Rosa's DopplerBench

Percentage Accurate: 73.1% → 98.0%

Time: 16.3s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% 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: 98.0% 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 75.7%

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

   1. times-frac 98.6%

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

   2. distribute-frac-neg 98.6%

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

   3. distribute-neg-frac2 98.6%

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

   4. +-commutative 98.6%

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

   5. distribute-neg-in 98.6%

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

   6. unsub-neg 98.6%

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

3. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -12200:\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{elif}\;u \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -12200.0)
   (/ (/ v (- t1 u)) (/ u t1))
   (if (<= u 7.5e-5) (/ (- v) t1) (* (/ v (+ t1 u)) (/ t1 (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -12200.0) {
		tmp = (v / (t1 - u)) / (u / t1);
	} else if (u <= 7.5e-5) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-12200.0d0)) then
        tmp = (v / (t1 - u)) / (u / t1)
    else if (u <= 7.5d-5) then
        tmp = -v / t1
    else
        tmp = (v / (t1 + u)) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -12200.0) {
		tmp = (v / (t1 - u)) / (u / t1);
	} else if (u <= 7.5e-5) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -12200.0:
		tmp = (v / (t1 - u)) / (u / t1)
	elif u <= 7.5e-5:
		tmp = -v / t1
	else:
		tmp = (v / (t1 + u)) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -12200.0)
		tmp = Float64(Float64(v / Float64(t1 - u)) / Float64(u / t1));
	elseif (u <= 7.5e-5)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -12200.0)
		tmp = (v / (t1 - u)) / (u / t1);
	elseif (u <= 7.5e-5)
		tmp = -v / t1;
	else
		tmp = (v / (t1 + u)) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -12200.0], N[(N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / N[(u / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.5e-5], N[((-v) / t1), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -12200

   1. Initial program 87.0%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around 0 91.5%

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

      1. associate-*r/ 91.5%

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

      2. mul-1-neg 91.5%

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

   7. Simplified 91.5%

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

      1. *-commutative 91.5%

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

      2. clear-num 91.5%

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

      3. un-div-inv 91.5%

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

      4. frac-2neg 91.5%

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

      5. add-sqr-sqrt 50.0%

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

      6. sqrt-unprod 69.7%

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

      7. sqr-neg 69.7%

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

      8. sqrt-unprod 31.7%

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

      9. add-sqr-sqrt 67.2%

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

      10. distribute-neg-in 67.2%

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

      11. add-sqr-sqrt 34.5%

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

      12. sqrt-unprod 67.2%

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

      13. sqr-neg 67.2%

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

      14. sqrt-unprod 32.5%

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

      15. add-sqr-sqrt 67.1%

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

      16. sub-neg 67.1%

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

      17. add-sqr-sqrt 34.6%

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

      18. sqrt-unprod 68.2%

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

      19. sqr-neg 68.2%

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

      20. sqrt-unprod 39.9%

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

      21. add-sqr-sqrt 91.6%

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

   9. Applied egg-rr 91.6%

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

   if -12200 < u < 7.49999999999999934e-5

   1. Initial program 66.6%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in t1 around inf 77.4%

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

      1. associate-*r/ 77.4%

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

      2. neg-mul-1 77.4%

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

   7. Simplified 77.4%

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

   if 7.49999999999999934e-5 < u

   1. Initial program 82.6%

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

      1. times-frac 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t1 around 0 92.6%

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

      1. associate-*r/ 92.6%

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

      2. mul-1-neg 92.6%

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

   7. Simplified 92.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -12200:\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{elif}\;u \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2750 \lor \neg \left(u \leq 6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2750.0) (not (<= u 6e-6)))
   (/ (/ v (- t1 u)) (/ u t1))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2750.0) || !(u <= 6e-6)) {
		tmp = (v / (t1 - u)) / (u / t1);
	} 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 <= (-2750.0d0)) .or. (.not. (u <= 6d-6))) then
        tmp = (v / (t1 - u)) / (u / t1)
    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 <= -2750.0) || !(u <= 6e-6)) {
		tmp = (v / (t1 - u)) / (u / t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2750.0) or not (u <= 6e-6):
		tmp = (v / (t1 - u)) / (u / t1)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2750.0) || !(u <= 6e-6))
		tmp = Float64(Float64(v / Float64(t1 - u)) / Float64(u / t1));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2750.0) || ~((u <= 6e-6)))
		tmp = (v / (t1 - u)) / (u / t1);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2750.0], N[Not[LessEqual[u, 6e-6]], $MachinePrecision]], N[(N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / N[(u / t1), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2750 or 6.0000000000000002e-6 < u

