Rosa's DopplerBench

Percentage Accurate: 77.2% → 97.4%

Time: 29.1s

Alternatives: 12

Speedup: 1.1×

• could not determine a ground truth

  1. u = -6.345523484422262e+49
  2. v = 4.1334891586105655e+249
  3. t1 = 6.759880953206116e+186

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: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. *-commutative 79.7%

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

   2. times-frac 96.7%

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

   3. neg-mul-1 96.7%

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

   4. associate-/l* 96.7%

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

   5. associate-*r/ 96.7%

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

   6. associate-/l* 96.7%

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

   7. associate-/l/ 96.7%

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

   8. neg-mul-1 96.7%

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

   9. *-lft-identity 96.7%

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

   10. metadata-eval 96.7%

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

   11. times-frac 96.7%

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

   12. neg-mul-1 96.7%

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

   13. remove-double-neg 96.7%

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

   14. neg-mul-1 96.7%

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

   15. sub0-neg 96.7%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

   16. associate--r+ 96.7%

       \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

   17. neg-sub0 96.7%

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

   18. div-sub 96.7%

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

   19. distribute-frac-neg 96.7%

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

   20. *-inverses 96.7%

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

   21. metadata-eval 96.7%

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

3. Simplified 96.7%

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

4. Final simplification 96.7%

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

Alternative 2: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{-43} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.5e-43) (not (<= t1 650000000000.0)))
   (/ (- v) t1)
   (* t1 (/ (- v) (* u u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.5e-43) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = t1 * (-v / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.5d-43)) .or. (.not. (t1 <= 650000000000.0d0))) then
        tmp = -v / t1
    else
        tmp = t1 * (-v / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.5e-43) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = t1 * (-v / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.5e-43) or not (t1 <= 650000000000.0):
		tmp = -v / t1
	else:
		tmp = t1 * (-v / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.5e-43) || !(t1 <= 650000000000.0))
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 * Float64(Float64(-v) / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.5e-43) || ~((t1 <= 650000000000.0)))
		tmp = -v / t1;
	else
		tmp = t1 * (-v / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.5e-43], N[Not[LessEqual[t1, 650000000000.0]], $MachinePrecision]], N[((-v) / t1), $MachinePrecision], N[(t1 * N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.50000000000000013e-43 or 6.5e11 < t1

   1. Initial program 65.6%

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

      1. associate-/l* 60.6%

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

      2. neg-mul-1 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*r/ 60.6%

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

      5. associate-/l* 61.8%

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

      6. neg-mul-1 61.8%

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

      7. associate-/r* 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in t1 around inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. neg-mul-1 94.6%

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

   6. Simplified 94.6%

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

   if -5.50000000000000013e-43 < t1 < 6.5e11

   1. Initial program 90.3%

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

      1. associate-/l* 87.5%

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

      2. neg-mul-1 87.5%

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

      3. *-commutative 87.5%

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

      4. associate-*r/ 86.8%

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

      5. associate-/l* 86.8%

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

      6. neg-mul-1 86.8%

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

      7. associate-/r* 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in t1 around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

      3. unpow2 74.3%

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

   6. Simplified 74.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{-43} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 3: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{-43} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.45e-43) (not (<= t1 650000000000.0)))
   (/ (- v) t1)
   (* (/ v u) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.45e-43) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.45d-43)) .or. (.not. (t1 <= 650000000000.0d0))) then
        tmp = -v / t1
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.45e-43) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.45e-43) or not (t1 <= 650000000000.0):
		tmp = -v / t1
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.45e-43) || !(t1 <= 650000000000.0))
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.45e-43) || ~((t1 <= 650000000000.0)))
		tmp = -v / t1;
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.45e-43], N[Not[LessEqual[t1, 650000000000.0]], $MachinePrecision]], N[((-v) / t1), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.4500000000000001e-43 or 6.5e11 < t1

   1. Initial program 65.6%

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

      1. associate-/l* 60.6%

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

      2. neg-mul-1 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*r/ 60.6%

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

      5. associate-/l* 61.8%

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

      6. neg-mul-1 61.8%

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

      7. associate-/r* 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in t1 around inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. neg-mul-1 94.6%

