Rosa's DopplerBench

Percentage Accurate: 73.0% → 97.8%

Time: 8.9s

Alternatives: 17

Speedup: 1.1×

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: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% 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 75.9%

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

   1. times-frac 97.8%

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

3. Simplified 97.8%

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

4. Final simplification 97.8%

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

Alternative 2: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-67} \lor \neg \left(u \leq 1.3 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 - u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.5e-67) (not (<= u 1.3e-79)))
   (/ (* t1 (/ v (- t1 u))) (+ t1 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.5e-67) || !(u <= 1.3e-79)) {
		tmp = (t1 * (v / (t1 - u))) / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.5d-67)) .or. (.not. (u <= 1.3d-79))) then
        tmp = (t1 * (v / (t1 - u))) / (t1 + u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.5e-67) || !(u <= 1.3e-79)) {
		tmp = (t1 * (v / (t1 - u))) / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.5e-67) or not (u <= 1.3e-79):
		tmp = (t1 * (v / (t1 - u))) / (t1 + u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.5e-67) || !(u <= 1.3e-79))
		tmp = Float64(Float64(t1 * Float64(v / Float64(t1 - u))) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.5e-67) || ~((u <= 1.3e-79)))
		tmp = (t1 * (v / (t1 - u))) / (t1 + u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.5e-67], N[Not[LessEqual[u, 1.3e-79]], $MachinePrecision]], N[(N[(t1 * N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.5000000000000003e-67 or 1.29999999999999997e-79 < u

   1. Initial program 81.8%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

      1. associate-*l/ 99.8%

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

      2. add-sqr-sqrt 53.6%

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

      3. sqrt-unprod 60.3%

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

      4. sqr-neg 60.3%

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

      5. sqrt-unprod 26.9%

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

      6. add-sqr-sqrt 55.6%

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

      7. frac-2neg 55.6%

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

      8. add-sqr-sqrt 27.0%

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

      9. sqrt-unprod 62.5%

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

      10. sqr-neg 62.5%

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

      11. sqrt-unprod 46.0%

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

      12. add-sqr-sqrt 99.8%

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

      13. distribute-neg-in 99.8%

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

      14. add-sqr-sqrt 53.7%

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

      15. sqrt-unprod 89.4%

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

      16. sqr-neg 89.4%

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

      17. sqrt-unprod 40.3%

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

      18. add-sqr-sqrt 85.9%

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

      19. sub-neg 85.9%

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

   5. Applied egg-rr 85.9%

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

   if -5.5000000000000003e-67 < u < 1.29999999999999997e-79

   1. Initial program 66.7%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 87.8%

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

      1. associate-*r/ 87.8%

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

      2. neg-mul-1 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-67} \lor \neg \left(u \leq 1.3 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{v}{t1 - u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 3: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5e-67)
   (/ (/ v (- t1 u)) (/ u t1))
   (if (<= u 7.2e-88) (/ (- v) t1) (* (/ v (+ t1 u)) (/ (- t1) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5e-67) {
		tmp = (v / (t1 - u)) / (u / t1);
	} else if (u <= 7.2e-88) {
		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 <= (-5d-67)) then
        tmp = (v / (t1 - u)) / (u / t1)
    else if (u <= 7.2d-88) 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 <= -5e-67) {
		tmp = (v / (t1 - u)) / (u / t1);
	} else if (u <= 7.2e-88) {
		tmp = -v / t1;
	} else {
		tmp = (v / (t1 + u)) * (-t1 / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5e-67:
		tmp = (v / (t1 - u)) / (u / t1)
	elif u <= 7.2e-88:
		tmp = -v / t1
	else:
		tmp = (v / (t1 + u)) * (-t1 / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5e-67)
		tmp = Float64(Float64(v / Float64(t1 - u)) / Float64(u / t1));
	elseif (u <= 7.2e-88)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / Float64(t1 + u)) * Float64(Float64(-t1) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5e-67)
		tmp = (v / (t1 - u)) / (u / t1);
	elseif (u <= 7.2e-88)
		tmp = -v / t1;
	else
		tmp = (v / (t1 + u)) * (-t1 / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5e-67], N[(N[(v / N[(t1 - u), $MachinePrecision]), $MachinePrecision] / N[(u / t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 7.2e-88], 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 < -4.9999999999999999e-67

   1. Initial program 78.5%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in t1 around 0 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   6. Simplified 81.5%

