Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.2%

Time: 9.5s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.6%

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

   1. times-frac 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 2: 66.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{if}\;u \leq -1.06 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -3.75 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -6.8 \cdot 10^{+64} \lor \neg \left(u \leq 1.7 \cdot 10^{+126}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ (/ v u) u))))
   (if (<= u -1.06e+219)
     t_1
     (if (<= u -3.75e+118)
       (/ v (- (* u -2.0) t1))
       (if (or (<= u -6.8e+64) (not (<= u 1.7e+126))) t_1 (/ (- v) t1))))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * ((v / u) / u);
	double tmp;
	if (u <= -1.06e+219) {
		tmp = t_1;
	} else if (u <= -3.75e+118) {
		tmp = v / ((u * -2.0) - t1);
	} else if ((u <= -6.8e+64) || !(u <= 1.7e+126)) {
		tmp = t_1;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t1 * ((v / u) / u)
    if (u <= (-1.06d+219)) then
        tmp = t_1
    else if (u <= (-3.75d+118)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if ((u <= (-6.8d+64)) .or. (.not. (u <= 1.7d+126))) then
        tmp = t_1
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * ((v / u) / u);
	double tmp;
	if (u <= -1.06e+219) {
		tmp = t_1;
	} else if (u <= -3.75e+118) {
		tmp = v / ((u * -2.0) - t1);
	} else if ((u <= -6.8e+64) || !(u <= 1.7e+126)) {
		tmp = t_1;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * ((v / u) / u)
	tmp = 0
	if u <= -1.06e+219:
		tmp = t_1
	elif u <= -3.75e+118:
		tmp = v / ((u * -2.0) - t1)
	elif (u <= -6.8e+64) or not (u <= 1.7e+126):
		tmp = t_1
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(Float64(v / u) / u))
	tmp = 0.0
	if (u <= -1.06e+219)
		tmp = t_1;
	elseif (u <= -3.75e+118)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif ((u <= -6.8e+64) || !(u <= 1.7e+126))
		tmp = t_1;
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * ((v / u) / u);
	tmp = 0.0;
	if (u <= -1.06e+219)
		tmp = t_1;
	elseif (u <= -3.75e+118)
		tmp = v / ((u * -2.0) - t1);
	elseif ((u <= -6.8e+64) || ~((u <= 1.7e+126)))
		tmp = t_1;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -1.06e+219], t$95$1, If[LessEqual[u, -3.75e+118], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[u, -6.8e+64], N[Not[LessEqual[u, 1.7e+126]], $MachinePrecision]], t$95$1, N[((-v) / t1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.06e219 or -3.75000000000000001e118 < u < -6.8000000000000003e64 or 1.69999999999999995e126 < u

   1. Initial program 82.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. mul-1-neg 94.9%

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

   7. Simplified 94.9%

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

      1. *-commutative 94.9%

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

      2. clear-num 94.9%

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

      3. frac-2neg 94.9%

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

      4. frac-times 88.4%

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

      5. *-un-lft-identity 88.4%

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

      6. remove-double-neg 88.4%

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

   9. Applied egg-rr 88.4%

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

   10. Taylor expanded in t1 around 0 88.4%

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

       1. clear-num 88.4%

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

       2. associate-/r/ 88.3%

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

       3. associate-/r* 88.4%

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

       4. clear-num 88.4%

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

       5. add-sqr-sqrt 46.6%

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

       6. sqrt-unprod 80.2%

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

       7. sqr-neg 80.2%

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

       8. sqrt-unprod 36.1%

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

       9. add-sqr-sqrt 74.8%

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

   12. Applied egg-rr 74.8%

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

   if -1.06e219 < u < -3.75000000000000001e118

   1. Initial program 47.5%

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

      1. associate-/r* 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 82.8%

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

      5. +-commutative 82.8%

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

      6. remove-double-neg 82.8%

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

      7. unsub-neg 82.8%

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

      8. div-sub 82.8%

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

      9. sub-neg 82.8%

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

      10. *-inverses 82.8%

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

      11. metadata-eval 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in t1 around inf 53.8%

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

      1. mul-1-neg 53.8%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 53.8%

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

      3. *-commutative 53.8%

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

   7. Simplified 53.8%

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

   if -6.8000000000000003e64 < u < 1.69999999999999995e126

   1. Initial program 67.0%

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

      1. times-frac 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in t1 around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. neg-mul-1 71.8%