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

   6. Simplified 94.6%

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

   if -1.4500000000000001e-43 < t1 < 6.5e11

   1. Initial program 90.3%

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

      1. associate-/l* 87.5%

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

      2. neg-mul-1 87.5%

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

      3. *-commutative 87.5%

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

      4. associate-*r/ 86.8%

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

      5. associate-/l* 86.8%

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

      6. neg-mul-1 86.8%

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

      7. associate-/r* 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in t1 around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

      3. unpow2 74.3%

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

   6. Simplified 74.3%

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

      1. associate-*r/ 78.1%

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

      2. frac-2neg 78.1%

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

      3. add-sqr-sqrt 38.5%

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

      4. sqrt-unprod 49.7%

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

      5. sqr-neg 49.7%

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

      6. sqrt-unprod 20.6%

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

      7. add-sqr-sqrt 41.0%

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

      8. distribute-rgt-neg-out 41.0%

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

      9. add-sqr-sqrt 20.4%

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

      10. sqrt-unprod 53.2%

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

      11. sqr-neg 53.2%

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

      12. sqrt-unprod 39.4%

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

      13. add-sqr-sqrt 78.1%

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

      14. distribute-rgt-neg-in 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. *-commutative 78.1%

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

      2. times-frac 79.2%

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

   10. Simplified 79.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{-43} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-44} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.6e-44) (not (<= t1 650000000000.0)))
   (/ (- v) t1)
   (/ (- v) (* u (/ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-44) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = -v / (u * (u / t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.6d-44)) .or. (.not. (t1 <= 650000000000.0d0))) then
        tmp = -v / t1
    else
        tmp = -v / (u * (u / t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-44) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = -v / (u * (u / t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.6e-44) or not (t1 <= 650000000000.0):
		tmp = -v / t1
	else:
		tmp = -v / (u * (u / t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.6e-44) || !(t1 <= 650000000000.0))
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(-v) / Float64(u * Float64(u / t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.6e-44) || ~((t1 <= 650000000000.0)))
		tmp = -v / t1;
	else
		tmp = -v / (u * (u / t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.6e-44], N[Not[LessEqual[t1, 650000000000.0]], $MachinePrecision]], N[((-v) / t1), $MachinePrecision], N[((-v) / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.59999999999999997e-44 or 6.5e11 < t1

   1. Initial program 65.6%

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

      1. associate-/l* 60.6%

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

      2. neg-mul-1 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*r/ 60.6%

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

      5. associate-/l* 61.8%

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

      6. neg-mul-1 61.8%

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

      7. associate-/r* 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in t1 around inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. neg-mul-1 94.6%

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

   6. Simplified 94.6%

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

   if -1.59999999999999997e-44 < t1 < 6.5e11

   1. Initial program 90.3%

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

      1. *-commutative 90.3%

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

      2. times-frac 94.2%

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

      3. neg-mul-1 94.2%

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

      4. associate-/l* 94.2%

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

      5. associate-*r/ 94.2%

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

      6. associate-/l* 94.2%

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

      7. associate-/l/ 94.2%

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

      8. neg-mul-1 94.2%

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

      9. *-lft-identity 94.2%

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

      10. metadata-eval 94.2%

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

      11. times-frac 94.2%

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

      12. neg-mul-1 94.2%

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

      13. remove-double-neg 94.2%

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

      14. neg-mul-1 94.2%

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

      15. sub0-neg 94.2%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]

      16. associate--r+ 94.2%

          \[\leadsto \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]

      17. neg-sub0 94.2%

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

      18. div-sub 94.2%

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

      19. distribute-frac-neg 94.2%

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

      20. *-inverses 94.2%

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

      21. metadata-eval 94.2%

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

   3. Simplified 94.2%

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

      1. frac-2neg 94.2%

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

      2. distribute-frac-neg 94.2%

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

      3. add-sqr-sqrt 48.7%

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

      4. sqrt-unprod 59.9%

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

      5. sqr-neg 59.9%

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

      6. sqrt-unprod 20.4%

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

      7. add-sqr-sqrt 41.0%

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

      8. distribute-frac-neg 41.0%

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

      9. frac-2neg 41.0%

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

      10. associate-/l/ 41.1%

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

      11. sub-neg 41.1%

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

      12. add-sqr-sqrt 15.1%

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

      13. sqrt-unprod 45.8%

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

      14. sqr-neg 45.8%

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

      15. sqrt-unprod 40.0%

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

      16. add-sqr-sqrt 72.9%

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

      17. distribute-frac-neg 72.9%

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

      18. frac-2neg 72.9%

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

   5. Applied egg-rr 72.9%

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

      1. distribute-neg-frac 72.9%

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

      2. *-commutative 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in t1 around 0 77.1%

