Rosa's DopplerBench

Percentage Accurate: 72.4% → 98.1%

Time: 9.4s

Alternatives: 13

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: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. associate-*l/ 75.1%

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

   2. *-commutative 75.1%

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

3. Simplified 75.1%

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

   1. associate-*r/ 74.0%

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

   2. times-frac 97.9%

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

   3. *-commutative 97.9%

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

   4. frac-2neg 97.9%

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

   5. +-commutative 97.9%

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

   6. distribute-neg-in 97.9%

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

   7. sub-neg 97.9%

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

   8. associate-*r/ 98.5%

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

   9. add-sqr-sqrt 50.3%

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

   10. sqrt-unprod 42.9%

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

   11. sqr-neg 42.9%

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

   12. sqrt-unprod 20.1%

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

   13. add-sqr-sqrt 38.4%

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

   14. sub-neg 38.4%

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

   15. +-commutative 38.4%

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

   16. add-sqr-sqrt 18.3%

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

   17. sqrt-unprod 49.9%

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

   18. sqr-neg 49.9%

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

   19. sqrt-unprod 36.4%

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

   20. add-sqr-sqrt 19.9%

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

   21. sqrt-unprod 39.9%

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

   22. sqr-neg 39.9%

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

   23. sqrt-unprod 22.9%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{+43}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 7.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{\frac{t1}{u}}{\frac{-1}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -6e+43)
   (/ (- v) (+ t1 u))
   (if (<= t1 7.5e+40)
     (/ (/ (/ t1 u) (/ -1.0 v)) u)
     (* (/ (- t1) (+ t1 u)) (/ v t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -6e+43) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 7.5e+40) {
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	} else {
		tmp = (-t1 / (t1 + u)) * (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 <= (-6d+43)) then
        tmp = -v / (t1 + u)
    else if (t1 <= 7.5d+40) then
        tmp = ((t1 / u) / ((-1.0d0) / v)) / u
    else
        tmp = (-t1 / (t1 + u)) * (v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -6e+43) {
		tmp = -v / (t1 + u);
	} else if (t1 <= 7.5e+40) {
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	} else {
		tmp = (-t1 / (t1 + u)) * (v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -6e+43:
		tmp = -v / (t1 + u)
	elif t1 <= 7.5e+40:
		tmp = ((t1 / u) / (-1.0 / v)) / u
	else:
		tmp = (-t1 / (t1 + u)) * (v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -6e+43)
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	elseif (t1 <= 7.5e+40)
		tmp = Float64(Float64(Float64(t1 / u) / Float64(-1.0 / v)) / u);
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(t1 + u)) * Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -6e+43)
		tmp = -v / (t1 + u);
	elseif (t1 <= 7.5e+40)
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	else
		tmp = (-t1 / (t1 + u)) * (v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -6e+43], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 7.5e+40], N[(N[(N[(t1 / u), $MachinePrecision] / N[(-1.0 / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -6.00000000000000033e43

   1. Initial program 51.9%

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

      1. associate-*l/ 56.3%

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

      2. *-commutative 56.3%

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

   3. Simplified 56.3%

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

      1. associate-*r/ 51.9%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 99.3%

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

      10. sqrt-unprod 40.2%

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

      11. sqr-neg 40.2%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 24.1%

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

      14. sub-neg 24.1%

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

      15. +-commutative 24.1%

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

      16. add-sqr-sqrt 24.1%

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

      17. sqrt-unprod 24.8%

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

      18. sqr-neg 24.8%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 0.0%

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

      22. sqr-neg 0.0%

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

      23. sqrt-unprod 0.0%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 90.2%

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

      1. mul-1-neg 90.2%

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

   9. Simplified 90.2%

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

   if -6.00000000000000033e43 < t1 < 7.4999999999999996e40

   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 96.3%

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

      2. distribute-frac-neg 96.3%

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

      3. distribute-neg-frac2 96.3%

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

      4. +-commutative 96.3%

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

      5. distribute-neg-in 96.3%

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

      6. unsub-neg 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in t1 around 0 75.5%