   1. Initial program 84.9%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 92.0%

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

      1. associate-*r/ 92.0%

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

      2. mul-1-neg 92.0%

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

   7. Simplified 92.0%

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

      1. *-commutative 92.0%

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

      2. clear-num 91.5%

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

      3. un-div-inv 91.5%

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

      4. frac-2neg 91.5%

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

      5. add-sqr-sqrt 47.5%

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

      6. sqrt-unprod 65.0%

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

      7. sqr-neg 65.0%

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

      8. sqrt-unprod 30.6%

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

      9. add-sqr-sqrt 63.5%

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

      10. distribute-neg-in 63.5%

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

      11. add-sqr-sqrt 30.6%

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

      12. sqrt-unprod 63.7%

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

      13. sqr-neg 63.7%

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

      14. sqrt-unprod 32.9%

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

      15. add-sqr-sqrt 63.5%

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

      16. sub-neg 63.5%

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

      17. add-sqr-sqrt 30.7%

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

      18. sqrt-unprod 65.1%

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

      19. sqr-neg 65.1%

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

      20. sqrt-unprod 45.5%

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

      21. add-sqr-sqrt 91.4%

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

   9. Applied egg-rr 91.4%

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

   if -2750 < u < 6.0000000000000002e-6

   1. Initial program 66.6%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in t1 around inf 77.4%

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

      1. associate-*r/ 77.4%

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

      2. neg-mul-1 77.4%

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

   7. Simplified 77.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2750 \lor \neg \left(u \leq 6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.1 \cdot 10^{-5} \lor \neg \left(u \leq 0.0205\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.1e-5) (not (<= u 0.0205)))
   (/ t1 (* u (/ (- t1 u) v)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.1e-5) || !(u <= 0.0205)) {
		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 <= (-1.1d-5)) .or. (.not. (u <= 0.0205d0))) 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 <= -1.1e-5) || !(u <= 0.0205)) {
		tmp = t1 / (u * ((t1 - u) / v));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.1e-5) or not (u <= 0.0205):
		tmp = t1 / (u * ((t1 - u) / v))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.1e-5) || !(u <= 0.0205))
		tmp = Float64(t1 / Float64(u * Float64(Float64(t1 - u) / v)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.1e-5) || ~((u <= 0.0205)))
		tmp = t1 / (u * ((t1 - u) / v));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.1e-5], N[Not[LessEqual[u, 0.0205]], $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 < -1.1e-5 or 0.0205000000000000009 < u