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

      1. unpow2 77.1%

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

   10. Simplified 77.1%

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

   11. Taylor expanded in u around 0 77.1%

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

       1. unpow2 77.1%

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

       2. associate-*r/ 79.8%

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

   13. Simplified 79.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-44} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \end{array} \]

Alternative 5: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-44} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.6e-44) (not (<= t1 650000000000.0)))
   (/ (- v) t1)
   (/ (* v (/ t1 u)) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-44) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.6d-44)) .or. (.not. (t1 <= 650000000000.0d0))) then
        tmp = -v / t1
    else
        tmp = (v * (t1 / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.6e-44) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = (v * (t1 / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.6e-44) or not (t1 <= 650000000000.0):
		tmp = -v / t1
	else:
		tmp = (v * (t1 / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.6e-44) || !(t1 <= 650000000000.0))
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.6e-44) || ~((t1 <= 650000000000.0)))
		tmp = -v / t1;
	else
		tmp = (v * (t1 / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.6e-44], N[Not[LessEqual[t1, 650000000000.0]], $MachinePrecision]], N[((-v) / t1), $MachinePrecision], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.59999999999999997e-44 or 6.5e11 < t1

   1. Initial program 65.6%

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

      1. associate-/l* 60.6%

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

      2. neg-mul-1 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*r/ 60.6%

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

      5. associate-/l* 61.8%

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

      6. neg-mul-1 61.8%

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

      7. associate-/r* 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in t1 around inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. neg-mul-1 94.6%

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

   6. Simplified 94.6%

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

   if -1.59999999999999997e-44 < t1 < 6.5e11

   1. Initial program 90.3%

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

      1. associate-/l* 87.5%

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

      2. neg-mul-1 87.5%

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

      3. *-commutative 87.5%

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

      4. associate-*r/ 86.8%

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

      5. associate-/l* 86.8%

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

      6. neg-mul-1 86.8%

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

      7. associate-/r* 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in t1 around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

      3. unpow2 74.3%

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

   6. Simplified 74.3%

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

      1. clear-num 74.4%

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

      2. un-div-inv 74.5%

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

      3. add-sqr-sqrt 36.0%

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

      4. sqrt-unprod 49.8%

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

      5. sqr-neg 49.8%

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

      6. sqrt-unprod 20.5%

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

      7. add-sqr-sqrt 40.9%

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

   8. Applied egg-rr 40.9%

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

      1. unpow2 40.9%

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

      2. associate-/l* 41.0%

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

      3. unpow2 41.0%

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

      4. times-frac 40.9%

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

   10. Simplified 40.9%

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

       1. *-commutative 40.9%

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

       2. frac-2neg 40.9%

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

       3. associate-*l/ 41.0%

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

       4. add-sqr-sqrt 20.4%

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

       5. sqrt-unprod 56.1%

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

       6. sqr-neg 56.1%

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

       7. sqrt-unprod 42.4%

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

       8. add-sqr-sqrt 82.4%

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

   12. Applied egg-rr 82.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-44} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 6: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-41} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{\frac{u}{t1}}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.15e-41) (not (<= t1 650000000000.0)))
   (/ (- v) t1)
   (/ (/ v (/ u t1)) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.15e-41) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = (v / (u / t1)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.15d-41)) .or. (.not. (t1 <= 650000000000.0d0))) then
        tmp = -v / t1
    else
        tmp = (v / (u / t1)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.15e-41) || !(t1 <= 650000000000.0)) {
		tmp = -v / t1;
	} else {
		tmp = (v / (u / t1)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.15e-41) or not (t1 <= 650000000000.0):
		tmp = -v / t1
	else:
		tmp = (v / (u / t1)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.15e-41) || !(t1 <= 650000000000.0))
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / Float64(u / t1)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.15e-41) || ~((t1 <= 650000000000.0)))
		tmp = -v / t1;
	else
		tmp = (v / (u / t1)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.15e-41], N[Not[LessEqual[t1, 650000000000.0]], $MachinePrecision]], N[((-v) / t1), $MachinePrecision], N[(N[(v / N[(u / t1), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.15000000000000005e-41 or 6.5e11 < t1