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

   6. Taylor expanded in t1 around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. mul-1-neg 78.0%

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

   8. Simplified 78.0%

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

      1. distribute-frac-neg 78.0%

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

      2. distribute-frac-neg2 78.0%

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

      3. associate-*l/ 78.9%

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

      4. add-sqr-sqrt 39.6%

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

      5. sqrt-unprod 56.1%

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

      6. sqr-neg 56.1%

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

      7. sqrt-unprod 19.0%

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

      8. add-sqr-sqrt 40.1%

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

   10. Applied egg-rr 40.1%

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

       1. clear-num 40.1%

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

       2. un-div-inv 40.1%

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

       3. add-sqr-sqrt 17.8%

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

       4. sqrt-unprod 48.1%

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

       5. sqr-neg 48.1%

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

       6. sqrt-unprod 42.4%

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

       7. add-sqr-sqrt 79.1%

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

       8. div-inv 79.1%

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

       9. associate-/r* 79.2%

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

       10. add-sqr-sqrt 42.5%

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

       11. sqrt-unprod 48.8%

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

       12. sqr-neg 48.8%

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

       13. sqrt-unprod 17.8%

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

       14. add-sqr-sqrt 40.2%

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

       15. frac-2neg 40.2%

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

       16. metadata-eval 40.2%

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

       17. add-sqr-sqrt 17.0%

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

       18. sqrt-unprod 48.0%

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

       19. sqr-neg 48.0%

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

       20. sqrt-unprod 41.9%

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

       21. add-sqr-sqrt 79.2%

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

   12. Applied egg-rr 79.2%

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

   if 7.4999999999999996e40 < t1

   1. Initial program 65.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 89.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{+43}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 7.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{\frac{t1}{u}}{\frac{-1}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+43}:\\ \;\;\;\;\left(-1 + \frac{u}{t1}\right) \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.35 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\frac{t1}{u}}{\frac{-1}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.8e+43)
   (* (+ -1.0 (/ u t1)) (/ v (+ t1 u)))
   (if (<= t1 3.35e+38)
     (/ (/ (/ t1 u) (/ -1.0 v)) u)
     (* (/ (- t1) (+ t1 u)) (/ v t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.8e+43) {
		tmp = (-1.0 + (u / t1)) * (v / (t1 + u));
	} else if (t1 <= 3.35e+38) {
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	} else {
		tmp = (-t1 / (t1 + u)) * (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 <= (-1.8d+43)) then
        tmp = ((-1.0d0) + (u / t1)) * (v / (t1 + u))
    else if (t1 <= 3.35d+38) then
        tmp = ((t1 / u) / ((-1.0d0) / v)) / u
    else
        tmp = (-t1 / (t1 + u)) * (v / t1)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.8e+43) {
		tmp = (-1.0 + (u / t1)) * (v / (t1 + u));
	} else if (t1 <= 3.35e+38) {
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	} else {
		tmp = (-t1 / (t1 + u)) * (v / t1);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.8e+43:
		tmp = (-1.0 + (u / t1)) * (v / (t1 + u))
	elif t1 <= 3.35e+38:
		tmp = ((t1 / u) / (-1.0 / v)) / u
	else:
		tmp = (-t1 / (t1 + u)) * (v / t1)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.8e+43)
		tmp = Float64(Float64(-1.0 + Float64(u / t1)) * Float64(v / Float64(t1 + u)));
	elseif (t1 <= 3.35e+38)
		tmp = Float64(Float64(Float64(t1 / u) / Float64(-1.0 / v)) / u);
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(t1 + u)) * Float64(v / t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -1.8e+43)
		tmp = (-1.0 + (u / t1)) * (v / (t1 + u));
	elseif (t1 <= 3.35e+38)
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	else
		tmp = (-t1 / (t1 + u)) * (v / t1);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.8e+43], N[(N[(-1.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.35e+38], N[(N[(N[(t1 / u), $MachinePrecision] / N[(-1.0 / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision], N[(N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / t1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.80000000000000005e43

   1. Initial program 51.9%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 90.4%

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

   if -1.80000000000000005e43 < t1 < 3.35000000000000012e38

   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 96.3%

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

      2. distribute-frac-neg 96.3%

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

      3. distribute-neg-frac2 96.3%

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

      4. +-commutative 96.3%

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

      5. distribute-neg-in 96.3%

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

      6. unsub-neg 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in t1 around 0 75.5%