   1. Initial program 84.9%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 92.0%

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

      1. associate-*r/ 92.0%

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

      2. mul-1-neg 92.0%

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

   7. Simplified 92.0%

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

      1. *-commutative 92.0%

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

      2. clear-num 91.9%

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

      3. frac-times 90.3%

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

      4. *-un-lft-identity 90.3%

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

      5. add-sqr-sqrt 44.9%

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

      6. sqrt-unprod 68.5%

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

      7. sqr-neg 68.5%

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

      8. sqrt-unprod 32.9%

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

      9. add-sqr-sqrt 63.6%

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

      10. frac-2neg 63.6%

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

      11. distribute-neg-in 63.6%

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

      12. add-sqr-sqrt 30.7%

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

      13. sqrt-unprod 63.7%

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

      14. sqr-neg 63.7%

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

      15. sqrt-unprod 33.0%

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

      16. add-sqr-sqrt 63.7%

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

      17. sub-neg 63.7%

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

      18. add-sqr-sqrt 33.0%

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

      19. sqrt-unprod 60.7%

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

      20. sqr-neg 60.7%

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

      21. sqrt-unprod 42.9%

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

      22. add-sqr-sqrt 90.2%

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

   9. Applied egg-rr 90.2%

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

   if -1.1e-5 < u < 0.0205000000000000009

   1. Initial program 66.6%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in t1 around inf 77.4%

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

      1. associate-*r/ 77.4%

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

      2. neg-mul-1 77.4%

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

   7. Simplified 77.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.1 \cdot 10^{-5} \lor \neg \left(u \leq 0.0205\right):\\ \;\;\;\;\frac{t1}{u \cdot \frac{t1 - u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.7 \lor \neg \left(u \leq 4800000000000\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.7) (not (<= u 4800000000000.0)))
   (* (/ t1 (- u)) (/ v u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.7) || !(u <= 4800000000000.0)) {
		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 <= (-1.7d0)) .or. (.not. (u <= 4800000000000.0d0))) 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 <= -1.7) || !(u <= 4800000000000.0)) {
		tmp = (t1 / -u) * (v / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.7) or not (u <= 4800000000000.0):
		tmp = (t1 / -u) * (v / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.7) || !(u <= 4800000000000.0))
		tmp = Float64(Float64(t1 / Float64(-u)) * Float64(v / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.7) || ~((u <= 4800000000000.0)))
		tmp = (t1 / -u) * (v / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.7], N[Not[LessEqual[u, 4800000000000.0]], $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 < -1.69999999999999996 or 4.8e12 < u

   1. Initial program 84.5%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around 0 91.8%

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

      1. associate-*r/ 91.8%

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

      2. mul-1-neg 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in t1 around 0 90.2%

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

   if -1.69999999999999996 < u < 4.8e12

   1. Initial program 67.6%

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

      1. associate-*l/ 77.2%

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

      2. *-commutative 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in t1 around inf 76.6%

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

      1. associate-*r/ 76.6%

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

      2. neg-mul-1 76.6%

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

   7. Simplified 76.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.7 \lor \neg \left(u \leq 4800000000000\right):\\ \;\;\;\;\frac{t1}{-u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{+30} \lor \neg \left(u \leq 8200000000000\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.9e+30) (not (<= u 8200000000000.0)))
   (/ v (* u (/ u t1)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.9e+30) || !(u <= 8200000000000.0)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.9d+30)) .or. (.not. (u <= 8200000000000.0d0))) then
        tmp = v / (u * (u / t1))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.9e+30) || !(u <= 8200000000000.0)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.9e+30) or not (u <= 8200000000000.0):
		tmp = v / (u * (u / t1))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.9e+30) || !(u <= 8200000000000.0))
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.9e+30) || ~((u <= 8200000000000.0)))
		tmp = v / (u * (u / t1));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.9e+30], N[Not[LessEqual[u, 8200000000000.0]], $MachinePrecision]], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.89999999999999984e30 or 8.2e12 < u

   1. Initial program 83.8%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around 0 92.3%

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

      1. associate-*r/ 92.3%

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

      2. mul-1-neg 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in t1 around 0 91.4%

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

      1. clear-num 90.8%

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

      2. frac-times 81.5%

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

      3. *-un-lft-identity 81.5%

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

      4. add-sqr-sqrt 41.0%

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

      5. sqrt-unprod 67.3%

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

      6. sqr-neg 67.3%

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

      7. sqrt-unprod 33.7%

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

      8. add-sqr-sqrt 63.2%

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

   10. Applied egg-rr 63.2%

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

   if -4.89999999999999984e30 < u < 8.2e12

   1. Initial program 68.7%

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

      1. associate-*l/ 78.0%

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

      2. *-commutative 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in t1 around inf 75.3%

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

      1. associate-*r/ 75.3%

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

      2. neg-mul-1 75.3%

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

   7. Simplified 75.3%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.9 \cdot 10^{+30} \lor \neg \left(u \leq 8200000000000\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 3.9 \cdot 10^{+180}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 3.9e+180)
   (/ v (* (+ t1 u) (- -1.0 (/ u t1))))
   (* (- t1) (/ (/ v (+ t1 u)) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 3.9e+180) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (v <= 3.9d+180) then
        tmp = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
    else
        tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 3.9e+180) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 3.9e+180:
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)))
	else:
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 3.9e+180)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 - Float64(u / t1))));
	else
		tmp = Float64(Float64(-t1) * Float64(Float64(v / Float64(t1 + u)) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 3.9e+180)
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	else
		tmp = -t1 * ((v / (t1 + u)) / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 3.9e+180], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t1) * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 3.9000000000000001e180