   1. Initial program 65.6%

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

      1. associate-/l* 60.6%

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

      2. neg-mul-1 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*r/ 60.6%

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

      5. associate-/l* 61.8%

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

      6. neg-mul-1 61.8%

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

      7. associate-/r* 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in t1 around inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. neg-mul-1 94.6%

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

   6. Simplified 94.6%

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

   if -1.15000000000000005e-41 < t1 < 6.5e11

   1. Initial program 90.3%

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

      1. associate-/l* 87.5%

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

      2. neg-mul-1 87.5%

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

      3. *-commutative 87.5%

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

      4. associate-*r/ 86.8%

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

      5. associate-/l* 86.8%

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

      6. neg-mul-1 86.8%

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

      7. associate-/r* 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in t1 around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

      3. unpow2 74.3%

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

   6. Simplified 74.3%

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

      1. clear-num 74.4%

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

      2. un-div-inv 74.5%

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

      3. add-sqr-sqrt 36.0%

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

      4. sqrt-unprod 49.8%

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

      5. sqr-neg 49.8%

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

      6. sqrt-unprod 20.5%

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

      7. add-sqr-sqrt 40.9%

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

   8. Applied egg-rr 40.9%

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

      1. unpow2 40.9%

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

      2. associate-/l* 41.0%

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

      3. unpow2 41.0%

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

      4. times-frac 40.9%

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

   10. Simplified 40.9%

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

       1. *-commutative 40.9%

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

       2. frac-2neg 40.9%

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

       3. associate-*l/ 41.0%

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

       4. add-sqr-sqrt 20.4%

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

       5. sqrt-unprod 56.1%

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

       6. sqr-neg 56.1%

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

       7. sqrt-unprod 42.4%

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

       8. add-sqr-sqrt 82.4%

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

   12. Applied egg-rr 82.4%

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

       1. clear-num 82.4%

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

       2. div-inv 82.5%

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

   14. Applied egg-rr 82.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-41} \lor \neg \left(t1 \leq 650000000000\right):\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{\frac{u}{t1}}}{-u}\\ \end{array} \]

Alternative 7: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+23} \lor \neg \left(u \leq 1150000000000\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.4e+23) (not (<= u 1150000000000.0)))
   (* t1 (/ v (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.4e+23) || !(u <= 1150000000000.0)) {
		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 <= (-3.4d+23)) .or. (.not. (u <= 1150000000000.0d0))) 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 <= -3.4e+23) || !(u <= 1150000000000.0)) {
		tmp = t1 * (v / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.4e+23) or not (u <= 1150000000000.0):
		tmp = t1 * (v / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.4e+23) || !(u <= 1150000000000.0))
		tmp = Float64(t1 * Float64(v / Float64(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 <= -3.4e+23) || ~((u <= 1150000000000.0)))
		tmp = t1 * (v / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.4e+23], N[Not[LessEqual[u, 1150000000000.0]], $MachinePrecision]], N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.39999999999999992e23 or 1.15e12 < u

   1. Initial program 95.8%

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

      1. associate-/l* 95.9%

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

      2. neg-mul-1 95.9%

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

      3. *-commutative 95.9%

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

      4. associate-*r/ 95.9%

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

      5. associate-/l* 95.9%

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

      6. neg-mul-1 95.9%

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

      7. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

      3. unpow2 95.9%

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

   6. Simplified 95.9%

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

      1. add-sqr-sqrt 45.3%

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

      2. sqrt-unprod 77.0%

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

      3. sqr-neg 77.0%

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

      4. sqrt-unprod 41.8%

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

      5. times-frac 41.7%

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

   8. Applied egg-rr 41.7%

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

      1. times-frac 41.8%

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

      2. rem-square-sqrt 78.0%

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

   10. Simplified 78.0%

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

   if -3.39999999999999992e23 < u < 1.15e12

   1. Initial program 73.3%

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

      1. associate-/l* 68.0%

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

      2. neg-mul-1 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*r/ 67.4%

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

      5. associate-/l* 68.1%

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

      6. neg-mul-1 68.1%

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

      7. associate-/r* 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in t1 around inf 75.7%