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

   6. Taylor expanded in t1 around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. mul-1-neg 78.0%

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

   8. Simplified 78.0%

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

      1. distribute-frac-neg 78.0%

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

      2. distribute-frac-neg2 78.0%

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

      3. associate-*l/ 78.9%

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

      4. add-sqr-sqrt 39.6%

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

      5. sqrt-unprod 56.1%

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

      6. sqr-neg 56.1%

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

      7. sqrt-unprod 19.0%

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

      8. add-sqr-sqrt 40.1%

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

   10. Applied egg-rr 40.1%

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

       1. clear-num 40.1%

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

       2. un-div-inv 40.1%

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

       3. add-sqr-sqrt 17.8%

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

       4. sqrt-unprod 48.1%

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

       5. sqr-neg 48.1%

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

       6. sqrt-unprod 42.4%

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

       7. add-sqr-sqrt 79.1%

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

       8. div-inv 79.1%

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

       9. associate-/r* 79.2%

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

       10. add-sqr-sqrt 42.5%

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

       11. sqrt-unprod 48.8%

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

       12. sqr-neg 48.8%

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

       13. sqrt-unprod 17.8%

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

       14. add-sqr-sqrt 40.2%

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

       15. frac-2neg 40.2%

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

       16. metadata-eval 40.2%

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

       17. add-sqr-sqrt 17.0%

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

       18. sqrt-unprod 48.0%

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

       19. sqr-neg 48.0%

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

       20. sqrt-unprod 41.9%

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

       21. add-sqr-sqrt 79.2%

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

   12. Applied egg-rr 79.2%

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

   if 3.35000000000000012e38 < t1

   1. Initial program 65.7%

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

      1. times-frac 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t1 around inf 89.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+43}:\\ \;\;\;\;\left(-1 + \frac{u}{t1}\right) \cdot \frac{v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.35 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{\frac{t1}{u}}{\frac{-1}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+43} \lor \neg \left(t1 \leq 2.4 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t1}{u}}{\frac{-1}{v}}}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.8e+43) (not (<= t1 2.4e+41)))
   (/ (- v) (+ t1 u))
   (/ (/ (/ t1 u) (/ -1.0 v)) u)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.8e+43) || !(t1 <= 2.4e+41)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-1.8d+43)) .or. (.not. (t1 <= 2.4d+41))) then
        tmp = -v / (t1 + u)
    else
        tmp = ((t1 / u) / ((-1.0d0) / v)) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.8e+43) || !(t1 <= 2.4e+41)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -1.8e+43) or not (t1 <= 2.4e+41):
		tmp = -v / (t1 + u)
	else:
		tmp = ((t1 / u) / (-1.0 / v)) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.8e+43) || !(t1 <= 2.4e+41))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(Float64(t1 / u) / Float64(-1.0 / v)) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -1.8e+43) || ~((t1 <= 2.4e+41)))
		tmp = -v / (t1 + u);
	else
		tmp = ((t1 / u) / (-1.0 / v)) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.8e+43], N[Not[LessEqual[t1, 2.4e+41]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] / N[(-1.0 / v), $MachinePrecision]), $MachinePrecision] / u), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.80000000000000005e43 or 2.4000000000000002e41 < t1

   1. Initial program 58.8%

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

      1. associate-*l/ 64.7%

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

      2. *-commutative 64.7%

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

   3. Simplified 64.7%

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

      1. associate-*r/ 58.8%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 49.2%

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

      10. sqrt-unprod 25.1%

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

      11. sqr-neg 25.1%

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

      12. sqrt-unprod 23.1%

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

      13. add-sqr-sqrt 35.0%

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

      14. sub-neg 35.0%

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

      15. +-commutative 35.0%

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

      16. add-sqr-sqrt 11.9%

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

      17. sqrt-unprod 47.9%

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

      18. sqr-neg 47.9%

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

      19. sqrt-unprod 45.3%

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

      20. add-sqr-sqrt 18.7%

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

      21. sqrt-unprod 44.5%

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

      22. sqr-neg 44.5%

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

      23. sqrt-unprod 31.0%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 89.6%

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

      1. mul-1-neg 89.6%

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

   9. Simplified 89.6%

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

   if -1.80000000000000005e43 < t1 < 2.4000000000000002e41

   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 96.3%

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

      2. distribute-frac-neg 96.3%

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

      3. distribute-neg-frac2 96.3%

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

      4. +-commutative 96.3%

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

      5. distribute-neg-in 96.3%

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

      6. unsub-neg 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in t1 around 0 75.5%

