Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.1%

Time: 9.3s

Alternatives: 11

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: 98.1% 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 69.5%

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

   1. times-frac 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-neg-frac2 99.5%

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

   4. +-commutative 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. unsub-neg 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.02 \cdot 10^{+108} \lor \neg \left(t1 \leq 2.3 \cdot 10^{+153}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.02e+108) (not (<= t1 2.3e+153)))
   (/ v (- (- t1) (* u 2.0)))
   (* t1 (/ (/ v (+ t1 u)) (- (- u) t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.02e+108) || !(t1 <= 2.3e+153)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * ((v / (t1 + 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 <= (-2.02d+108)) .or. (.not. (t1 <= 2.3d+153))) then
        tmp = v / (-t1 - (u * 2.0d0))
    else
        tmp = t1 * ((v / (t1 + u)) / (-u - t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.02e+108) || !(t1 <= 2.3e+153)) {
		tmp = v / (-t1 - (u * 2.0));
	} else {
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.02e+108) or not (t1 <= 2.3e+153):
		tmp = v / (-t1 - (u * 2.0))
	else:
		tmp = t1 * ((v / (t1 + u)) / (-u - t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.02e+108) || !(t1 <= 2.3e+153))
		tmp = Float64(v / Float64(Float64(-t1) - Float64(u * 2.0)));
	else
		tmp = Float64(t1 * Float64(Float64(v / Float64(t1 + u)) / Float64(Float64(-u) - t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -2.02e+108) || ~((t1 <= 2.3e+153)))
		tmp = v / (-t1 - (u * 2.0));
	else
		tmp = t1 * ((v / (t1 + u)) / (-u - t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.02e+108], N[Not[LessEqual[t1, 2.3e+153]], $MachinePrecision]], N[(v / N[((-t1) - N[(u * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.02000000000000007e108 or 2.3000000000000001e153 < t1

   1. Initial program 43.3%

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

      1. associate-/l* 44.0%

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

      2. distribute-lft-neg-out 44.0%

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

      3. distribute-rgt-neg-in 44.0%

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

      4. associate-/r* 67.6%

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

      5. distribute-neg-frac2 67.6%

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

   3. Simplified 67.6%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-*l/ 100.0%

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

      6. clear-num 100.0%

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

      7. frac-2neg 100.0%

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

      8. frac-times 98.7%

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

      9. *-un-lft-identity 98.7%

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

      10. frac-2neg 98.7%

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

      11. sub-neg 98.7%

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

      12. distribute-neg-in 98.7%

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

      13. +-commutative 98.7%

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

      14. remove-double-neg 98.7%

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

      15. add-sqr-sqrt 52.9%

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

      16. sqrt-unprod 13.1%

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

      17. sqr-neg 13.1%

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

      18. sqrt-unprod 20.1%

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

      19. add-sqr-sqrt 37.8%

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

      20. add-sqr-sqrt 16.5%

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

      21. sqrt-unprod 38.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in u around 0 95.8%

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

      1. *-commutative 95.8%

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

   9. Simplified 95.8%

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

   if -2.02000000000000007e108 < t1 < 2.3000000000000001e153

   1. Initial program 80.8%

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

      1. associate-/l* 83.5%

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

      2. distribute-lft-neg-out 83.5%

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

      3. distribute-rgt-neg-in 83.5%

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

      4. associate-/r* 91.4%

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

      5. distribute-neg-frac2 91.4%

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

   3. Simplified 91.4%

      \[\leadsto \color{blue}{t1 \cdot \frac{\frac{v}{t1 + u}}{-\left(t1 + u\right)}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.02 \cdot 10^{+108} \lor \neg \left(t1 \leq 2.3 \cdot 10^{+153}\right):\\ \;\;\;\;\frac{v}{\left(-t1\right) - u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{t1 + u}}{\left(-u\right) - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{-47} \lor \neg \left(u \leq 2 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.2e-47) (not (<= u 2e-55)))
   (* (/ v (+ t1 u)) (/ t1 (- u)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.2e-47) || !(u <= 2e-55)) {
		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 <= (-5.2d-47)) .or. (.not. (u <= 2d-55))) 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 <= -5.2e-47) || !(u <= 2e-55)) {
		tmp = (v / (t1 + u)) * (t1 / -u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.2e-47) or not (u <= 2e-55):
		tmp = (v / (t1 + u)) * (t1 / -u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.2e-47) || !(u <= 2e-55))
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.2e-47) || ~((u <= 2e-55)))
		tmp = (v / (t1 + u)) * (t1 / -u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.2e-47], N[Not[LessEqual[u, 2e-55]], $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 < -5.2e-47 or 1.99999999999999999e-55 < u