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

   7. Simplified 71.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.06 \cdot 10^{+219}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq -3.75 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq -6.8 \cdot 10^{+64} \lor \neg \left(u \leq 1.7 \cdot 10^{+126}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{-78} \lor \neg \left(u \leq 2.2 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3e-78) (not (<= u 2.2e+17)))
   (* (/ v (+ t1 u)) (/ (- t1) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e-78) || !(u <= 2.2e+17)) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3d-78)) .or. (.not. (u <= 2.2d+17))) then
        tmp = (v / (t1 + u)) * (-t1 / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e-78) || !(u <= 2.2e+17)) {
		tmp = (v / (t1 + u)) * (-t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3e-78) or not (u <= 2.2e+17):
		tmp = (v / (t1 + u)) * (-t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3e-78) || !(u <= 2.2e+17))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3e-78) || ~((u <= 2.2e+17)))
		tmp = (v / (t1 + u)) * (-t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3e-78], N[Not[LessEqual[u, 2.2e+17]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.99999999999999988e-78 or 2.2e17 < u

   1. Initial program 72.6%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 79.4%

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

      1. associate-*r/ 79.4%

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

      2. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

   if -2.99999999999999988e-78 < u < 2.2e17

   1. Initial program 65.7%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{-78} \lor \neg \left(u \leq 2.2 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{-34} \lor \neg \left(u \leq 4.4 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2e-34) (not (<= u 4.4e+19)))
   (/ (* t1 (/ v (+ t1 u))) (- u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e-34) || !(u <= 4.4e+19)) {
		tmp = (t1 * (v / (t1 + u))) / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2d-34)) .or. (.not. (u <= 4.4d+19))) then
        tmp = (t1 * (v / (t1 + u))) / -u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2e-34) || !(u <= 4.4e+19)) {
		tmp = (t1 * (v / (t1 + u))) / -u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2e-34) or not (u <= 4.4e+19):
		tmp = (t1 * (v / (t1 + u))) / -u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2e-34) || !(u <= 4.4e+19))
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 + u))) / Float64(-u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2e-34) || ~((u <= 4.4e+19)))
		tmp = (t1 * (v / (t1 + u))) / -u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2e-34], N[Not[LessEqual[u, 4.4e+19]], $MachinePrecision]], N[(N[(t1 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.99999999999999986e-34 or 4.4e19 < u

   1. Initial program 74.5%

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

      1. times-frac 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in t1 around 0 82.9%

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

      1. associate-*r/ 82.9%

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

      2. mul-1-neg 82.9%

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

   7. Simplified 82.9%

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

      1. frac-2neg 82.9%

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

      2. remove-double-neg 82.9%

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

      3. associate-*l/ 84.3%

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

   9. Applied egg-rr 84.3%

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

   if -1.99999999999999986e-34 < u < 4.4e19

   1. Initial program 64.7%

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

      1. times-frac 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in t1 around inf 80.4%

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

      1. associate-*r/ 80.4%

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

      2. neg-mul-1 80.4%

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

   7. Simplified 80.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{-34} \lor \neg \left(u \leq 4.4 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{\left(-t1\right) - u}{v}}\\ \mathbf{elif}\;u \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.5e-78)
   (/ t1 (* u (/ (- (- t1) u) v)))
   (if (<= u 7.5e+18) (/ (- v) t1) (* (/ v (+ t1 u)) (/ (- t1) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.5e-78) {
		tmp = t1 / (u * ((-t1 - u) / v));
	} else if (u <= 7.5e+18) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.5d-78)) then
        tmp = t1 / (u * ((-t1 - u) / v))
    else if (u <= 7.5d+18) then
        tmp = -v / t1
    else
        tmp = (v / (t1 + u)) * (-t1 / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.5e-78) {
		tmp = t1 / (u * ((-t1 - u) / v));
	} else if (u <= 7.5e+18) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.5e-78:
		tmp = t1 / (u * ((-t1 - u) / v))
	elif u <= 7.5e+18:
		tmp = -v / t1
	else:
		tmp = (v / (t1 + u)) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.5e-78)
		tmp = Float64(t1 / Float64(u * Float64(Float64(Float64(-t1) - u) / v)));
	elseif (u <= 7.5e+18)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.5e-78)
		tmp = t1 / (u * ((-t1 - u) / v));
	elseif (u <= 7.5e+18)
		tmp = -v / t1;
	else
		tmp = (v / (t1 + u)) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.5e-78], N[(t1 / N[(u * N[(N[((-t1) - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.5e+18], N[((-v) / t1), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[((-t1) / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.4999999999999998e-78

   1. Initial program 70.5%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in t1 around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. mul-1-neg 77.1%