   1. Initial program 77.3%

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

      1. times-frac 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t1 around -inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. mul-1-neg 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in v around 0 97.1%

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

      1. associate-*r/ 97.1%

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

      2. mul-1-neg 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in v around 0 97.1%

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

       1. mul-1-neg 97.1%

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

       2. associate-/r* 97.6%

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

       3. +-commutative 97.6%

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

       4. associate-/r* 97.1%

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

       5. distribute-neg-frac2 97.1%

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

       6. *-commutative 97.1%

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

       7. distribute-rgt-neg-in 97.1%

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

       8. distribute-neg-in 97.1%

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

       9. metadata-eval 97.1%

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

       10. unsub-neg 97.1%

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

   13. Simplified 97.1%

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

   if 3.9000000000000001e180 < v

   1. Initial program 63.1%

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

      1. associate-/l* 67.0%

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

   3. Simplified 67.0%

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

      1. associate-/r* 92.2%

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

      2. div-inv 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. associate-*r/ 92.2%

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

      2. *-rgt-identity 92.2%

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

   8. Simplified 92.2%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 3.9 \cdot 10^{+180}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.125:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u 0.125)
   (/ v (* (+ t1 u) (- -1.0 (/ u t1))))
   (* (/ v (+ t1 u)) (/ t1 (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= 0.125) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (v / (t1 + u)) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= 0.125d0) then
        tmp = v / ((t1 + u) * ((-1.0d0) - (u / t1)))
    else
        tmp = (v / (t1 + u)) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= 0.125) {
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	} else {
		tmp = (v / (t1 + u)) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= 0.125:
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)))
	else:
		tmp = (v / (t1 + u)) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= 0.125)
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 - Float64(u / t1))));
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= 0.125)
		tmp = v / ((t1 + u) * (-1.0 - (u / t1)));
	else
		tmp = (v / (t1 + u)) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, 0.125], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < 0.125

   1. Initial program 73.6%

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

      1. times-frac 98.5%

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

      2. distribute-frac-neg 98.5%

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

      3. distribute-neg-frac2 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-neg-in 98.5%

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

      6. unsub-neg 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in t1 around -inf 98.4%

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

      1. associate-*r* 98.4%

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

      2. mul-1-neg 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in v around 0 97.5%

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

      1. associate-*r/ 97.5%

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

      2. mul-1-neg 97.5%

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

   10. Simplified 97.5%

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

   11. Taylor expanded in v around 0 97.5%

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

       1. mul-1-neg 97.5%

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

       2. associate-/r* 97.8%

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

       3. +-commutative 97.8%

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

       4. associate-/r* 97.5%

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

       5. distribute-neg-frac2 97.5%

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

       6. *-commutative 97.5%

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

       7. distribute-rgt-neg-in 97.5%

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

       8. distribute-neg-in 97.5%

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

       9. metadata-eval 97.5%

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

       10. unsub-neg 97.5%

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

   13. Simplified 97.5%

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

   if 0.125 < u

   1. Initial program 82.6%

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

      1. times-frac 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t1 around 0 92.6%

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

      1. associate-*r/ 92.6%

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

      2. mul-1-neg 92.6%

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

   7. Simplified 92.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq 0.125:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.05 \cdot 10^{+75}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.1e+65)
   (/ 1.0 (/ u v))
   (if (<= u 2.05e+75) (/ (- v) t1) (/ v (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+65) {
		tmp = 1.0 / (u / v);
	} else if (u <= 2.05e+75) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.1d+65)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 2.05d+75) then
        tmp = -v / t1
    else
        tmp = v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+65) {
		tmp = 1.0 / (u / v);
	} else if (u <= 2.05e+75) {
		tmp = -v / t1;
	} else {
		tmp = v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.1e+65:
		tmp = 1.0 / (u / v)
	elif u <= 2.05e+75:
		tmp = -v / t1
	else:
		tmp = v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.1e+65)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 2.05e+75)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.1e+65)
		tmp = 1.0 / (u / v);
	elseif (u <= 2.05e+75)
		tmp = -v / t1;
	else
		tmp = v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.1e+65], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.05e+75], N[((-v) / t1), $MachinePrecision], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.09999999999999991e65