   1. Initial program 72.0%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 81.3%

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

      1. associate-*r/ 81.3%

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

      2. mul-1-neg 81.3%

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

   7. Simplified 81.3%

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

   if -5.2e-47 < u < 1.99999999999999999e-55

   1. Initial program 66.0%

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

      1. associate-/l* 69.2%

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

      2. distribute-lft-neg-out 69.2%

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

      3. distribute-rgt-neg-in 69.2%

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

      4. associate-/r* 79.2%

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

      5. distribute-neg-frac2 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in t1 around inf 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{-47} \lor \neg \left(u \leq 2 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-51} \lor \neg \left(u \leq 1.25 \cdot 10^{-123}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.5e-51) (not (<= u 1.25e-123)))
   (/ t1 (* u (/ (- t1 u) v)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.5e-51) || !(u <= 1.25e-123)) {
		tmp = t1 / (u * ((t1 - u) / v));
	} 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 <= (-5.5d-51)) .or. (.not. (u <= 1.25d-123))) then
        tmp = t1 / (u * ((t1 - u) / v))
    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 <= -5.5e-51) || !(u <= 1.25e-123)) {
		tmp = t1 / (u * ((t1 - u) / v));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.5e-51) or not (u <= 1.25e-123):
		tmp = t1 / (u * ((t1 - u) / v))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.5e-51) || !(u <= 1.25e-123))
		tmp = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.5e-51) || ~((u <= 1.25e-123)))
		tmp = t1 / (u * ((t1 - u) / v));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.5e-51], N[Not[LessEqual[u, 1.25e-123]], $MachinePrecision]], N[(t1 / N[(u * N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.4999999999999997e-51 or 1.25000000000000007e-123 < 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 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. mul-1-neg 80.0%