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

   7. Simplified 77.1%

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

      1. *-commutative 77.1%

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

      2. clear-num 78.0%

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

      3. frac-2neg 78.0%

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

      4. frac-times 77.2%

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

      5. *-un-lft-identity 77.2%

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

      6. remove-double-neg 77.2%

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

   9. Applied egg-rr 77.2%

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

   if -2.4999999999999998e-78 < u < 7.5e18

   1. Initial program 65.7%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   7. Simplified 85.4%

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

   if 7.5e18 < u

   1. Initial program 75.7%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 82.6%

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

      1. associate-*r/ 82.6%

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

      2. mul-1-neg 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{\left(-t1\right) - u}{v}}\\ \mathbf{elif}\;u \leq 7.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 - u}{t1}}}{t1 + u}\\ \mathbf{elif}\;u \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.7e-78)
   (/ (/ v (/ (- t1 u) t1)) (+ t1 u))
   (if (<= u 8e+20) (/ (- v) t1) (/ (* t1 (/ v (+ t1 u))) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.7e-78) {
		tmp = (v / ((t1 - u) / t1)) / (t1 + u);
	} else if (u <= 8e+20) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / (t1 + u))) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.7d-78)) then
        tmp = (v / ((t1 - u) / t1)) / (t1 + u)
    else if (u <= 8d+20) then
        tmp = -v / t1
    else
        tmp = (t1 * (v / (t1 + u))) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.7e-78) {
		tmp = (v / ((t1 - u) / t1)) / (t1 + u);
	} else if (u <= 8e+20) {
		tmp = -v / t1;
	} else {
		tmp = (t1 * (v / (t1 + u))) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.7e-78:
		tmp = (v / ((t1 - u) / t1)) / (t1 + u)
	elif u <= 8e+20:
		tmp = -v / t1
	else:
		tmp = (t1 * (v / (t1 + u))) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.7e-78)
		tmp = Float64(Float64(v / Float64(Float64(t1 - u) / t1)) / Float64(t1 + u));
	elseif (u <= 8e+20)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 + u))) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.7e-78)
		tmp = (v / ((t1 - u) / t1)) / (t1 + u);
	elseif (u <= 8e+20)
		tmp = -v / t1;
	else
		tmp = (t1 * (v / (t1 + u))) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.7e-78], N[(N[(v / N[(N[(t1 - u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 8e+20], N[((-v) / t1), $MachinePrecision], N[(N[(t1 * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.69999999999999994e-78

   1. Initial program 70.5%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

      1. associate-*r/ 96.4%

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

      2. clear-num 96.3%

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

      3. associate-*l/ 96.4%

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

      4. *-un-lft-identity 96.4%

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

      5. frac-2neg 96.4%

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

      6. distribute-neg-in 96.4%

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

      7. add-sqr-sqrt 41.0%

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

      8. sqrt-unprod 78.2%

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

      9. sqr-neg 78.2%

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

      10. sqrt-unprod 43.6%

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

      11. add-sqr-sqrt 77.9%

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

      12. sub-neg 77.9%

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

      13. remove-double-neg 77.9%

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

   6. Applied egg-rr 77.9%

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

   if -2.69999999999999994e-78 < u < 8e20

   1. Initial program 65.7%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   7. Simplified 85.4%

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

   if 8e20 < u

   1. Initial program 75.7%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 82.6%

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

      1. associate-*r/ 82.6%

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

      2. mul-1-neg 82.6%

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

   7. Simplified 82.6%

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

      1. frac-2neg 82.6%

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

      2. remove-double-neg 82.6%

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

      3. associate-*l/ 84.0%

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

   9. Applied egg-rr 84.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{v}{\frac{t1 - u}{t1}}}{t1 + u}\\ \mathbf{elif}\;u \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 + u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.5e-78)
   (* (/ (- t1) u) (/ v u))
   (if (<= u 2.5e+20) (/ (- v) t1) (/ (* v (/ t1 u)) (- t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.5e-78) {
		tmp = (-t1 / u) * (v / u);
	} else if (u <= 2.5e+20) {
		tmp = -v / t1;
	} else {
		tmp = (v * (t1 / u)) / (t1 - u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.5d-78)) then
        tmp = (-t1 / u) * (v / u)
    else if (u <= 2.5d+20) then
        tmp = -v / t1
    else
        tmp = (v * (t1 / u)) / (t1 - u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.5e-78) {
		tmp = (-t1 / u) * (v / u);
	} else if (u <= 2.5e+20) {
		tmp = -v / t1;
	} else {
		tmp = (v * (t1 / u)) / (t1 - u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.5e-78:
		tmp = (-t1 / u) * (v / u)
	elif u <= 2.5e+20:
		tmp = -v / t1
	else:
		tmp = (v * (t1 / u)) / (t1 - u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.5e-78)
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	elseif (u <= 2.5e+20)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(t1 - u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.5e-78)
		tmp = (-t1 / u) * (v / u);
	elseif (u <= 2.5e+20)
		tmp = -v / t1;
	else
		tmp = (v * (t1 / u)) / (t1 - u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.5e-78], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.5e+20], N[((-v) / t1), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / N[(t1 - u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.4999999999999998e-78