   1. Initial program 87.6%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 37.5%

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

      1. mul-1-neg 37.5%

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

      2. neg-sub0 37.5%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. frac-2neg 37.5%

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

      4. add-sqr-sqrt 22.9%

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

      5. sqrt-unprod 35.2%

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

      6. sqr-neg 35.2%

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

      7. sqrt-unprod 12.8%

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

      8. add-sqr-sqrt 33.4%

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

      9. distribute-neg-in 33.4%

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

      10. add-sqr-sqrt 18.1%

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

      11. sqrt-unprod 35.2%

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

      12. sqr-neg 35.2%

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

      13. sqrt-unprod 17.1%

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

      14. add-sqr-sqrt 37.0%

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

      15. sub-neg 37.0%

          \[\leadsto 0 - \frac{v}{\color{blue}{t1 - u}} \]

   7. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
   8. Step-by-step derivation

      1. neg-sub0 37.0%

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

      2. distribute-frac-neg2 37.0%

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

   9. Simplified 37.0%

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

   10. Taylor expanded in t1 around 0 33.5%

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

       1. clear-num 34.2%

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

       2. inv-pow 34.2%

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

   12. Applied egg-rr 34.2%

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

       1. unpow-1 34.2%

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

   14. Simplified 34.2%

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

   if -2.09999999999999991e65 < u < 2.0499999999999999e75

   1. Initial program 70.0%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in t1 around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   7. Simplified 70.5%

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

   if 2.0499999999999999e75 < u

   1. Initial program 80.4%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around inf 27.5%

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

      1. add-sqr-sqrt 25.9%

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

      2. sqrt-unprod 32.1%

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

      3. mul-1-neg 32.1%

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

      4. mul-1-neg 32.1%

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

      5. sqr-neg 32.1%

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

      6. sqrt-unprod 23.1%

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

      7. add-sqr-sqrt 24.9%

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

   7. Applied egg-rr 24.9%

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

Alternative 10: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{+64} \lor \neg \left(u \leq 1.1 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -3.7e+64) (not (<= u 1.1e+79))) (/ v u) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -3.7e+64) || !(u <= 1.1e+79)) {
		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 <= (-3.7d+64)) .or. (.not. (u <= 1.1d+79))) 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 <= -3.7e+64) || !(u <= 1.1e+79)) {
		tmp = v / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -3.7e+64) or not (u <= 1.1e+79):
		tmp = v / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -3.7e+64) || !(u <= 1.1e+79))
		tmp = Float64(v / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -3.7e+64) || ~((u <= 1.1e+79)))
		tmp = v / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -3.7e+64], N[Not[LessEqual[u, 1.1e+79]], $MachinePrecision]], N[(v / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -3.69999999999999983e64 or 1.0999999999999999e79 < u

   1. Initial program 84.3%

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

      1. times-frac 98.4%

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

      2. distribute-frac-neg 98.4%

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

      3. distribute-neg-frac2 98.4%

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

      4. +-commutative 98.4%

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

      5. distribute-neg-in 98.4%

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

      6. unsub-neg 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in t1 around inf 32.9%

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

      1. mul-1-neg 32.9%

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

      2. neg-sub0 32.9%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. frac-2neg 32.9%

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

      4. add-sqr-sqrt 21.3%

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

      5. sqrt-unprod 33.2%

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

      6. sqr-neg 33.2%

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

      7. sqrt-unprod 10.6%

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

      8. add-sqr-sqrt 29.5%

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

      9. distribute-neg-in 29.5%

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

      10. add-sqr-sqrt 15.5%

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

      11. sqrt-unprod 30.9%

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

      12. sqr-neg 30.9%

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

      13. sqrt-unprod 16.2%

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

      14. add-sqr-sqrt 32.7%

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

      15. sub-neg 32.7%

          \[\leadsto 0 - \frac{v}{\color{blue}{t1 - u}} \]