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

      1. *-commutative 81.5%

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

      2. clear-num 81.5%

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

      3. un-div-inv 81.5%

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

      4. frac-2neg 81.5%

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

      5. add-sqr-sqrt 41.4%

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

      6. sqrt-unprod 61.6%

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

      7. sqr-neg 61.6%

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

      8. sqrt-unprod 31.6%

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

      9. add-sqr-sqrt 56.6%

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

      10. distribute-neg-in 56.6%

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

      11. add-sqr-sqrt 25.6%

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

      12. sqrt-unprod 57.0%

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

      13. sqr-neg 57.0%

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

      14. sqrt-unprod 30.8%

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

      15. add-sqr-sqrt 56.7%

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

      16. sub-neg 56.7%

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

      17. add-sqr-sqrt 25.8%

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

      18. sqrt-unprod 51.5%

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

      19. sqr-neg 51.5%

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

      20. sqrt-unprod 42.2%

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

      21. add-sqr-sqrt 81.5%

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

   8. Applied egg-rr 81.5%

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

   if -4.9999999999999999e-67 < u < 7.1999999999999999e-88

   1. Initial program 66.1%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

   6. Simplified 88.6%

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

   if 7.1999999999999999e-88 < u

   1. Initial program 84.8%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around 0 85.2%

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

      1. associate-*r/ 85.2%

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

      2. neg-mul-1 85.2%

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

   6. Simplified 85.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 4: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{-67} \lor \neg \left(u \leq 1.02 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.2e-67) (not (<= u 1.02e-81)))
   (/ (/ v (- t1 u)) (/ u t1))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.2e-67) || !(u <= 1.02e-81)) {
		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 <= (-6.2d-67)) .or. (.not. (u <= 1.02d-81))) 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 <= -6.2e-67) || !(u <= 1.02e-81)) {
		tmp = (v / (t1 - u)) / (u / t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.2e-67) or not (u <= 1.02e-81):
		tmp = (v / (t1 - u)) / (u / t1)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.2e-67) || !(u <= 1.02e-81))
		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 <= -6.2e-67) || ~((u <= 1.02e-81)))
		tmp = (v / (t1 - u)) / (u / t1);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.2e-67], N[Not[LessEqual[u, 1.02e-81]], $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 < -6.2000000000000005e-67 or 1.01999999999999998e-81 < u

   1. Initial program 81.8%

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

      1. times-frac 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in t1 around 0 83.4%

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

      1. associate-*r/ 83.4%

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

      2. neg-mul-1 83.4%

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

   6. Simplified 83.4%

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

      1. *-commutative 83.4%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

      4. frac-2neg 82.9%

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

      5. add-sqr-sqrt 43.2%

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

      6. sqrt-unprod 62.4%

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

      7. sqr-neg 62.4%

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

      8. sqrt-unprod 28.6%

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

      9. add-sqr-sqrt 55.5%

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

      10. distribute-neg-in 55.5%

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

      11. add-sqr-sqrt 28.7%

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

      12. sqrt-unprod 55.8%

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

      13. sqr-neg 55.8%

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

      14. sqrt-unprod 26.7%

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

      15. add-sqr-sqrt 55.4%

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

      16. sub-neg 55.4%

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

      17. add-sqr-sqrt 28.7%

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

      18. sqrt-unprod 51.6%

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

      19. sqr-neg 51.6%

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

      20. sqrt-unprod 39.2%

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

      21. add-sqr-sqrt 83.0%

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

   8. Applied egg-rr 83.0%

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

   if -6.2000000000000005e-67 < u < 1.01999999999999998e-81

   1. Initial program 66.7%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 87.8%

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

      1. associate-*r/ 87.8%

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

      2. neg-mul-1 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{-67} \lor \neg \left(u \leq 1.02 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{\frac{v}{t1 - u}}{\frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 5: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{-29} \lor \neg \left(t1 \leq 4.6 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -6.5e-29) (not (<= t1 4.6e-94)))
   (/ (- v) (+ t1 u))
   (* v (/ (/ (- t1) u) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6.5e-29) || !(t1 <= 4.6e-94)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-6.5d-29)) .or. (.not. (t1 <= 4.6d-94))) then
        tmp = -v / (t1 + u)
    else
        tmp = v * ((-t1 / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -6.5e-29) || !(t1 <= 4.6e-94)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = v * ((-t1 / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -6.5e-29) or not (t1 <= 4.6e-94):
		tmp = -v / (t1 + u)
	else:
		tmp = v * ((-t1 / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -6.5e-29) || !(t1 <= 4.6e-94))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(v * Float64(Float64(Float64(-t1) / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -6.5e-29) || ~((t1 <= 4.6e-94)))
		tmp = -v / (t1 + u);
	else
		tmp = v * ((-t1 / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -6.5e-29], N[Not[LessEqual[t1, 4.6e-94]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v * N[(N[((-t1) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.5e-29 or 4.5999999999999999e-94 < t1

   1. Initial program 69.6%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t1 around inf 83.0%

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

   if -6.5e-29 < t1 < 4.5999999999999999e-94

   1. Initial program 83.5%

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

      1. associate-/l* 82.2%

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

      2. neg-mul-1 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-*r/ 81.1%

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

      5. associate-/l* 81.1%

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

      6. neg-mul-1 81.1%

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

      7. associate-/r* 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in t1 around 0 75.0%