   1. Initial program 70.5%

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

      1. times-frac 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in t1 around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. mul-1-neg 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in t1 around 0 76.3%

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

   if -2.4999999999999998e-78 < u < 2.5e20

   1. Initial program 65.7%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   7. Simplified 85.4%

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

   if 2.5e20 < u

   1. Initial program 75.7%

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

      1. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around 0 82.6%

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

      1. associate-*r/ 82.6%

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

      2. mul-1-neg 82.6%

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

   7. Simplified 82.6%

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

      1. associate-*r/ 80.9%

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

      2. frac-2neg 80.9%

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

      3. add-sqr-sqrt 34.0%

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

      4. sqrt-unprod 55.2%

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

      5. sqr-neg 55.2%

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

      6. sqrt-unprod 33.0%

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

      7. add-sqr-sqrt 54.4%

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

      8. distribute-lft-neg-out 54.4%

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

      9. distribute-frac-neg 54.4%

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

      10. *-commutative 54.4%

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

      11. add-sqr-sqrt 21.4%

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

      12. sqrt-unprod 49.9%

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

      13. sqr-neg 49.9%

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

      14. sqrt-unprod 46.8%

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

      15. add-sqr-sqrt 80.9%

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

      16. distribute-neg-in 80.9%

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

   9. Applied egg-rr 81.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{t1 - u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{-78} \lor \neg \left(u \leq 2.25 \cdot 10^{+19}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3e-78) (not (<= u 2.25e+19)))
   (* t1 (/ (/ (- v) u) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e-78) || !(u <= 2.25e+19)) {
		tmp = t1 * ((-v / u) / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3d-78)) .or. (.not. (u <= 2.25d+19))) then
        tmp = t1 * ((-v / u) / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3e-78) || !(u <= 2.25e+19)) {
		tmp = t1 * ((-v / u) / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3e-78) or not (u <= 2.25e+19):
		tmp = t1 * ((-v / u) / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3e-78) || !(u <= 2.25e+19))
		tmp = Float64(t1 * Float64(Float64(Float64(-v) / u) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3e-78) || ~((u <= 2.25e+19)))
		tmp = t1 * ((-v / u) / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3e-78], N[Not[LessEqual[u, 2.25e+19]], $MachinePrecision]], N[(t1 * N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.99999999999999988e-78 or 2.25e19 < u

   1. Initial program 72.6%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 79.4%

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

      1. associate-*r/ 79.4%

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

      2. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

      1. *-commutative 79.4%

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

      2. clear-num 79.9%

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

      3. frac-2neg 79.9%

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

      4. frac-times 77.1%

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

      5. *-un-lft-identity 77.1%

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

      6. remove-double-neg 77.1%

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

   9. Applied egg-rr 77.1%

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

   10. Taylor expanded in t1 around 0 76.0%

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

       1. frac-2neg 76.0%

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

       2. div-inv 76.0%

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

       3. distribute-rgt-neg-out 76.0%

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

       4. remove-double-neg 76.0%

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

       5. add-sqr-sqrt 31.9%

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

       6. sqrt-unprod 56.6%

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

       7. sqr-neg 56.6%

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

       8. sqrt-unprod 25.9%

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

       9. add-sqr-sqrt 49.0%

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

       10. associate-/r* 49.0%

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

       11. clear-num 49.0%

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

       12. add-sqr-sqrt 25.9%

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

       13. sqrt-unprod 56.6%

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

       14. sqr-neg 56.6%

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

       15. sqrt-unprod 31.9%

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

       16. add-sqr-sqrt 76.0%

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

   12. Applied egg-rr 76.0%

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

   if -2.99999999999999988e-78 < u < 2.25e19

   1. Initial program 65.7%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{-78} \lor \neg \left(u \leq 2.25 \cdot 10^{+19}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{-78} \lor \neg \left(u \leq 8.5 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.7e-78) (not (<= u 8.5e+21)))
   (* (/ (- t1) u) (/ v u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e-78) || !(u <= 8.5e+21)) {
		tmp = (-t1 / u) * (v / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.7d-78)) .or. (.not. (u <= 8.5d+21))) then
        tmp = (-t1 / u) * (v / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e-78) || !(u <= 8.5e+21)) {
		tmp = (-t1 / u) * (v / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.7e-78) or not (u <= 8.5e+21):
		tmp = (-t1 / u) * (v / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.7e-78) || !(u <= 8.5e+21))
		tmp = Float64(Float64(Float64(-t1) / u) * Float64(v / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.7e-78) || ~((u <= 8.5e+21)))
		tmp = (-t1 / u) * (v / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.7e-78], N[Not[LessEqual[u, 8.5e+21]], $MachinePrecision]], N[(N[((-t1) / u), $MachinePrecision] * N[(v / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.69999999999999994e-78 or 8.5e21 < u