   7. Applied egg-rr 32.7%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
   8. Step-by-step derivation

      1. neg-sub0 32.7%

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

      2. distribute-frac-neg2 32.7%

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

   9. Simplified 32.7%

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

   10. Taylor expanded in t1 around 0 29.5%

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

   if -3.69999999999999983e64 < u < 1.0999999999999999e79

   1. Initial program 70.0%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in t1 around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   7. Simplified 70.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.7 \cdot 10^{+64} \lor \neg \left(u \leq 1.1 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.1e+65) (/ 1.0 (/ u v)) (if (<= u 2.2e+78) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+65) {
		tmp = 1.0 / (u / v);
	} else if (u <= 2.2e+78) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.1d+65)) then
        tmp = 1.0d0 / (u / v)
    else if (u <= 2.2d+78) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.1e+65) {
		tmp = 1.0 / (u / v);
	} else if (u <= 2.2e+78) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.1e+65:
		tmp = 1.0 / (u / v)
	elif u <= 2.2e+78:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.1e+65)
		tmp = Float64(1.0 / Float64(u / v));
	elseif (u <= 2.2e+78)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.1e+65)
		tmp = 1.0 / (u / v);
	elseif (u <= 2.2e+78)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.1e+65], N[(1.0 / N[(u / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 2.2e+78], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.09999999999999991e65

   1. Initial program 87.6%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 37.5%

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

      1. mul-1-neg 37.5%

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

      2. neg-sub0 37.5%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. frac-2neg 37.5%

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

      4. add-sqr-sqrt 22.9%

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

      5. sqrt-unprod 35.2%

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

      6. sqr-neg 35.2%

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

      7. sqrt-unprod 12.8%

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

      8. add-sqr-sqrt 33.4%

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

      9. distribute-neg-in 33.4%

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

      10. add-sqr-sqrt 18.1%

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

      11. sqrt-unprod 35.2%

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

      12. sqr-neg 35.2%

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

      13. sqrt-unprod 17.1%

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

      14. add-sqr-sqrt 37.0%

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

      15. sub-neg 37.0%

          \[\leadsto 0 - \frac{v}{\color{blue}{t1 - u}} \]

   7. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
   8. Step-by-step derivation

      1. neg-sub0 37.0%

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

      2. distribute-frac-neg2 37.0%

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

   9. Simplified 37.0%

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

   10. Taylor expanded in t1 around 0 33.5%

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

       1. clear-num 34.2%

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

       2. inv-pow 34.2%

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

   12. Applied egg-rr 34.2%

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

       1. unpow-1 34.2%

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

   14. Simplified 34.2%

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

   if -2.09999999999999991e65 < u < 2.20000000000000014e78

   1. Initial program 70.0%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in t1 around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   7. Simplified 70.5%

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

   if 2.20000000000000014e78 < u

   1. Initial program 80.4%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around inf 27.5%

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

      1. mul-1-neg 27.5%

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

      2. neg-sub0 27.5%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. frac-2neg 27.5%

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

      4. add-sqr-sqrt 19.5%

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

      5. sqrt-unprod 30.9%

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

      6. sqr-neg 30.9%

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

      7. sqrt-unprod 8.2%

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

      8. add-sqr-sqrt 24.9%

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

      9. distribute-neg-in 24.9%

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

      10. add-sqr-sqrt 12.6%

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

      11. sqrt-unprod 25.8%

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

      12. sqr-neg 25.8%

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

      13. sqrt-unprod 15.2%

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

      14. add-sqr-sqrt 27.8%

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

      15. sub-neg 27.8%

          \[\leadsto 0 - \frac{v}{\color{blue}{t1 - u}} \]

   7. Applied egg-rr 27.8%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
   8. Step-by-step derivation

      1. neg-sub0 27.8%

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

      2. distribute-frac-neg2 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in t1 around 0 24.8%

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

Alternative 12: 57.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 10^{+68}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.05e+65) (/ v (- u)) (if (<= u 1e+68) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.05e+65) {
		tmp = v / -u;
	} else if (u <= 1e+68) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-2.05d+65)) then
        tmp = v / -u
    else if (u <= 1d+68) then
        tmp = -v / t1
    else
        tmp = v / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.05e+65) {
		tmp = v / -u;
	} else if (u <= 1e+68) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.05e+65:
		tmp = v / -u
	elif u <= 1e+68:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.05e+65)
		tmp = Float64(v / Float64(-u));
	elseif (u <= 1e+68)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(v / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -2.05e+65)
		tmp = v / -u;
	elseif (u <= 1e+68)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.05e+65], N[(v / (-u)), $MachinePrecision], If[LessEqual[u, 1e+68], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -2.0500000000000001e65