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

      1. associate-*r/ 75.0%

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

      2. neg-mul-1 75.0%

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

      3. unpow2 75.0%

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

   6. Simplified 75.0%

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

      1. associate-*r/ 77.1%

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

      2. clear-num 75.9%

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

      3. distribute-rgt-neg-out 75.9%

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

      4. distribute-lft-neg-out 75.9%

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

      5. associate-/l/ 75.1%

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

      6. associate-/l* 78.6%

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

      7. associate-/l/ 82.3%

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

      8. add-sqr-sqrt 44.5%

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

      9. sqrt-unprod 54.8%

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

      10. sqr-neg 54.8%

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

      11. sqrt-unprod 21.4%

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

      12. add-sqr-sqrt 40.5%

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

   8. Applied egg-rr 40.5%

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

      1. associate-*r/ 40.6%

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

      2. associate-/r/ 40.6%

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

      3. associate-*r/ 40.4%

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

      4. *-commutative 40.4%

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

      5. associate-*r* 40.4%

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

      6. associate-*l/ 40.4%

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

      7. *-lft-identity 40.4%

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

   10. Simplified 40.4%

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

       1. associate-/l/ 40.4%

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

       2. associate-/r/ 40.5%

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

       3. associate-*l/ 40.5%

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

       4. add-sqr-sqrt 22.4%

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

       5. sqrt-unprod 50.0%

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

       6. sqr-neg 50.0%

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

       7. sqrt-unprod 39.9%

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

       8. add-sqr-sqrt 79.8%

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

       9. distribute-frac-neg 79.8%

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

       10. div-inv 79.7%

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

       11. associate-/r* 79.8%

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

       12. clear-num 79.9%

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

   12. Applied egg-rr 79.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{-29} \lor \neg \left(t1 \leq 4.6 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{u}}{u}\\ \end{array} \]

Alternative 6: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-67} \lor \neg \left(u \leq 3.8 \cdot 10^{-85}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -5.5e-67) (not (<= u 3.8e-85)))
   (* t1 (/ (/ (- v) u) u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.5e-67) || !(u <= 3.8e-85)) {
		tmp = t1 * ((-v / u) / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-5.5d-67)) .or. (.not. (u <= 3.8d-85))) then
        tmp = t1 * ((-v / u) / u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -5.5e-67) || !(u <= 3.8e-85)) {
		tmp = t1 * ((-v / u) / u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -5.5e-67) or not (u <= 3.8e-85):
		tmp = t1 * ((-v / u) / u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -5.5e-67) || !(u <= 3.8e-85))
		tmp = Float64(t1 * Float64(Float64(Float64(-v) / u) / u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -5.5e-67) || ~((u <= 3.8e-85)))
		tmp = t1 * ((-v / u) / u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -5.5e-67], N[Not[LessEqual[u, 3.8e-85]], $MachinePrecision]], N[(t1 * N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -5.5000000000000003e-67 or 3.7999999999999999e-85 < u

   1. Initial program 82.1%

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

      1. associate-/l* 80.9%

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

      2. neg-mul-1 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-*r/ 80.6%

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

      5. associate-/l* 81.0%

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

      6. neg-mul-1 81.0%

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

      7. associate-/r* 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in t1 around 0 74.7%

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

      1. associate-*r/ 74.7%

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

      2. neg-mul-1 74.7%

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

      3. unpow2 74.7%

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

   6. Simplified 74.7%

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

      1. frac-2neg 74.7%

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

      2. remove-double-neg 74.7%

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

      3. associate-*r/ 73.4%

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

   8. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. distribute-rgt-neg-in 73.4%

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

      3. times-frac 79.3%

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

   10. Simplified 79.3%

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

   11. Taylor expanded in v around 0 73.4%

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

       1. unpow2 73.4%

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

       2. associate-*r/ 74.7%

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

       3. associate-*r* 74.7%

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

       4. neg-mul-1 74.7%

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

       5. associate-/r* 78.9%

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

   13. Simplified 78.9%

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

   if -5.5000000000000003e-67 < u < 3.7999999999999999e-85

   1. Initial program 66.1%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

   6. Simplified 88.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{-67} \lor \neg \left(u \leq 3.8 \cdot 10^{-85}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 7: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{-67} \lor \neg \left(u \leq 3.8 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -8.2e-67) (not (<= u 3.8e-85)))
   (/ (* t1 (/ (- v) u)) u)
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.2e-67) || !(u <= 3.8e-85)) {
		tmp = (t1 * (-v / u)) / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-8.2d-67)) .or. (.not. (u <= 3.8d-85))) then
        tmp = (t1 * (-v / u)) / u
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -8.2e-67) || !(u <= 3.8e-85)) {
		tmp = (t1 * (-v / u)) / u;
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -8.2e-67) or not (u <= 3.8e-85):
		tmp = (t1 * (-v / u)) / u
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -8.2e-67) || !(u <= 3.8e-85))
		tmp = Float64(Float64(t1 * Float64(Float64(-v) / u)) / u);
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -8.2e-67) || ~((u <= 3.8e-85)))
		tmp = (t1 * (-v / u)) / u;
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -8.2e-67], N[Not[LessEqual[u, 3.8e-85]], $MachinePrecision]], N[(N[(t1 * N[((-v) / u), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -8.1999999999999994e-67 or 3.7999999999999999e-85 < u