   1. Initial program 72.6%

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

      1. times-frac 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in t1 around 0 79.4%

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

      1. associate-*r/ 79.4%

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

      2. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in t1 around 0 78.3%

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

   if -2.69999999999999994e-78 < u < 8.5e21

   1. Initial program 65.7%

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

      1. times-frac 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in t1 around inf 85.4%

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

      1. associate-*r/ 85.4%

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

      2. neg-mul-1 85.4%

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

   7. Simplified 85.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{-78} \lor \neg \left(u \leq 8.5 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+73} \lor \neg \left(u \leq 1.35 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.1e+73) (not (<= u 1.35e+140)))
   (* (/ v u) (/ t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.1e+73) || !(u <= 1.35e+140)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-3.1d+73)) .or. (.not. (u <= 1.35d+140))) then
        tmp = (v / u) * (t1 / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.1e+73) || !(u <= 1.35e+140)) {
		tmp = (v / u) * (t1 / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.1e+73) or not (u <= 1.35e+140):
		tmp = (v / u) * (t1 / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.1e+73) || !(u <= 1.35e+140))
		tmp = Float64(Float64(v / u) * Float64(t1 / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.1e+73) || ~((u <= 1.35e+140)))
		tmp = (v / u) * (t1 / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.1e+73], N[Not[LessEqual[u, 1.35e+140]], $MachinePrecision]], N[(N[(v / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.1e73 or 1.35000000000000009e140 < u

   1. Initial program 73.1%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 89.5%

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

      1. associate-*r/ 89.5%

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

      2. mul-1-neg 89.5%

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

   7. Simplified 89.5%

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

      1. *-commutative 89.5%

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

      2. clear-num 89.4%

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

      3. frac-2neg 89.4%

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

      4. frac-times 83.3%

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

      5. *-un-lft-identity 83.3%

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

      6. remove-double-neg 83.3%

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

   9. Applied egg-rr 83.3%

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

   10. Taylor expanded in t1 around 0 83.4%

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

       1. *-un-lft-identity 83.4%

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

       2. times-frac 89.6%

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

       3. clear-num 89.7%

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

       4. add-sqr-sqrt 55.3%

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

       5. sqrt-unprod 72.5%

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

       6. sqr-neg 72.5%

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

       7. sqrt-unprod 27.5%

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

       8. add-sqr-sqrt 65.5%

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

   12. Applied egg-rr 65.5%

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

   if -3.1e73 < u < 1.35000000000000009e140

   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 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in t1 around inf 71.3%

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

      1. associate-*r/ 71.3%

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

      2. neg-mul-1 71.3%

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

   7. Simplified 71.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+73} \lor \neg \left(u \leq 1.35 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+64} \lor \neg \left(u \leq 2.85 \cdot 10^{+127}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.4e+64) (not (<= u 2.85e+127)))
   (* t1 (/ (/ v u) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+64) || !(u <= 2.85e+127)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.4d+64)) .or. (.not. (u <= 2.85d+127))) then
        tmp = t1 * ((v / u) / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.4e+64) || !(u <= 2.85e+127)) {
		tmp = t1 * ((v / u) / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.4e+64) or not (u <= 2.85e+127):
		tmp = t1 * ((v / u) / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.4e+64) || !(u <= 2.85e+127))
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.4e+64) || ~((u <= 2.85e+127)))
		tmp = t1 * ((v / u) / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.4e+64], N[Not[LessEqual[u, 2.85e+127]], $MachinePrecision]], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.40000000000000012e64 or 2.85000000000000021e127 < u

   1. Initial program 74.3%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 89.9%

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

      1. associate-*r/ 89.9%

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

      2. mul-1-neg 89.9%

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

   7. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. clear-num 89.9%

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

      3. frac-2neg 89.9%

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

      4. frac-times 84.1%

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

      5. *-un-lft-identity 84.1%

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

      6. remove-double-neg 84.1%

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

   9. Applied egg-rr 84.1%

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

   10. Taylor expanded in t1 around 0 84.1%

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

       1. clear-num 84.0%

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

       2. associate-/r/ 84.0%

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

       3. associate-/r* 84.1%

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

       4. clear-num 84.1%

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

       5. add-sqr-sqrt 52.1%

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

       6. sqrt-unprod 72.6%

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

       7. sqr-neg 72.6%

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

       8. sqrt-unprod 27.6%

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

       9. add-sqr-sqrt 65.2%

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

   12. Applied egg-rr 65.2%

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

   if -1.40000000000000012e64 < u < 2.85000000000000021e127

   1. Initial program 67.0%

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

      1. times-frac 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in t1 around inf 71.8%