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

   7. Simplified 80.0%

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

      1. *-commutative 80.0%

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

      2. clear-num 80.3%

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

      3. frac-times 77.1%

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

      4. *-un-lft-identity 77.1%

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

      5. add-sqr-sqrt 36.5%

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

      6. sqrt-unprod 49.1%

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

      7. sqr-neg 49.1%

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

      8. sqrt-unprod 25.6%

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

      9. add-sqr-sqrt 49.5%

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

      10. frac-2neg 49.5%

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

      11. distribute-neg-in 49.5%

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

      12. add-sqr-sqrt 23.9%

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

      13. sqrt-unprod 49.6%

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

      14. sqr-neg 49.6%

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

      15. sqrt-unprod 25.6%

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

      16. add-sqr-sqrt 48.9%

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

      17. sub-neg 48.9%

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

      18. add-sqr-sqrt 25.7%

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

      19. sqrt-unprod 50.8%

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

      20. sqr-neg 50.8%

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

      21. sqrt-unprod 37.6%

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

      22. add-sqr-sqrt 77.7%

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

   9. Applied egg-rr 77.7%

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

   if -5.4999999999999997e-51 < u < 1.25000000000000007e-123

   1. Initial program 64.4%

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

      1. associate-/l* 66.1%

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

      2. distribute-lft-neg-out 66.1%

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

      3. distribute-rgt-neg-in 66.1%

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

      4. associate-/r* 77.1%

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

      5. distribute-neg-frac2 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in t1 around inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-51} \lor \neg \left(u \leq 1.25 \cdot 10^{-123}\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{-48}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;u \leq 1.44 \cdot 10^{-123}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -4.6e-48)
   (* t1 (/ (/ v (- u)) (+ t1 u)))
   (if (<= u 1.44e-123) (/ v (- t1)) (/ t1 (* u (/ (- t1 u) v))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.6e-48) {
		tmp = t1 * ((v / -u) / (t1 + u));
	} else if (u <= 1.44e-123) {
		tmp = v / -t1;
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-4.6d-48)) then
        tmp = t1 * ((v / -u) / (t1 + u))
    else if (u <= 1.44d-123) then
        tmp = v / -t1
    else
        tmp = t1 / (u * ((t1 - u) / v))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -4.6e-48) {
		tmp = t1 * ((v / -u) / (t1 + u));
	} else if (u <= 1.44e-123) {
		tmp = v / -t1;
	} else {
		tmp = t1 / (u * ((t1 - u) / v));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -4.6e-48:
		tmp = t1 * ((v / -u) / (t1 + u))
	elif u <= 1.44e-123:
		tmp = v / -t1
	else:
		tmp = t1 / (u * ((t1 - u) / v))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -4.6e-48)
		tmp = Float64(t1 * Float64(Float64(v / Float64(-u)) / Float64(t1 + u)));
	elseif (u <= 1.44e-123)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -4.6e-48)
		tmp = t1 * ((v / -u) / (t1 + u));
	elseif (u <= 1.44e-123)
		tmp = v / -t1;
	else
		tmp = t1 / (u * ((t1 - u) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -4.6e-48], N[(t1 * N[(N[(v / (-u)), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.44e-123], N[(v / (-t1)), $MachinePrecision], N[(t1 / N[(u * N[(N[(t1 - u), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.6000000000000001e-48

   1. Initial program 73.4%

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

      1. associate-/l* 75.1%

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

      2. distribute-lft-neg-out 75.1%

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

      3. distribute-rgt-neg-in 75.1%

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

      4. associate-/r* 92.2%

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

      5. distribute-neg-frac2 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in t1 around 0 80.4%

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

   if -4.6000000000000001e-48 < u < 1.44000000000000004e-123

   1. Initial program 64.4%

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

      1. associate-/l* 66.1%

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

      2. distribute-lft-neg-out 66.1%

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

      3. distribute-rgt-neg-in 66.1%

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

      4. associate-/r* 77.1%

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

      5. distribute-neg-frac2 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in t1 around inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

   7. Simplified 88.6%

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

   if 1.44000000000000004e-123 < u

   1. Initial program 71.8%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. mul-1-neg 78.3%

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

   7. Simplified 78.3%

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

      1. *-commutative 78.3%

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

      2. clear-num 78.3%

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

      3. frac-times 74.5%

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

      4. *-un-lft-identity 74.5%

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

      5. add-sqr-sqrt 27.8%

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

      6. sqrt-unprod 45.7%

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

      7. sqr-neg 45.7%

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

      8. sqrt-unprod 31.9%

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

      9. add-sqr-sqrt 51.8%

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

      10. frac-2neg 51.8%

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

      11. distribute-neg-in 51.8%

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

      12. add-sqr-sqrt 19.9%

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

      13. sqrt-unprod 52.1%

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

      14. sqr-neg 52.1%

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

      15. sqrt-unprod 32.3%

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

      16. add-sqr-sqrt 50.7%

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

      17. sub-neg 50.7%

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

      18. add-sqr-sqrt 30.6%

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

      19. sqrt-unprod 56.2%

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

      20. sqr-neg 56.2%

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

      21. sqrt-unprod 35.6%

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

      22. add-sqr-sqrt 75.5%

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

   9. Applied egg-rr 75.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{-48}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{-u}}{t1 + u}\\ \mathbf{elif}\;u \leq 1.44 \cdot 10^{-123}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+20} \lor \neg \left(u \leq 3.9 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.8e+20) (not (<= u 3.9e-54)))
   (* (/ t1 (- u)) (/ v u))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.8e+20) || !(u <= 3.9e-54)) {
		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 <= (-5.8d+20)) .or. (.not. (u <= 3.9d-54))) 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 <= -5.8e+20) || !(u <= 3.9e-54)) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.8e+20) or not (u <= 3.9e-54):
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.8e+20) || !(u <= 3.9e-54))
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.8e+20) || ~((u <= 3.9e-54)))
		tmp = (t1 / -u) * (v / u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.8e+20], N[Not[LessEqual[u, 3.9e-54]], $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 < -5.8e20 or 3.9e-54 < u