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

      1. associate-*r/ 75.7%

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

      2. neg-mul-1 75.7%

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

   6. Simplified 75.7%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+23} \lor \neg \left(u \leq 1150000000000\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+23}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1100000000000:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.4e+23)
   (* t1 (/ v (* u u)))
   (if (<= u 1100000000000.0) (/ (- v) t1) (* v (/ t1 (* u u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e+23) {
		tmp = t1 * (v / (u * u));
	} else if (u <= 1100000000000.0) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.4d+23)) then
        tmp = t1 * (v / (u * u))
    else if (u <= 1100000000000.0d0) then
        tmp = -v / t1
    else
        tmp = v * (t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e+23) {
		tmp = t1 * (v / (u * u));
	} else if (u <= 1100000000000.0) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.4e+23:
		tmp = t1 * (v / (u * u))
	elif u <= 1100000000000.0:
		tmp = -v / t1
	else:
		tmp = v * (t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.4e+23)
		tmp = Float64(t1 * Float64(v / Float64(u * u)));
	elseif (u <= 1100000000000.0)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.4e+23)
		tmp = t1 * (v / (u * u));
	elseif (u <= 1100000000000.0)
		tmp = -v / t1;
	else
		tmp = v * (t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.4e+23], N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1100000000000.0], N[((-v) / t1), $MachinePrecision], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.39999999999999992e23

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. neg-mul-1 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-*r/ 96.7%

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

      5. associate-/l* 96.7%

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

      6. neg-mul-1 96.7%

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

      7. associate-/r* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around 0 96.7%

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

      1. associate-*r/ 96.7%

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

      2. neg-mul-1 96.7%

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

      3. unpow2 96.7%

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

   6. Simplified 96.7%

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

      1. add-sqr-sqrt 42.5%

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

      2. sqrt-unprod 85.0%

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

      3. sqr-neg 85.0%

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

      4. sqrt-unprod 48.5%

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

      5. times-frac 48.5%

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

   8. Applied egg-rr 48.5%

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

      1. times-frac 48.5%

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

      2. rem-square-sqrt 88.1%

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

   10. Simplified 88.1%

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

   if -3.39999999999999992e23 < u < 1.1e12

   1. Initial program 73.3%

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

      1. associate-/l* 68.0%

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

      2. neg-mul-1 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*r/ 67.4%

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

      5. associate-/l* 68.1%

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

      6. neg-mul-1 68.1%

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

      7. associate-/r* 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in t1 around inf 75.7%