   1. Initial program 82.1%

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

      1. associate-/l* 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in t1 around 0 75.7%

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

      1. unpow2 75.7%

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

   6. Simplified 75.7%

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

      1. add-sqr-sqrt 39.5%

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

      2. sqrt-unprod 50.4%

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

      3. sqr-neg 50.4%

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

      4. sqrt-unprod 25.3%

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

      5. add-sqr-sqrt 52.9%

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

      6. associate-/r/ 50.6%

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

   8. Applied egg-rr 50.6%

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

      1. associate-*l/ 50.4%

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

      2. times-frac 50.2%

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

      3. associate-*l/ 50.1%

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

      4. add-sqr-sqrt 26.9%

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

      5. sqrt-prod 58.8%

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

      6. sqr-neg 58.8%

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

      7. sqrt-unprod 35.2%

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

      8. add-sqr-sqrt 81.6%

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

      9. associate-*l/ 79.3%

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

      10. *-commutative 79.3%

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

      11. frac-2neg 79.3%

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

      12. remove-double-neg 79.3%

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

      13. associate-*r/ 81.6%

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

   10. Applied egg-rr 81.6%

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

   if -8.1999999999999994e-67 < u < 3.7999999999999999e-85

   1. Initial program 66.1%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

   6. Simplified 88.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{-67} \lor \neg \left(u \leq 3.8 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{-67}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -8.2e-67)
   (* t1 (/ (/ (- v) u) u))
   (if (<= u 3.8e-85) (/ (- v) t1) (* (/ v u) (/ t1 (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.2e-67) {
		tmp = t1 * ((-v / u) / u);
	} else if (u <= 3.8e-85) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-8.2d-67)) then
        tmp = t1 * ((-v / u) / u)
    else if (u <= 3.8d-85) then
        tmp = -v / t1
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8.2e-67) {
		tmp = t1 * ((-v / u) / u);
	} else if (u <= 3.8e-85) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -8.2e-67:
		tmp = t1 * ((-v / u) / u)
	elif u <= 3.8e-85:
		tmp = -v / t1
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -8.2e-67)
		tmp = Float64(t1 * Float64(Float64(Float64(-v) / u) / u));
	elseif (u <= 3.8e-85)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -8.2e-67)
		tmp = t1 * ((-v / u) / u);
	elseif (u <= 3.8e-85)
		tmp = -v / t1;
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -8.2e-67], N[(t1 * N[(N[((-v) / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.8e-85], N[((-v) / t1), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.1999999999999994e-67

   1. Initial program 78.5%

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

      1. associate-/l* 80.1%

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

      2. neg-mul-1 80.1%

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

      3. *-commutative 80.1%

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

      4. associate-*r/ 80.0%

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

      5. associate-/l* 80.1%

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

      6. neg-mul-1 80.1%

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

      7. associate-/r* 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in t1 around 0 73.0%

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

      1. associate-*r/ 73.0%

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

      2. neg-mul-1 73.0%

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

      3. unpow2 73.0%

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

   6. Simplified 73.0%

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

      1. frac-2neg 73.0%

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

      2. remove-double-neg 73.0%

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

      3. associate-*r/ 71.6%

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

   8. Applied egg-rr 71.6%

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

      1. *-commutative 71.6%

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

      2. distribute-rgt-neg-in 71.6%

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

      3. times-frac 78.5%

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

   10. Simplified 78.5%

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

   11. Taylor expanded in v around 0 71.6%

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

       1. unpow2 71.6%

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

       2. associate-*r/ 73.0%

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

       3. associate-*r* 73.0%

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

       4. neg-mul-1 73.0%

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

       5. associate-/r* 78.5%

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

   13. Simplified 78.5%

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

   if -8.1999999999999994e-67 < u < 3.7999999999999999e-85

   1. Initial program 66.1%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

   6. Simplified 88.6%

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

   if 3.7999999999999999e-85 < u

   1. Initial program 84.8%

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

      1. associate-/l* 81.6%

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

      2. neg-mul-1 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*r/ 81.0%

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

      5. associate-/l* 81.8%

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

      6. neg-mul-1 81.8%

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

      7. associate-/r* 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in t1 around 0 76.0%