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

      1. associate-*r/ 71.8%

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

      2. neg-mul-1 71.8%

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

   7. Simplified 71.8%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+64} \lor \neg \left(u \leq 2.85 \cdot 10^{+127}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot 0.5}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.4e+146)
   (/ -0.5 (/ u v))
   (if (<= u 2.2e+140) (/ (- v) t1) (/ (* v 0.5) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+146) {
		tmp = -0.5 / (u / v);
	} else if (u <= 2.2e+140) {
		tmp = -v / t1;
	} else {
		tmp = (v * 0.5) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.4d+146)) then
        tmp = (-0.5d0) / (u / v)
    else if (u <= 2.2d+140) then
        tmp = -v / t1
    else
        tmp = (v * 0.5d0) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+146) {
		tmp = -0.5 / (u / v);
	} else if (u <= 2.2e+140) {
		tmp = -v / t1;
	} else {
		tmp = (v * 0.5) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.4e+146:
		tmp = -0.5 / (u / v)
	elif u <= 2.2e+140:
		tmp = -v / t1
	else:
		tmp = (v * 0.5) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.4e+146)
		tmp = Float64(-0.5 / Float64(u / v));
	elseif (u <= 2.2e+140)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v * 0.5) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.4e+146)
		tmp = -0.5 / (u / v);
	elseif (u <= 2.2e+140)
		tmp = -v / t1;
	else
		tmp = (v * 0.5) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.4e+146], N[(-0.5 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.2e+140], N[((-v) / t1), $MachinePrecision], N[(N[(v * 0.5), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.4e146

   1. Initial program 64.5%

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

      1. associate-/r* 88.8%

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

      2. *-commutative 88.8%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 81.3%

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

      5. +-commutative 81.3%

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

      6. remove-double-neg 81.3%

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

      7. unsub-neg 81.3%

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

      8. div-sub 81.3%

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

      9. sub-neg 81.3%

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

      10. *-inverses 81.3%

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

      11. metadata-eval 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in t1 around inf 39.4%

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

      1. mul-1-neg 39.4%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 39.4%

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

      3. *-commutative 39.4%

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

   7. Simplified 39.4%

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

   8. Taylor expanded in u around inf 32.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   9. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   10. Simplified 32.3%

       \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   11. Taylor expanded in v around 0 32.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   12. Step-by-step derivation

       1. associate-*r/ 32.3%

          \[\leadsto \color{blue}{\frac{-0.5 \cdot v}{u}} \]

       2. *-rgt-identity 32.3%

          \[\leadsto \frac{-0.5 \cdot v}{\color{blue}{u \cdot 1}} \]

       3. times-frac 32.3%

          \[\leadsto \color{blue}{\frac{-0.5}{u} \cdot \frac{v}{1}} \]

       4. /-rgt-identity 32.3%

          \[\leadsto \frac{-0.5}{u} \cdot \color{blue}{v} \]

       5. associate-/r/ 34.9%

          \[\leadsto \color{blue}{\frac{-0.5}{\frac{u}{v}}} \]

   13. Simplified 34.9%

       \[\leadsto \color{blue}{\frac{-0.5}{\frac{u}{v}}} \]

   if -1.4e146 < u < 2.1999999999999998e140

   1. Initial program 69.6%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t1 around inf 67.7%

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

      1. associate-*r/ 67.7%

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

      2. neg-mul-1 67.7%

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

   7. Simplified 67.7%

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

   if 2.1999999999999998e140 < u

   1. Initial program 74.7%

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

      1. associate-/r* 88.4%

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

      2. *-commutative 88.4%

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

      3. associate-/l* 99.8%

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

      4. associate-/l/ 88.9%

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

      5. +-commutative 88.9%

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

      6. remove-double-neg 88.9%

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

      7. unsub-neg 88.9%

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

      8. div-sub 88.9%

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

      9. sub-neg 88.9%

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

      10. *-inverses 88.9%

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

      11. metadata-eval 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in t1 around inf 42.3%

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

      1. mul-1-neg 42.3%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 42.3%

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

      3. *-commutative 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in u around inf 35.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   9. Step-by-step derivation

      1. *-commutative 35.8%

         \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   10. Simplified 35.8%

       \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]
   11. Step-by-step derivation

       1. associate-*l/ 35.8%

          \[\leadsto \color{blue}{\frac{v \cdot -0.5}{u}} \]

       2. frac-2neg 35.8%

          \[\leadsto \color{blue}{\frac{-v \cdot -0.5}{-u}} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto \frac{-v \cdot -0.5}{\color{blue}{\sqrt{-u} \cdot \sqrt{-u}}} \]

       4. sqrt-unprod 64.6%

          \[\leadsto \frac{-v \cdot -0.5}{\color{blue}{\sqrt{\left(-u\right) \cdot \left(-u\right)}}} \]