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

      1. associate-*r/ 75.7%

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

      2. neg-mul-1 75.7%

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

   6. Simplified 75.7%

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

   if 1.1e12 < u

   1. Initial program 95.2%

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

      1. associate-/l* 95.3%

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

      2. neg-mul-1 95.3%

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

      3. *-commutative 95.3%

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

      4. associate-*r/ 95.3%

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

      5. associate-/l* 95.2%

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

      6. neg-mul-1 95.2%

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

      7. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 95.2%

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

      1. associate-*r/ 95.2%

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

      2. neg-mul-1 95.2%

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

      3. unpow2 95.2%

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

   6. Simplified 95.2%

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

      1. clear-num 95.3%

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

      2. un-div-inv 95.3%

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

      3. add-sqr-sqrt 47.5%

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

      4. sqrt-unprod 70.5%

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

      5. sqr-neg 70.5%

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

      6. sqrt-unprod 36.2%

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

      7. add-sqr-sqrt 69.6%

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

   8. Applied egg-rr 69.6%

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

      1. unpow2 69.6%

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

      2. associate-/l* 69.6%

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

      3. unpow2 69.6%

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

      4. times-frac 69.6%

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

   10. Simplified 69.6%

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

   11. Taylor expanded in t1 around 0 69.6%

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

       1. *-commutative 69.6%

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

       2. unpow2 69.6%

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

       3. associate-*r/ 69.8%

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

   13. Simplified 69.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+23}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 1100000000000:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 9: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 1080000000000:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.4e+23)
   (/ v (* u (/ u t1)))
   (if (<= u 1080000000000.0) (/ (- v) t1) (* v (/ t1 (* u u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e+23) {
		tmp = v / (u * (u / t1));
	} else if (u <= 1080000000000.0) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (u * u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-3.4d+23)) then
        tmp = v / (u * (u / t1))
    else if (u <= 1080000000000.0d0) then
        tmp = -v / t1
    else
        tmp = v * (t1 / (u * u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e+23) {
		tmp = v / (u * (u / t1));
	} else if (u <= 1080000000000.0) {
		tmp = -v / t1;
	} else {
		tmp = v * (t1 / (u * u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -3.4e+23:
		tmp = v / (u * (u / t1))
	elif u <= 1080000000000.0:
		tmp = -v / t1
	else:
		tmp = v * (t1 / (u * u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.4e+23)
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	elseif (u <= 1080000000000.0)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -3.4e+23)
		tmp = v / (u * (u / t1));
	elseif (u <= 1080000000000.0)
		tmp = -v / t1;
	else
		tmp = v * (t1 / (u * u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.4e+23], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1080000000000.0], N[((-v) / t1), $MachinePrecision], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.39999999999999992e23

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. neg-mul-1 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-*r/ 96.7%

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

      5. associate-/l* 96.7%

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

      6. neg-mul-1 96.7%

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

      7. associate-/r* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t1 around 0 96.7%

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

      1. associate-*r/ 96.7%

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

      2. neg-mul-1 96.7%

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

      3. unpow2 96.7%

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

   6. Simplified 96.7%

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

      1. clear-num 96.7%

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

      2. un-div-inv 96.6%

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

      3. add-sqr-sqrt 42.5%

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

      4. sqrt-unprod 85.0%

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

      5. sqr-neg 85.0%

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

      6. sqrt-unprod 48.5%

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

      7. add-sqr-sqrt 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. unpow2 88.1%

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

      2. associate-/l* 88.1%

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

      3. unpow2 88.1%

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

      4. times-frac 88.0%

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

   10. Simplified 88.0%

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

       1. clear-num 88.0%

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

       2. frac-times 88.1%

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

       3. *-un-lft-identity 88.1%

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

   12. Applied egg-rr 88.1%

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

   if -3.39999999999999992e23 < u < 1.08e12

   1. Initial program 73.3%

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

      1. associate-/l* 68.0%

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

      2. neg-mul-1 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*r/ 67.4%

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

      5. associate-/l* 68.1%

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

      6. neg-mul-1 68.1%

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

      7. associate-/r* 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in t1 around inf 75.7%

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

      1. associate-*r/ 75.7%

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

      2. neg-mul-1 75.7%

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

   6. Simplified 75.7%

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

   if 1.08e12 < u

   1. Initial program 95.2%

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

      1. associate-/l* 95.3%

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

      2. neg-mul-1 95.3%

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

      3. *-commutative 95.3%

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

      4. associate-*r/ 95.3%

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

      5. associate-/l* 95.2%

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

      6. neg-mul-1 95.2%

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

      7. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around 0 95.2%

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

      1. associate-*r/ 95.2%

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

      2. neg-mul-1 95.2%

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

      3. unpow2 95.2%

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

   6. Simplified 95.2%

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

      1. clear-num 95.3%

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

      2. un-div-inv 95.3%

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

      3. add-sqr-sqrt 47.5%

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

      4. sqrt-unprod 70.5%

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

      5. sqr-neg 70.5%

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

      6. sqrt-unprod 36.2%

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

      7. add-sqr-sqrt 69.6%

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

   8. Applied egg-rr 69.6%

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

      1. unpow2 69.6%

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

      2. associate-/l* 69.6%

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

      3. unpow2 69.6%

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

      4. times-frac 69.6%

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

   10. Simplified 69.6%

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

   11. Taylor expanded in t1 around 0 69.6%

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

       1. *-commutative 69.6%

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

       2. unpow2 69.6%

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

       3. associate-*r/ 69.8%

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

   13. Simplified 69.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 1080000000000:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 10: 59.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+15} \lor \neg \left(u \leq 1150000000000\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.7e+15) (not (<= u 1150000000000.0)))
   (/ (- v) u)
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e+15) || !(u <= 1150000000000.0)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.7d+15)) .or. (.not. (u <= 1150000000000.0d0))) then
        tmp = -v / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e+15) || !(u <= 1150000000000.0)) {
		tmp = -v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.7e+15) or not (u <= 1150000000000.0):
		tmp = -v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.7e+15) || !(u <= 1150000000000.0))
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.7e+15) || ~((u <= 1150000000000.0)))
		tmp = -v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.7e+15], N[Not[LessEqual[u, 1150000000000.0]], $MachinePrecision]], N[((-v) / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.7e15 or 1.15e12 < u