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

      1. associate-*r/ 76.0%

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

      2. neg-mul-1 76.0%

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

      3. unpow2 76.0%

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

   6. Simplified 76.0%

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

      1. frac-2neg 76.0%

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

      2. remove-double-neg 76.0%

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

      3. associate-*r/ 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. *-commutative 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. times-frac 79.9%

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

   10. Simplified 79.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{-67}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]

Alternative 9: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{-67}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -8e-67)
   (/ (- t1) (* u (/ u v)))
   (if (<= u 3.8e-85) (/ (- v) t1) (* (/ v u) (/ t1 (- u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8e-67) {
		tmp = -t1 / (u * (u / v));
	} else if (u <= 3.8e-85) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-8d-67)) then
        tmp = -t1 / (u * (u / v))
    else if (u <= 3.8d-85) then
        tmp = -v / t1
    else
        tmp = (v / u) * (t1 / -u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -8e-67) {
		tmp = -t1 / (u * (u / v));
	} else if (u <= 3.8e-85) {
		tmp = -v / t1;
	} else {
		tmp = (v / u) * (t1 / -u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -8e-67:
		tmp = -t1 / (u * (u / v))
	elif u <= 3.8e-85:
		tmp = -v / t1
	else:
		tmp = (v / u) * (t1 / -u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -8e-67)
		tmp = Float64(Float64(-t1) / Float64(u * Float64(u / v)));
	elseif (u <= 3.8e-85)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(v / u) * Float64(t1 / Float64(-u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -8e-67)
		tmp = -t1 / (u * (u / v));
	elseif (u <= 3.8e-85)
		tmp = -v / t1;
	else
		tmp = (v / u) * (t1 / -u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -8e-67], N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.8e-85], N[((-v) / t1), $MachinePrecision], N[(N[(v / u), $MachinePrecision] * N[(t1 / (-u)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -7.99999999999999954e-67

   1. Initial program 78.5%

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

      1. associate-/l* 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in t1 around 0 74.4%

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

      1. unpow2 74.4%

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

   6. Simplified 74.4%

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

      1. associate-/l* 79.9%

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

      2. associate-/r/ 79.9%

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

   8. Applied egg-rr 79.9%

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

   if -7.99999999999999954e-67 < u < 3.7999999999999999e-85

   1. Initial program 66.1%

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

      1. times-frac 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in t1 around inf 88.6%

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

      1. associate-*r/ 88.6%

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

      2. neg-mul-1 88.6%

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

   6. Simplified 88.6%

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

   if 3.7999999999999999e-85 < u

   1. Initial program 84.8%

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

      1. associate-/l* 81.6%

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

      2. neg-mul-1 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*r/ 81.0%

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

      5. associate-/l* 81.8%

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

      6. neg-mul-1 81.8%

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

      7. associate-/r* 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in t1 around 0 76.0%

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

      1. associate-*r/ 76.0%

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

      2. neg-mul-1 76.0%

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

      3. unpow2 76.0%

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

   6. Simplified 76.0%

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

      1. frac-2neg 76.0%

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

      2. remove-double-neg 76.0%

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

      3. associate-*r/ 74.8%

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

   8. Applied egg-rr 74.8%

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

      1. *-commutative 74.8%

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

      2. distribute-rgt-neg-in 74.8%

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

      3. times-frac 79.9%

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

   10. Simplified 79.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{-67}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]

Alternative 10: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+106} \lor \neg \left(u \leq 5 \cdot 10^{+18}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -1.15e+106) (not (<= u 5e+18)))
   (* v (/ t1 (* u u)))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.15e+106) || !(u <= 5e+18)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-1.15d+106)) .or. (.not. (u <= 5d+18))) then
        tmp = v * (t1 / (u * u))
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -1.15e+106) || !(u <= 5e+18)) {
		tmp = v * (t1 / (u * u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -1.15e+106) or not (u <= 5e+18):
		tmp = v * (t1 / (u * u))
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -1.15e+106) || !(u <= 5e+18))
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -1.15e+106) || ~((u <= 5e+18)))
		tmp = v * (t1 / (u * u));
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -1.15e+106], N[Not[LessEqual[u, 5e+18]], $MachinePrecision]], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.1500000000000001e106 or 5e18 < u

   1. Initial program 81.8%

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

      1. associate-/l* 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in t1 around 0 81.9%