   1. Initial program 87.6%

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

      1. times-frac 98.3%

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

      2. distribute-frac-neg 98.3%

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

      3. distribute-neg-frac2 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-neg-in 98.3%

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

      6. unsub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in t1 around inf 37.5%

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

   6. Taylor expanded in t1 around 0 34.1%

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

      1. associate-*r/ 34.1%

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

      2. mul-1-neg 34.1%

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

   8. Simplified 34.1%

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

   if -2.0500000000000001e65 < u < 9.99999999999999953e67

   1. Initial program 70.0%

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

      1. associate-*l/ 77.3%

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

      2. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in t1 around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   7. Simplified 70.5%

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

   if 9.99999999999999953e67 < u

   1. Initial program 80.4%

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

      1. times-frac 98.6%

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

      2. distribute-frac-neg 98.6%

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

      3. distribute-neg-frac2 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-neg-in 98.6%

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

      6. unsub-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in t1 around inf 27.5%

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

      1. mul-1-neg 27.5%

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

      2. neg-sub0 27.5%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. frac-2neg 27.5%

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

      4. add-sqr-sqrt 19.5%

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

      5. sqrt-unprod 30.9%

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

      6. sqr-neg 30.9%

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

      7. sqrt-unprod 8.2%

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

      8. add-sqr-sqrt 24.9%

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

      9. distribute-neg-in 24.9%

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

      10. add-sqr-sqrt 12.6%

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

      11. sqrt-unprod 25.8%

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

      12. sqr-neg 25.8%

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

      13. sqrt-unprod 15.2%

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

      14. add-sqr-sqrt 27.8%

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

      15. sub-neg 27.8%

          \[\leadsto 0 - \frac{v}{\color{blue}{t1 - u}} \]

   7. Applied egg-rr 27.8%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
   8. Step-by-step derivation

      1. neg-sub0 27.8%

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

      2. distribute-frac-neg2 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in t1 around 0 24.8%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{v}{-u}\\ \mathbf{elif}\;u \leq 10^{+68}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 23.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+144} \lor \neg \left(t1 \leq 3.95 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -4.5e+144) (not (<= t1 3.95e+102))) (/ v t1) (/ v u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -4.5e+144) || !(t1 <= 3.95e+102)) {
		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 <= (-4.5d+144)) .or. (.not. (t1 <= 3.95d+102))) 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 <= -4.5e+144) || !(t1 <= 3.95e+102)) {
		tmp = v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -4.5e+144) or not (t1 <= 3.95e+102):
		tmp = v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -4.5e+144) || !(t1 <= 3.95e+102))
		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 <= -4.5e+144) || ~((t1 <= 3.95e+102)))
		tmp = v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -4.5e+144], N[Not[LessEqual[t1, 3.95e+102]], $MachinePrecision]], N[(v / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.49999999999999967e144 or 3.9500000000000001e102 < t1

   1. Initial program 51.4%

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

      1. associate-*l/ 55.3%

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

      2. *-commutative 55.3%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in t1 around inf 85.7%

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

      1. associate-*r/ 85.7%

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

      2. neg-mul-1 85.7%

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

   7. Simplified 85.7%

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

      1. add-sqr-sqrt 44.7%

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

      2. sqrt-unprod 53.1%

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

      3. sqr-neg 53.1%

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

      4. sqrt-unprod 14.6%

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

      5. add-sqr-sqrt 33.6%

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

      6. *-un-lft-identity 33.6%

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

   9. Applied egg-rr 33.6%

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

       1. *-lft-identity 33.6%

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

   11. Simplified 33.6%

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

   if -4.49999999999999967e144 < t1 < 3.9500000000000001e102

   1. Initial program 84.3%

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

      1. times-frac 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. distribute-neg-frac2 98.1%

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

      4. +-commutative 98.1%

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

      5. distribute-neg-in 98.1%

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

      6. unsub-neg 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in t1 around inf 43.4%