   1. Initial program 95.9%

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

      1. associate-/l* 95.9%

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

      2. neg-mul-1 95.9%

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

      3. *-commutative 95.9%

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

      4. associate-*r/ 96.0%

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

      5. associate-/l* 95.9%

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

      6. neg-mul-1 95.9%

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

      7. associate-/r* 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l/ 95.9%

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

      2. associate-*r/ 95.9%

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

      3. distribute-rgt-neg-in 95.9%

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

      4. distribute-lft-neg-out 95.9%

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

      5. times-frac 97.7%

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

      6. frac-2neg 97.7%

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

      7. associate-*r/ 97.7%

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

      8. add-sqr-sqrt 58.6%

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

      9. sqrt-unprod 82.5%

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

      10. sqr-neg 82.5%

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

      11. sqrt-unprod 26.3%

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

      12. add-sqr-sqrt 76.9%

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

      13. add-sqr-sqrt 35.7%

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

      14. sqrt-unprod 74.9%

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

      15. sqr-neg 74.9%

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

      16. sqrt-unprod 50.3%

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

      17. add-sqr-sqrt 97.7%

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

      18. distribute-neg-in 97.7%

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

   5. Applied egg-rr 97.7%

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

   6. Taylor expanded in t1 around inf 33.8%

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

   7. Taylor expanded in t1 around 0 33.8%

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

      1. associate-*r/ 33.8%

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

      2. neg-mul-1 33.8%

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

   9. Simplified 33.8%

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

   if -2.7e15 < u < 1.15e12

   1. Initial program 73.2%

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

      1. associate-/l* 67.9%

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

      2. neg-mul-1 67.9%

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

      3. *-commutative 67.9%

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

      4. associate-*r/ 67.2%

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

      5. associate-/l* 68.0%

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

      6. neg-mul-1 68.0%

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

      7. associate-/r* 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in t1 around inf 76.1%

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

      1. associate-*r/ 76.1%

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

      2. neg-mul-1 76.1%

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

   6. Simplified 76.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+15} \lor \neg \left(u \leq 1150000000000\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 51.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. associate-/l* 76.0%

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

   2. neg-mul-1 76.0%

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

   3. *-commutative 76.0%

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

   4. associate-*r/ 75.5%

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

   5. associate-/l* 76.0%

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

   6. neg-mul-1 76.0%

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

   7. associate-/r* 82.7%

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

3. Simplified 82.7%

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

4. Taylor expanded in t1 around inf 55.0%

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

   1. associate-*r/ 55.0%

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

   2. neg-mul-1 55.0%

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

6. Simplified 55.0%

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

7. Final simplification 55.0%

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

Alternative 12: 10.0% 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 79.7%

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

   1. associate-/l* 76.0%

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

   2. neg-mul-1 76.0%

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

   3. *-commutative 76.0%

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

   4. associate-*r/ 75.5%

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

   5. associate-/l* 76.0%

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

   6. neg-mul-1 76.0%

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

   7. associate-/r* 82.7%

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

3. Simplified 82.7%

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

   1. associate-/l/ 76.0%

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

   2. associate-*r/ 79.7%

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

   3. distribute-rgt-neg-in 79.7%

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

   4. distribute-lft-neg-out 79.7%

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

   5. times-frac 96.7%

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

   6. frac-2neg 96.7%

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

   7. associate-*r/ 98.5%

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

   8. add-sqr-sqrt 53.5%

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

   9. sqrt-unprod 47.3%

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

   10. sqr-neg 47.3%

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

   11. sqrt-unprod 12.2%

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

   12. add-sqr-sqrt 35.5%

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

   13. add-sqr-sqrt 19.8%

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

   14. sqrt-unprod 53.0%

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

   15. sqr-neg 53.0%

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

   16. sqrt-unprod 44.9%

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

   17. add-sqr-sqrt 98.5%

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

   18. distribute-neg-in 98.5%

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

5. Applied egg-rr 56.0%

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

6. Taylor expanded in t1 around inf 11.7%

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

7. Final simplification 11.7%

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

Reproduce

herbie shell --seed 2023278 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))