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

      1. unpow2 81.9%

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

   6. Simplified 81.9%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 57.9%

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

      3. sqr-neg 57.9%

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

      4. sqrt-unprod 30.0%

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

      5. add-sqr-sqrt 70.3%

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

      6. associate-/r/ 67.7%

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

   8. Applied egg-rr 67.7%

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

   if -1.1500000000000001e106 < u < 5e18

   1. Initial program 71.7%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in t1 around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   6. Simplified 71.6%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+106} \lor \neg \left(u \leq 5 \cdot 10^{+18}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11: 67.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+105}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -5.5e+105)
   (* v (/ t1 (* u u)))
   (if (<= u 6.5e+18) (/ (- v) t1) (* t1 (/ (/ v u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e+105) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 6.5e+18) {
		tmp = -v / t1;
	} else {
		tmp = t1 * ((v / u) / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-5.5d+105)) then
        tmp = v * (t1 / (u * u))
    else if (u <= 6.5d+18) then
        tmp = -v / t1
    else
        tmp = t1 * ((v / u) / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -5.5e+105) {
		tmp = v * (t1 / (u * u));
	} else if (u <= 6.5e+18) {
		tmp = -v / t1;
	} else {
		tmp = t1 * ((v / u) / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -5.5e+105:
		tmp = v * (t1 / (u * u))
	elif u <= 6.5e+18:
		tmp = -v / t1
	else:
		tmp = t1 * ((v / u) / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -5.5e+105)
		tmp = Float64(v * Float64(t1 / Float64(u * u)));
	elseif (u <= 6.5e+18)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(t1 * Float64(Float64(v / u) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -5.5e+105)
		tmp = v * (t1 / (u * u));
	elseif (u <= 6.5e+18)
		tmp = -v / t1;
	else
		tmp = t1 * ((v / u) / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -5.5e+105], N[(v * N[(t1 / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 6.5e+18], N[((-v) / t1), $MachinePrecision], N[(t1 * N[(N[(v / u), $MachinePrecision] / u), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -5.49999999999999979e105

   1. Initial program 80.0%

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

      1. associate-/l* 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in t1 around 0 80.1%

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

      1. unpow2 80.1%

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

   6. Simplified 80.1%

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

      1. add-sqr-sqrt 40.2%

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

      2. sqrt-unprod 65.0%

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

      3. sqr-neg 65.0%

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

      4. sqrt-unprod 39.9%

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

      5. add-sqr-sqrt 78.4%

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

      6. associate-/r/ 78.4%

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

   8. Applied egg-rr 78.4%

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

   if -5.49999999999999979e105 < u < 6.5e18

   1. Initial program 71.7%

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

      1. times-frac 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in t1 around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   6. Simplified 71.6%

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

   if 6.5e18 < u

   1. Initial program 83.1%

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

      1. associate-/l* 83.4%

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

      2. neg-mul-1 83.4%

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

      3. *-commutative 83.4%

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

      4. associate-*r/ 83.4%

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

      5. associate-/l* 83.4%

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

      6. neg-mul-1 83.4%

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

      7. associate-/r* 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in t1 around 0 83.2%

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

      1. associate-*r/ 83.2%

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

      2. neg-mul-1 83.2%

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

      3. unpow2 83.2%

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

   6. Simplified 83.2%

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

      1. associate-*r/ 78.4%

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

      2. clear-num 78.3%

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

      3. distribute-rgt-neg-out 78.3%

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

      4. distribute-lft-neg-out 78.3%

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

      5. associate-/l/ 83.1%

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

      6. associate-/l* 87.7%

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

      7. associate-/l/ 87.5%

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

      8. add-sqr-sqrt 55.1%

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

      9. sqrt-unprod 58.9%

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

      10. sqr-neg 58.9%

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

      11. sqrt-unprod 21.4%

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

      12. add-sqr-sqrt 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. associate-*r/ 59.5%

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

      2. associate-/r/ 58.9%

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

      3. associate-*r/ 59.7%

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

      4. *-commutative 59.7%

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

      5. associate-*r* 64.4%

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

      6. associate-*l/ 64.4%

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

      7. *-lft-identity 64.4%

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

   10. Simplified 64.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -5.5 \cdot 10^{+105}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]

Alternative 12: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. *-commutative 75.9%

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

   2. times-frac 97.8%

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

   3. neg-mul-1 97.8%

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

   4. associate-/l* 97.5%

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

   5. associate-*r/ 97.5%

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

   6. associate-/l* 97.5%

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

   7. associate-/l/ 97.5%

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

   8. neg-mul-1 97.5%

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

   9. *-lft-identity 97.5%

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

   10. metadata-eval 97.5%

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

   11. times-frac 97.5%

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

   12. neg-mul-1 97.5%

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

   13. remove-double-neg 97.5%

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

   14. neg-mul-1 97.5%

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

   15. sub0-neg 97.5%

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

   16. associate--r+ 97.5%

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

   17. neg-sub0 97.5%

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

   18. div-sub 97.5%

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

   19. distribute-frac-neg 97.5%

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

   20. *-inverses 97.5%

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

   21. metadata-eval 97.5%

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

3. Simplified 97.5%

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

4. Final simplification 97.5%

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

Alternative 13: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -6.6 \cdot 10^{+126} \lor \neg \left(u \leq 7 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -6.6e+126) (not (<= u 7e+93))) (/ v (+ t1 u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.6e+126) || !(u <= 7e+93)) {
		tmp = v / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-6.6d+126)) .or. (.not. (u <= 7d+93))) then
        tmp = v / (t1 + u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -6.6e+126) || !(u <= 7e+93)) {
		tmp = v / (t1 + u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -6.6e+126) or not (u <= 7e+93):
		tmp = v / (t1 + u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -6.6e+126) || !(u <= 7e+93))
		tmp = Float64(v / Float64(t1 + u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -6.6e+126) || ~((u <= 7e+93)))
		tmp = v / (t1 + u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -6.6e+126], N[Not[LessEqual[u, 7e+93]], $MachinePrecision]], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -6.60000000000000026e126 or 6.99999999999999996e93 < u