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

      1. mul-1-neg 43.4%

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

      2. neg-sub0 43.4%

         \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

      3. frac-2neg 43.4%

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

      4. add-sqr-sqrt 24.7%

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

      5. sqrt-unprod 26.7%

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

      6. sqr-neg 26.7%

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

      7. sqrt-unprod 5.6%

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

      8. add-sqr-sqrt 15.1%

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

      9. distribute-neg-in 15.1%

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

      10. add-sqr-sqrt 8.8%

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

      11. sqrt-unprod 26.7%

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

      12. sqr-neg 26.7%

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

      13. sqrt-unprod 17.7%

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

      14. add-sqr-sqrt 43.6%

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

      15. sub-neg 43.6%

          \[\leadsto 0 - \frac{v}{\color{blue}{t1 - u}} \]

   7. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
   8. Step-by-step derivation

      1. neg-sub0 43.6%

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

      2. distribute-frac-neg2 43.6%

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

   9. Simplified 43.6%

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

   10. Taylor expanded in t1 around 0 17.1%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{+144} \lor \neg \left(t1 \leq 3.95 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. times-frac 98.6%

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

   2. distribute-frac-neg 98.6%

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

   3. distribute-neg-frac2 98.6%

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

   4. +-commutative 98.6%

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

   5. distribute-neg-in 98.6%

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

   6. unsub-neg 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in t1 around inf 55.1%

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

   1. clear-num 55.2%

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

   2. un-div-inv 55.2%

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

   3. frac-2neg 55.2%

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

   4. distribute-neg-in 55.2%

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

   5. add-sqr-sqrt 28.2%

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

   6. sqrt-unprod 33.1%

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

   7. sqr-neg 33.1%

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

   8. sqrt-unprod 11.5%

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

   9. add-sqr-sqrt 20.9%

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

   10. sub-neg 20.9%

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

   11. add-sqr-sqrt 12.5%

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

   12. sqrt-unprod 32.4%

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

   13. sqr-neg 32.4%

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

   14. sqrt-unprod 24.8%

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

   15. add-sqr-sqrt 55.3%

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

7. Applied egg-rr 55.3%

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

Alternative 15: 62.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. times-frac 98.6%

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

   2. distribute-frac-neg 98.6%

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

   3. distribute-neg-frac2 98.6%

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

   4. +-commutative 98.6%

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

   5. distribute-neg-in 98.6%

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

   6. unsub-neg 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in t1 around inf 55.1%

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

   1. mul-1-neg 55.1%

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

   2. neg-sub0 55.1%

      \[\leadsto \color{blue}{0 - \frac{v}{t1 + u}} \]

   3. frac-2neg 55.1%

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

   4. add-sqr-sqrt 30.2%

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

   5. sqrt-unprod 34.0%

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

   6. sqr-neg 34.0%

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

   7. sqrt-unprod 8.3%

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

   8. add-sqr-sqrt 20.6%

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

   9. distribute-neg-in 20.6%

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

   10. add-sqr-sqrt 10.0%

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

   11. sqrt-unprod 32.2%

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

   12. sqr-neg 32.2%

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

   13. sqrt-unprod 26.2%

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

   14. add-sqr-sqrt 55.2%

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

   15. sub-neg 55.2%

       \[\leadsto 0 - \frac{v}{\color{blue}{t1 - u}} \]

7. Applied egg-rr 55.2%

   \[\leadsto \color{blue}{0 - \frac{v}{t1 - u}} \]
8. Step-by-step derivation

   1. neg-sub0 55.2%

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

   2. distribute-frac-neg2 55.2%

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

9. Simplified 55.2%

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

10. Taylor expanded in v around 0 55.2%

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

Alternative 16: 13.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. associate-*l/ 77.9%

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

   2. *-commutative 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in t1 around inf 48.1%

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

   1. associate-*r/ 48.1%

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

   2. neg-mul-1 48.1%

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

7. Simplified 48.1%

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

   1. add-sqr-sqrt 25.1%

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

   2. sqrt-unprod 30.9%

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

   3. sqr-neg 30.9%

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

   4. sqrt-unprod 5.2%

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

   5. add-sqr-sqrt 11.4%

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

   6. *-un-lft-identity 11.4%

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

9. Applied egg-rr 11.4%

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

    1. *-lft-identity 11.4%

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

11. Simplified 11.4%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))