       5. sqr-neg 64.6%

          \[\leadsto \frac{-v \cdot -0.5}{\sqrt{\color{blue}{u \cdot u}}} \]

       6. sqrt-unprod 36.4%

          \[\leadsto \frac{-v \cdot -0.5}{\color{blue}{\sqrt{u} \cdot \sqrt{u}}} \]

       7. add-sqr-sqrt 36.4%

          \[\leadsto \frac{-v \cdot -0.5}{\color{blue}{u}} \]

   12. Applied egg-rr 36.4%

       \[\leadsto \color{blue}{\frac{-v \cdot -0.5}{u}} \]
   13. Step-by-step derivation

       1. distribute-rgt-neg-in 36.4%

          \[\leadsto \frac{\color{blue}{v \cdot \left(--0.5\right)}}{u} \]

       2. metadata-eval 36.4%

          \[\leadsto \frac{v \cdot \color{blue}{0.5}}{u} \]

   14. Simplified 36.4%

       \[\leadsto \color{blue}{\frac{v \cdot 0.5}{u}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot 0.5}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+167} \lor \neg \left(u \leq 3.8 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.9e+167) (not (<= u 3.8e+140))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+167) || !(u <= 3.8e+140)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.9d+167)) .or. (.not. (u <= 3.8d+140))) then
        tmp = v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.9e+167) || !(u <= 3.8e+140)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.9e+167) or not (u <= 3.8e+140):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.9e+167) || !(u <= 3.8e+140))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.9e+167) || ~((u <= 3.8e+140)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.9e+167], N[Not[LessEqual[u, 3.8e+140]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.89999999999999997e167 or 3.8000000000000001e140 < u

   1. Initial program 69.7%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around 0 91.6%

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

      1. associate-*r/ 91.6%

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

      2. mul-1-neg 91.6%

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

   7. Simplified 91.6%

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

      1. associate-*r/ 91.6%

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

      2. frac-2neg 91.6%

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

      3. add-sqr-sqrt 43.1%

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

      4. sqrt-unprod 66.4%

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

      5. sqr-neg 66.4%

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

      6. sqrt-unprod 34.2%

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

      7. add-sqr-sqrt 68.5%

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

      8. distribute-lft-neg-out 68.5%

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

      9. distribute-frac-neg 68.5%

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

      10. *-commutative 68.5%

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

      11. add-sqr-sqrt 34.3%

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

      12. sqrt-unprod 68.9%

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

      13. sqr-neg 68.9%

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

      14. sqrt-unprod 48.3%

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

      15. add-sqr-sqrt 91.6%

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

      16. distribute-neg-in 91.6%

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

   9. Applied egg-rr 91.9%

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

   10. Taylor expanded in t1 around inf 34.5%

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

   if -1.89999999999999997e167 < u < 3.8000000000000001e140

   1. Initial program 69.6%

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

      1. times-frac 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in t1 around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. neg-mul-1 67.2%

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

   7. Simplified 67.2%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+167} \lor \neg \left(u \leq 3.8 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+137}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.45e+137)
   (/ -0.5 (/ u v))
   (if (<= u 2.2e+140) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.45e+137) {
		tmp = -0.5 / (u / v);
	} else if (u <= 2.2e+140) {
		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 <= (-1.45d+137)) then
        tmp = (-0.5d0) / (u / v)
    else if (u <= 2.2d+140) 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 <= -1.45e+137) {
		tmp = -0.5 / (u / v);
	} else if (u <= 2.2e+140) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.45e+137:
		tmp = -0.5 / (u / v)
	elif u <= 2.2e+140:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.45e+137)
		tmp = Float64(-0.5 / Float64(u / v));
	elseif (u <= 2.2e+140)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.45e+137)
		tmp = -0.5 / (u / v);
	elseif (u <= 2.2e+140)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.45e+137], N[(-0.5 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.2e+140], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.44999999999999992e137

   1. Initial program 64.5%

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

      1. associate-/r* 88.8%

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

      2. *-commutative 88.8%

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

      3. associate-/l* 99.9%

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

      4. associate-/l/ 81.3%

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

      5. +-commutative 81.3%

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

      6. remove-double-neg 81.3%

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

      7. unsub-neg 81.3%

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

      8. div-sub 81.3%

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

      9. sub-neg 81.3%

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

      10. *-inverses 81.3%

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

      11. metadata-eval 81.3%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in t1 around inf 39.4%

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

      1. mul-1-neg 39.4%

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(-t1\right)}} \]

      2. unsub-neg 39.4%

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

      3. *-commutative 39.4%

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

   7. Simplified 39.4%

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

   8. Taylor expanded in u around inf 32.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   9. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   10. Simplified 32.3%

       \[\leadsto \color{blue}{\frac{v}{u} \cdot -0.5} \]