   1. Initial program 81.5%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*l/ 99.9%

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

      2. add-sqr-sqrt 57.5%

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

      3. sqrt-unprod 67.4%

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

      4. sqr-neg 67.4%

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

      5. sqrt-unprod 32.8%

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

      6. add-sqr-sqrt 78.4%

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

      7. frac-2neg 78.4%

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

      8. add-sqr-sqrt 36.5%

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

      9. sqrt-unprod 80.1%

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

      10. sqr-neg 80.1%

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

      11. sqrt-unprod 51.7%

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

      12. add-sqr-sqrt 99.9%

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

      13. distribute-neg-in 99.9%

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

      14. add-sqr-sqrt 57.6%

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

      15. sqrt-unprod 91.7%

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

      16. sqr-neg 91.7%

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

      17. sqrt-unprod 39.5%

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

      18. add-sqr-sqrt 94.9%

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

      19. sub-neg 94.9%

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

   5. Applied egg-rr 94.9%

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

   6. Taylor expanded in t1 around inf 50.2%

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

   if -6.60000000000000026e126 < u < 6.99999999999999996e93

   1. Initial program 73.2%

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

      1. times-frac 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in t1 around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

   6. Simplified 65.1%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -6.6 \cdot 10^{+126} \lor \neg \left(u \leq 7 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 14: 58.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.55 \cdot 10^{+222}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2.55e+222) (/ v u) (if (<= u 5.2e+123) (/ (- v) t1) (/ v u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2.55e+222) {
		tmp = v / u;
	} else if (u <= 5.2e+123) {
		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 <= (-2.55d+222)) then
        tmp = v / u
    else if (u <= 5.2d+123) 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 <= -2.55e+222) {
		tmp = v / u;
	} else if (u <= 5.2e+123) {
		tmp = -v / t1;
	} else {
		tmp = v / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -2.55e+222:
		tmp = v / u
	elif u <= 5.2e+123:
		tmp = -v / t1
	else:
		tmp = v / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2.55e+222)
		tmp = Float64(v / u);
	elseif (u <= 5.2e+123)
		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 <= -2.55e+222)
		tmp = v / u;
	elseif (u <= 5.2e+123)
		tmp = -v / t1;
	else
		tmp = v / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2.55e+222], N[(v / u), $MachinePrecision], If[LessEqual[u, 5.2e+123], N[((-v) / t1), $MachinePrecision], N[(v / u), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.55e222 or 5.19999999999999971e123 < u

   1. Initial program 85.2%

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

      1. times-frac 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t1 around 0 96.6%

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

      1. associate-*r/ 96.6%

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

      2. neg-mul-1 96.6%

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

   6. Simplified 96.6%

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

      1. *-commutative 96.6%

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

      2. clear-num 96.6%

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

      3. un-div-inv 96.7%

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

      4. frac-2neg 96.7%

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

      5. add-sqr-sqrt 41.3%

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

      6. sqrt-unprod 83.3%

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

      7. sqr-neg 83.3%

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

      8. sqrt-unprod 47.2%

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

      9. add-sqr-sqrt 82.0%

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

      10. distribute-neg-in 82.0%

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

      11. add-sqr-sqrt 49.1%

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

      12. sqrt-unprod 82.1%

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

      13. sqr-neg 82.1%

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

      14. sqrt-unprod 33.0%

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

      15. add-sqr-sqrt 81.9%

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

      16. sub-neg 81.9%

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

      17. add-sqr-sqrt 49.0%

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

      18. sqrt-unprod 67.7%

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

      19. sqr-neg 67.7%

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

      20. sqrt-unprod 37.9%

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

      21. add-sqr-sqrt 96.8%

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

   8. Applied egg-rr 96.8%

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

   9. Taylor expanded in t1 around inf 55.7%

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

   if -2.55e222 < u < 5.19999999999999971e123

   1. Initial program 73.2%

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

      1. times-frac 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in t1 around inf 60.8%