   1. Initial program 72.5%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 83.2%

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

      1. associate-*r/ 83.2%

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

      2. mul-1-neg 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in t1 around 0 79.1%

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

   if -5.8e20 < u < 3.9e-54

   1. Initial program 66.1%

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

      1. associate-/l* 69.1%

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

      2. distribute-lft-neg-out 69.1%

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

      3. distribute-rgt-neg-in 69.1%

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

      4. associate-/r* 79.6%

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

      5. distribute-neg-frac2 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in t1 around inf 82.7%

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

      1. associate-*r/ 82.7%

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

      2. neg-mul-1 82.7%

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

   7. Simplified 82.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+20} \lor \neg \left(u \leq 3.9 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+72} \lor \neg \left(u \leq 24500000000000\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.45e+72) (not (<= u 24500000000000.0)))
   (/ t1 (* u (/ u v)))
   (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.45e+72) || !(u <= 24500000000000.0)) {
		tmp = t1 / (u * (u / v));
	} 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 <= (-1.45d+72)) .or. (.not. (u <= 24500000000000.0d0))) then
        tmp = t1 / (u * (u / v))
    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 <= -1.45e+72) || !(u <= 24500000000000.0)) {
		tmp = t1 / (u * (u / v));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.45e+72) or not (u <= 24500000000000.0):
		tmp = t1 / (u * (u / v))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.45e+72) || !(u <= 24500000000000.0))
		tmp = Float64(t1 / Float64(u * Float64(u / v)));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.45e+72) || ~((u <= 24500000000000.0)))
		tmp = t1 / (u * (u / v));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.45e+72], N[Not[LessEqual[u, 24500000000000.0]], $MachinePrecision]], N[(t1 / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.45000000000000009e72 or 2.45e13 < u

   1. Initial program 71.1%

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

      1. times-frac 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t1 around 0 86.8%

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

      1. associate-*r/ 86.8%

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

      2. mul-1-neg 86.8%

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

   7. Simplified 86.8%

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

   8. Taylor expanded in t1 around 0 84.3%

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

      1. *-commutative 84.3%

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

      2. clear-num 85.1%

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

      3. frac-times 80.9%

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

      4. *-un-lft-identity 80.9%

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

      5. add-sqr-sqrt 39.8%

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

      6. sqrt-unprod 61.0%

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

      7. sqr-neg 61.0%

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

      8. sqrt-unprod 29.2%

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

      9. add-sqr-sqrt 55.7%

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

   10. Applied egg-rr 55.7%

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

   if -1.45000000000000009e72 < u < 2.45e13

   1. Initial program 68.2%

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

      1. associate-/l* 69.6%

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

      2. distribute-lft-neg-out 69.6%

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

      3. distribute-rgt-neg-in 69.6%

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

      4. associate-/r* 79.8%

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

      5. distribute-neg-frac2 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in t1 around inf 77.8%

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

      1. associate-*r/ 77.8%

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

      2. neg-mul-1 77.8%

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

   7. Simplified 77.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.45 \cdot 10^{+72} \lor \neg \left(u \leq 24500000000000\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 23.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.65 \cdot 10^{+180} \lor \neg \left(t1 \leq 2.7 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.65e+180) (not (<= t1 2.7e+49))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.65e+180) || !(t1 <= 2.7e+49)) {
		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 <= (-3.65d+180)) .or. (.not. (t1 <= 2.7d+49))) 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 <= -3.65e+180) || !(t1 <= 2.7e+49)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.65e+180) or not (t1 <= 2.7e+49):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.65e+180) || !(t1 <= 2.7e+49))
		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 <= -3.65e+180) || ~((t1 <= 2.7e+49)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.65e+180], N[Not[LessEqual[t1, 2.7e+49]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.65000000000000019e180 or 2.7000000000000001e49 < t1

