neg-in 85.9%

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

   6. sub-neg 85.9%

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

   7. associate-*l/ 97.8%

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

   8. div-inv 97.8%

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

   9. associate-*l* 97.8%

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

8. Applied egg-rr 97.8%

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

9. Final simplification 97.8%

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

Alternative 12: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{t1}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 6.5e+86) (/ v (- (- u) t1)) (/ (/ t1 u) (/ t1 v))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 6.5e+86) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / u) / (t1 / v);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 6.5d+86) then
        tmp = v / (-u - t1)
    else
        tmp = (t1 / u) / (t1 / v)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 6.5e+86) {
		tmp = v / (-u - t1);
	} else {
		tmp = (t1 / u) / (t1 / v);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 6.5e+86:
		tmp = v / (-u - t1)
	else:
		tmp = (t1 / u) / (t1 / v)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 6.5e+86)
		tmp = Float64(v / Float64(Float64(-u) - t1));
	else
		tmp = Float64(Float64(t1 / u) / Float64(t1 / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 6.5e+86)
		tmp = v / (-u - t1);
	else
		tmp = (t1 / u) / (t1 / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 6.5e+86], N[(v / N[((-u) - t1), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] / N[(t1 / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 6.49999999999999996e86

   1. Initial program 72.9%

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

      1. times-frac 97.5%

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

      2. distribute-frac-neg 97.5%

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

      3. distribute-neg-frac2 97.5%

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

      4. +-commutative 97.5%

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

      5. distribute-neg-in 97.5%

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

      6. unsub-neg 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in t1 around inf 64.7%

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

   6. Taylor expanded in v around 0 63.2%

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

      1. neg-mul-1 63.2%

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

      2. distribute-neg-frac2 63.2%

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

      3. distribute-neg-in 63.2%

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

      4. sub-neg 63.2%

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

   8. Simplified 63.2%

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

   if 6.49999999999999996e86 < u

   1. Initial program 86.2%

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

      1. times-frac 93.2%

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

      2. distribute-frac-neg 93.2%

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

      3. distribute-neg-frac2 93.2%

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

      4. +-commutative 93.2%

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

      5. distribute-neg-in 93.2%

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

      6. unsub-neg 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in t1 around inf 53.0%

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

      1. clear-num 53.0%

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

      2. un-div-inv 53.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 64.4%

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

      5. sqr-neg 64.4%

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

      6. sqrt-unprod 52.9%

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

      7. add-sqr-sqrt 52.9%

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

   7. Applied egg-rr 52.9%

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

   8. Taylor expanded in u around inf 52.9%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{t1}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. associate-/l* 74.5%

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

   2. distribute-lft-neg-out 74.5%

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

   3. distribute-rgt-neg-in 74.5%

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

   4. associate-/r* 86.2%

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

   5. distribute-neg-frac2 86.2%

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

3. Simplified 86.2%

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

   1. distribute-frac-neg2 86.2%

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

   2. distribute-rgt-neg-out 86.2%

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

   3. associate-/r* 74.5%

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

   4. distribute-lft-neg-out 74.5%

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

   5. associate-/l* 75.0%

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

   6. times-frac 96.8%

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

   7. frac-2neg 96.8%

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

   8. associate-*r/ 97.8%

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

   9. add-sqr-sqrt 49.3%

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

   10. sqrt-unprod 44.8%

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

   11. sqr-neg 44.8%

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

   12. sqrt-unprod 17.7%

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

   13. add-sqr-sqrt 36.4%

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

   14. add-sqr-sqrt 18.7%

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

   15. sqrt-unprod 57.8%

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

   16. sqr-neg 57.8%

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

   17. sqrt-prod 46.9%

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

   18. add-sqr-sqrt 97.8%

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

6. Applied egg-rr 97.8%

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

7. Final simplification 97.8%

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

Alternative 14: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. times-frac 96.8%

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

   2. distribute-frac-neg 96.8%

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

   3. distribute-neg-frac2 96.8%

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

   4. +-commutative 96.8%

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

   5. distribute-neg-in 96.8%

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

   6. unsub-neg 96.8%

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

3. Simplified 96.8%

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

5. Final simplification 96.8%

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

Alternative 15: 60.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

   1. times-frac 96.8%

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

   2. distribute-frac-neg 96.8%

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

   3. distribute-neg-frac2 96.8%

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

   4. +-commutative 96.8%

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

   5. distribute-neg-in 96.8%

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

   6. unsub-neg 96.8%

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

3. Simplified 96.8%

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

5. Taylor expanded in t1 around inf 62.9%

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

6. Taylor expanded in v around 0 58.7%

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

   1. neg-mul-1 58.7%

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

   2. distribute-neg-frac2 58.7%

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

   3. distribute-neg-in 58.7%

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

   4. sub-neg 58.7%

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

8. Simplified 58.7%

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

9. Final simplification 58.7%

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

Alternative 16: 60.9% 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 75.0%

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

   1. times-frac 96.8%

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

   2. distribute-frac-neg 96.8%

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

   3. distribute-neg-frac2 96.8%

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

   4. +-commutative 96.8%

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

   5. distribute-neg-in 96.8%

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

   6. unsub-neg 96.8%

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

3. Simplified 96.8%

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

5. Taylor expanded in t1 around inf 62.9%

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

   1. clear-num 63.2%

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

   2. clear-num 63.1%

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

   3. frac-times 63.8%

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

   4. metadata-eval 63.8%

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

   5. add-sqr-sqrt 28.9%

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

   6. sqrt-unprod 65.8%

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

   7. sqr-neg 65.8%

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

   8. sqrt-unprod 33.9%

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

   9. add-sqr-sqrt 62.4%

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

7. Applied egg-rr 62.4%

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

   1. *-commutative 62.4%

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

   2. associate-*l/ 65.3%

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

   3. associate-*r/ 44.7%

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

   4. associate-/r* 44.3%

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

   5. times-frac 57.4%

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

   6. *-inverses 57.4%

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

9. Simplified 57.4%

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

10. Taylor expanded in v around 0 57.3%

    \[\leadsto \color{blue}{\frac{v}{u - t1}} \]
11. Add Preprocessing

Alternative 17: 14.8% 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 75.0%

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

   1. times-frac 96.8%

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

   2. distribute-frac-neg 96.8%

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

   3. distribute-neg-frac2 96.8%

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

   4. +-commutative 96.8%

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

   5. distribute-neg-in 96.8%

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

   6. unsub-neg 96.8%

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

3. Simplified 96.8%

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

5. Taylor expanded in t1 around inf 53.8%

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

6. Taylor expanded in u around inf 12.4%

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

Reproduce

herbie shell --seed 2024114 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))