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

      1. associate-*r/ 60.8%

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

      2. neg-mul-1 60.8%

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

   6. Simplified 60.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.55 \cdot 10^{+222}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 15: 23.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{+96}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.05e+136) (/ v t1) (if (<= t1 1.42e+96) (/ v u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.05e+136) {
		tmp = v / t1;
	} else if (t1 <= 1.42e+96) {
		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 (t1 <= (-2.05d+136)) then
        tmp = v / t1
    else if (t1 <= 1.42d+96) 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 (t1 <= -2.05e+136) {
		tmp = v / t1;
	} else if (t1 <= 1.42e+96) {
		tmp = v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -2.05e+136:
		tmp = v / t1
	elif t1 <= 1.42e+96:
		tmp = v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.05e+136)
		tmp = Float64(v / t1);
	elseif (t1 <= 1.42e+96)
		tmp = Float64(v / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -2.05e+136)
		tmp = v / t1;
	elseif (t1 <= 1.42e+96)
		tmp = v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.05e+136], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 1.42e+96], N[(v / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.0499999999999999e136 or 1.41999999999999995e96 < t1

   1. Initial program 55.8%

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

      1. times-frac 99.9%

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

   3. Simplified 99.9%

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

      1. associate-*l/ 100.0%

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

      2. add-sqr-sqrt 47.1%

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

      3. sqrt-unprod 14.2%

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

      4. sqr-neg 14.2%

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

      5. sqrt-unprod 25.2%

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

      6. add-sqr-sqrt 47.8%

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

      7. frac-2neg 47.8%

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

      8. add-sqr-sqrt 27.3%

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

      9. sqrt-unprod 69.0%

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

      10. sqr-neg 69.0%

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

      11. sqrt-unprod 49.8%

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

      12. add-sqr-sqrt 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. add-sqr-sqrt 47.1%

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

      15. sqrt-unprod 53.7%

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

      16. sqr-neg 53.7%

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

      17. sqrt-unprod 29.1%

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

      18. add-sqr-sqrt 54.2%

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

      19. sub-neg 54.2%

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

   5. Applied egg-rr 54.2%

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

   6. Taylor expanded in t1 around inf 45.5%

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

   if -2.0499999999999999e136 < t1 < 1.41999999999999995e96

   1. Initial program 84.1%

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

      1. times-frac 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in t1 around 0 66.5%

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

      1. associate-*r/ 66.5%

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

      2. neg-mul-1 66.5%

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

   6. Simplified 66.5%

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

      1. *-commutative 66.5%

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

      2. clear-num 66.1%

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

      3. un-div-inv 66.1%

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

      4. frac-2neg 66.1%

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

      5. add-sqr-sqrt 32.8%

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

      6. sqrt-unprod 45.3%

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

      7. sqr-neg 45.3%

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

      8. sqrt-unprod 21.9%

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

      9. add-sqr-sqrt 38.4%

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

      10. distribute-neg-in 38.4%

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

      11. add-sqr-sqrt 20.1%

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

      12. sqrt-unprod 37.9%

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

      13. sqr-neg 37.9%

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

      14. sqrt-unprod 17.9%

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

      15. add-sqr-sqrt 38.5%

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

      16. sub-neg 38.5%

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

      17. add-sqr-sqrt 20.5%

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

      18. sqrt-unprod 43.2%

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

      19. sqr-neg 43.2%

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

      20. sqrt-unprod 30.7%

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

      21. add-sqr-sqrt 66.1%

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

   8. Applied egg-rr 66.1%

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

   9. Taylor expanded in t1 around inf 19.7%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 1.42 \cdot 10^{+96}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 16: 62.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return -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 = -v / (t1 + u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. times-frac 97.8%

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

3. Simplified 97.8%

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

4. Taylor expanded in t1 around inf 61.5%

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

5. Final simplification 61.5%

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

Alternative 17: 15.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. times-frac 97.8%

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

3. Simplified 97.8%

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

   1. associate-*l/ 98.8%

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

   2. add-sqr-sqrt 54.1%

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

   3. sqrt-unprod 50.0%

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

   4. sqr-neg 50.0%

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

   5. sqrt-unprod 19.5%

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

   6. add-sqr-sqrt 39.3%

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

   7. frac-2neg 39.3%

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

   8. add-sqr-sqrt 18.7%

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

   9. sqrt-unprod 56.5%

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

   10. sqr-neg 56.5%

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

   11. sqrt-unprod 49.0%

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

   12. add-sqr-sqrt 98.8%

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

   13. distribute-neg-in 98.8%

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

   14. add-sqr-sqrt 54.2%

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

   15. sqrt-unprod 75.9%

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

   16. sqr-neg 75.9%

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

   17. sqrt-unprod 30.1%

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

   18. add-sqr-sqrt 62.1%

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

   19. sub-neg 62.1%

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

5. Applied egg-rr 62.1%

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

6. Taylor expanded in t1 around inf 15.8%

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

7. Final simplification 15.8%

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

Reproduce

herbie shell --seed 2023257 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))