   1. Initial program 44.1%

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

      1. associate-/l* 46.9%

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

      2. distribute-lft-neg-out 46.9%

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

      3. distribute-rgt-neg-in 46.9%

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

      4. associate-/r* 69.3%

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

      5. distribute-neg-frac2 69.3%

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

   3. Simplified 69.3%

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

      1. distribute-frac-neg2 69.3%

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

      2. distribute-rgt-neg-out 69.3%

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

      3. associate-/r* 46.9%

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

      4. distribute-lft-neg-out 46.9%

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

      5. associate-/l* 44.1%

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

      6. times-frac 99.9%

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

      7. frac-2neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 31.2%

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

      10. sqrt-unprod 9.0%

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

      11. sqr-neg 9.0%

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

      12. sqrt-unprod 26.4%

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

      13. add-sqr-sqrt 37.7%

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

      14. add-sqr-sqrt 14.0%

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

      15. sqrt-unprod 48.0%

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

      16. sqr-neg 48.0%

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

      17. sqrt-prod 58.8%

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

      18. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 83.4%

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

      1. mul-1-neg 83.4%

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

   9. Simplified 83.4%

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

       1. clear-num 81.6%

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

       2. associate-/r/ 83.2%

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

       3. +-commutative 83.2%

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

       4. add-sqr-sqrt 45.3%

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

       5. sqrt-unprod 46.4%

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

       6. sqr-neg 46.4%

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

       7. sqrt-unprod 13.0%

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

       8. add-sqr-sqrt 34.7%

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

   11. Applied egg-rr 34.7%

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

   12. Taylor expanded in u around 0 34.1%

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

   if -3.65000000000000019e180 < t1 < 2.7000000000000001e49

   1. Initial program 81.7%

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

      1. times-frac 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-neg-frac2 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t1 around 0 67.1%

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

      1. associate-*r/ 67.1%

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

      2. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

      1. clear-num 67.1%

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

      2. frac-2neg 67.1%

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

      3. frac-times 60.6%

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

      4. *-un-lft-identity 60.6%

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

      5. add-sqr-sqrt 27.9%

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

      6. sqrt-unprod 38.2%

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

      7. sqr-neg 38.2%

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

      8. sqrt-unprod 18.9%

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

      9. add-sqr-sqrt 37.4%

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

      10. add-sqr-sqrt 20.0%

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

      11. sqrt-unprod 41.1%

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

      12. sqr-neg 41.1%

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

      13. sqrt-unprod 28.7%

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

      14. add-sqr-sqrt 60.6%

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

      15. distribute-neg-in 60.6%

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

      16. add-sqr-sqrt 31.8%

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

      17. sqrt-unprod 61.9%

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

      18. sqr-neg 61.9%

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

      19. sqrt-unprod 30.0%

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

      20. add-sqr-sqrt 62.6%

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

      21. sub-neg 62.6%

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

   9. Applied egg-rr 62.6%

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

   10. Taylor expanded in u around 0 16.6%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.65 \cdot 10^{+180} \lor \neg \left(t1 \leq 2.7 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.25e+145) (/ v u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.25e+145) {
		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 <= (-1.25d+145)) 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 <= -1.25e+145) {
		tmp = v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.25e+145:
		tmp = v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.25e+145)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.25e+145)
		tmp = v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.25e+145], N[(v / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.24999999999999992e145