   11. Taylor expanded in v around 0 32.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{v}{u}} \]
   12. Step-by-step derivation

       1. associate-*r/ 32.3%

          \[\leadsto \color{blue}{\frac{-0.5 \cdot v}{u}} \]

       2. *-rgt-identity 32.3%

          \[\leadsto \frac{-0.5 \cdot v}{\color{blue}{u \cdot 1}} \]

       3. times-frac 32.3%

          \[\leadsto \color{blue}{\frac{-0.5}{u} \cdot \frac{v}{1}} \]

       4. /-rgt-identity 32.3%

          \[\leadsto \frac{-0.5}{u} \cdot \color{blue}{v} \]

       5. associate-/r/ 34.9%

          \[\leadsto \color{blue}{\frac{-0.5}{\frac{u}{v}}} \]

   13. Simplified 34.9%

       \[\leadsto \color{blue}{\frac{-0.5}{\frac{u}{v}}} \]

   if -1.44999999999999992e137 < u < 2.1999999999999998e140

   1. Initial program 69.6%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t1 around inf 67.7%

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

      1. associate-*r/ 67.7%

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

      2. neg-mul-1 67.7%

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

   7. Simplified 67.7%

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

   if 2.1999999999999998e140 < u

   1. Initial program 74.7%

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

      1. times-frac 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around 0 89.5%

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

      1. associate-*r/ 89.5%

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

      2. mul-1-neg 89.5%

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

   7. Simplified 89.5%

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

      1. associate-*r/ 89.4%

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

      2. frac-2neg 89.4%

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

      3. add-sqr-sqrt 30.6%

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

      4. sqrt-unprod 66.2%

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

      5. sqr-neg 66.2%

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

      6. sqrt-unprod 47.6%

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

      7. add-sqr-sqrt 72.4%

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

      8. distribute-lft-neg-out 72.4%

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

      9. distribute-frac-neg 72.4%

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

      10. *-commutative 72.4%

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

      11. add-sqr-sqrt 24.8%

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

      12. sqrt-unprod 65.5%

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

      13. sqr-neg 65.5%

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

      14. sqrt-unprod 58.7%

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

      15. add-sqr-sqrt 89.4%

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

      16. distribute-neg-in 89.4%

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

   9. Applied egg-rr 89.7%

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

   10. Taylor expanded in t1 around inf 36.3%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+137}:\\ \;\;\;\;\frac{-0.5}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+140}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 22.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{+163} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.8e+163) (not (<= t1 1.3e+118))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.8e+163) || !(t1 <= 1.3e+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 <= (-5.8d+163)) .or. (.not. (t1 <= 1.3d+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 <= -5.8e+163) || !(t1 <= 1.3e+118)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.8e+163) or not (t1 <= 1.3e+118):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.8e+163) || !(t1 <= 1.3e+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 <= -5.8e+163) || ~((t1 <= 1.3e+118)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.8e+163], N[Not[LessEqual[t1, 1.3e+118]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.79999999999999996e163 or 1.30000000000000008e118 < t1

   1. Initial program 40.2%

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

      1. times-frac 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around inf 88.5%

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

   6. Taylor expanded in u around inf 35.1%

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

   if -5.79999999999999996e163 < t1 < 1.30000000000000008e118

   1. Initial program 80.0%

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

      1. times-frac 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in t1 around 0 59.7%

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

      1. associate-*r/ 59.7%

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

      2. mul-1-neg 59.7%

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

   7. Simplified 59.7%

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

      1. associate-*r/ 60.6%

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

      2. frac-2neg 60.6%

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

      3. add-sqr-sqrt 25.2%

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

      4. sqrt-unprod 40.1%

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

      5. sqr-neg 40.1%

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

      6. sqrt-unprod 20.4%

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

      7. add-sqr-sqrt 35.2%

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

      8. distribute-lft-neg-out 35.2%

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

      9. distribute-frac-neg 35.2%

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

      10. *-commutative 35.2%

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

      11. add-sqr-sqrt 14.8%

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

      12. sqrt-unprod 40.2%

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

      13. sqr-neg 40.2%

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

      14. sqrt-unprod 35.3%

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

      15. add-sqr-sqrt 60.6%

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

      16. distribute-neg-in 60.6%

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

   9. Applied egg-rr 62.9%

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

   10. Taylor expanded in t1 around inf 17.8%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{+163} \lor \neg \left(t1 \leq 1.3 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 13.6% 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 69.6%

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

   1. times-frac 97.3%

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

3. Simplified 97.3%

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

5. Taylor expanded in t1 around inf 55.9%

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

6. Taylor expanded in u around inf 12.7%

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

7. Final simplification 12.7%

   \[\leadsto \frac{v}{t1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))