   1. Initial program 68.9%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around 0 94.3%

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

      1. associate-*r/ 94.3%

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

      2. mul-1-neg 94.3%

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

   7. Simplified 94.3%

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

      1. clear-num 94.2%

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

      2. frac-2neg 94.2%

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

      3. frac-times 72.1%

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

      4. *-un-lft-identity 72.1%

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

      5. add-sqr-sqrt 34.7%

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

      6. sqrt-unprod 53.9%

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

      7. sqr-neg 53.9%

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

      8. sqrt-unprod 34.8%

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

      9. add-sqr-sqrt 69.8%

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

      10. add-sqr-sqrt 39.9%

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

      11. sqrt-unprod 60.8%

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

      12. sqr-neg 60.8%

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

      13. sqrt-unprod 32.3%

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

      14. add-sqr-sqrt 72.1%

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

      15. distribute-neg-in 72.1%

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

      16. add-sqr-sqrt 39.9%

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

      17. sqrt-unprod 69.5%

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

      18. sqr-neg 69.5%

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

      19. sqrt-unprod 32.6%

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

      20. add-sqr-sqrt 72.5%

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

      21. sub-neg 72.5%

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

   9. Applied egg-rr 72.5%

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

   10. Taylor expanded in u around 0 31.4%

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

   if -1.24999999999999992e145 < u

   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. associate-/l* 72.0%

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

      2. distribute-lft-neg-out 72.0%

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

      3. distribute-rgt-neg-in 72.0%

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

      4. associate-/r* 82.6%

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

      5. distribute-neg-frac2 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in t1 around inf 59.7%

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

      1. associate-*r/ 59.7%

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

      2. neg-mul-1 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.25 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. associate-/l* 71.7%

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

   2. distribute-lft-neg-out 71.7%

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

   3. distribute-rgt-neg-in 71.7%

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

   4. associate-/r* 84.3%

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

   5. distribute-neg-frac2 84.3%

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

3. Simplified 84.3%

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

   1. distribute-frac-neg2 84.3%

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

   2. distribute-rgt-neg-out 84.3%

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

   3. associate-/r* 71.7%

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

   4. distribute-lft-neg-out 71.7%

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

   5. associate-/l* 69.5%

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

   6. times-frac 99.5%

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

   7. frac-2neg 99.5%

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

   8. associate-*r/ 99.3%

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

   9. add-sqr-sqrt 48.4%

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

   10. sqrt-unprod 44.5%

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

   11. sqr-neg 44.5%

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

   12. sqrt-unprod 19.3%

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

   13. add-sqr-sqrt 36.0%

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

   14. add-sqr-sqrt 16.4%

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

   15. sqrt-unprod 54.4%

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

   16. sqr-neg 54.4%

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

   17. sqrt-prod 48.2%

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

   18. add-sqr-sqrt 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in t1 around inf 58.1%

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

   1. mul-1-neg 58.1%

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

9. Simplified 58.1%

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

10. Final simplification 58.1%

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

Alternative 11: 14.5% 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 69.5%

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

   1. associate-/l* 71.7%

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

   2. distribute-lft-neg-out 71.7%

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

   3. distribute-rgt-neg-in 71.7%

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

   4. associate-/r* 84.3%

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

   5. distribute-neg-frac2 84.3%

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

3. Simplified 84.3%

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

   1. distribute-frac-neg2 84.3%

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

   2. distribute-rgt-neg-out 84.3%

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

   3. associate-/r* 71.7%

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

   4. distribute-lft-neg-out 71.7%

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

   5. associate-/l* 69.5%

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

   6. times-frac 99.5%

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

   7. frac-2neg 99.5%

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

   8. associate-*r/ 99.3%

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

   9. add-sqr-sqrt 48.4%

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

   10. sqrt-unprod 44.5%

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

   11. sqr-neg 44.5%

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

   12. sqrt-unprod 19.3%

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

   13. add-sqr-sqrt 36.0%

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

   14. add-sqr-sqrt 16.4%

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

   15. sqrt-unprod 54.4%

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

   16. sqr-neg 54.4%

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

   17. sqrt-prod 48.2%

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

   18. add-sqr-sqrt 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in t1 around inf 58.1%

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

   1. mul-1-neg 58.1%

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

9. Simplified 58.1%

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

    1. clear-num 58.2%

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

    2. associate-/r/ 58.0%

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

    3. +-commutative 58.0%

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

    4. add-sqr-sqrt 29.4%

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

    5. sqrt-unprod 36.2%

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

    6. sqr-neg 36.2%

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

    7. sqrt-unprod 9.2%

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

    8. add-sqr-sqrt 21.1%

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

11. Applied egg-rr 21.1%

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

12. Taylor expanded in u around 0 13.2%

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

13. Final simplification 13.2%

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

Reproduce

herbie shell --seed 2024074 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))