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

   6. Taylor expanded in t1 around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. mul-1-neg 78.0%

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

   8. Simplified 78.0%

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

      1. distribute-frac-neg 78.0%

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

      2. distribute-frac-neg2 78.0%

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

      3. associate-*l/ 78.9%

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

      4. add-sqr-sqrt 39.6%

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

      5. sqrt-unprod 56.1%

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

      6. sqr-neg 56.1%

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

      7. sqrt-unprod 19.0%

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

      8. add-sqr-sqrt 40.1%

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

   10. Applied egg-rr 40.1%

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

       1. clear-num 40.1%

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

       2. un-div-inv 40.1%

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

       3. add-sqr-sqrt 17.8%

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

       4. sqrt-unprod 48.1%

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

       5. sqr-neg 48.1%

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

       6. sqrt-unprod 42.4%

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

       7. add-sqr-sqrt 79.1%

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

       8. div-inv 79.1%

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

       9. associate-/r* 79.2%

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

       10. add-sqr-sqrt 42.5%

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

       11. sqrt-unprod 48.8%

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

       12. sqr-neg 48.8%

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

       13. sqrt-unprod 17.8%

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

       14. add-sqr-sqrt 40.2%

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

       15. frac-2neg 40.2%

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

       16. metadata-eval 40.2%

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

       17. add-sqr-sqrt 17.0%

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

       18. sqrt-unprod 48.0%

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

       19. sqr-neg 48.0%

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

       20. sqrt-unprod 41.9%

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

       21. add-sqr-sqrt 79.2%

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

   12. Applied egg-rr 79.2%

       \[\leadsto \frac{\color{blue}{\frac{\frac{t1}{u}}{\frac{-1}{v}}}}{u} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.8 \cdot 10^{+43} \lor \neg \left(t1 \leq 2.4 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t1}{u}}{\frac{-1}{v}}}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{+43} \lor \neg \left(t1 \leq 9.5 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.3e+43) (not (<= t1 9.5e+38)))
   (/ (- v) (+ t1 u))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.3e+43) || !(t1 <= 9.5e+38)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-3.3d+43)) .or. (.not. (t1 <= 9.5d+38))) then
        tmp = -v / (t1 + u)
    else
        tmp = (t1 / u) * (-v / u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.3e+43) || !(t1 <= 9.5e+38)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3.3e+43) or not (t1 <= 9.5e+38):
		tmp = -v / (t1 + u)
	else:
		tmp = (t1 / u) * (-v / u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.3e+43) || !(t1 <= 9.5e+38))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -3.3e+43) || ~((t1 <= 9.5e+38)))
		tmp = -v / (t1 + u);
	else
		tmp = (t1 / u) * (-v / u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.3e+43], N[Not[LessEqual[t1, 9.5e+38]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.3000000000000001e43 or 9.4999999999999995e38 < t1

   1. Initial program 58.8%

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

      1. associate-*l/ 64.7%

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

      2. *-commutative 64.7%

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

   3. Simplified 64.7%

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

      1. associate-*r/ 58.8%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 49.2%

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

      10. sqrt-unprod 25.1%

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

      11. sqr-neg 25.1%

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

      12. sqrt-unprod 23.1%

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

      13. add-sqr-sqrt 35.0%

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

      14. sub-neg 35.0%

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

      15. +-commutative 35.0%

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

      16. add-sqr-sqrt 11.9%

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

      17. sqrt-unprod 47.9%

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

      18. sqr-neg 47.9%

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

      19. sqrt-unprod 45.3%

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

      20. add-sqr-sqrt 18.7%

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

      21. sqrt-unprod 44.5%

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

      22. sqr-neg 44.5%

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

      23. sqrt-unprod 31.0%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 89.6%

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

      1. mul-1-neg 89.6%

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

   9. Simplified 89.6%

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

   if -3.3000000000000001e43 < t1 < 9.4999999999999995e38

   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 96.3%

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

      2. distribute-frac-neg 96.3%

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

      3. distribute-neg-frac2 96.3%

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

      4. +-commutative 96.3%

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

      5. distribute-neg-in 96.3%

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

      6. unsub-neg 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in t1 around 0 75.5%

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

   6. Taylor expanded in t1 around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. mul-1-neg 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{+43} \lor \neg \left(t1 \leq 9.5 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{+43} \lor \neg \left(t1 \leq 6 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -5.2e+43) (not (<= t1 6e+38)))
   (/ (- v) (+ t1 u))
   (/ (* t1 (/ v u)) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.2e+43) || !(t1 <= 6e+38)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((t1 <= (-5.2d+43)) .or. (.not. (t1 <= 6d+38))) then
        tmp = -v / (t1 + u)
    else
        tmp = (t1 * (v / u)) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -5.2e+43) || !(t1 <= 6e+38)) {
		tmp = -v / (t1 + u);
	} else {
		tmp = (t1 * (v / u)) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (t1 <= -5.2e+43) or not (t1 <= 6e+38):
		tmp = -v / (t1 + u)
	else:
		tmp = (t1 * (v / u)) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -5.2e+43) || !(t1 <= 6e+38))
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	else
		tmp = Float64(Float64(t1 * Float64(v / u)) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((t1 <= -5.2e+43) || ~((t1 <= 6e+38)))
		tmp = -v / (t1 + u);
	else
		tmp = (t1 * (v / u)) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -5.2e+43], N[Not[LessEqual[t1, 6e+38]], $MachinePrecision]], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * N[(v / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.20000000000000042e43 or 6.0000000000000002e38 < t1

   1. Initial program 58.8%

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

      1. associate-*l/ 64.7%

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

      2. *-commutative 64.7%

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

   3. Simplified 64.7%

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

      1. associate-*r/ 58.8%

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

      2. times-frac 99.9%

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

      3. *-commutative 99.9%

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

      4. frac-2neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. sub-neg 99.9%

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

      8. associate-*r/ 99.9%

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

      9. add-sqr-sqrt 49.2%

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

      10. sqrt-unprod 25.1%

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

      11. sqr-neg 25.1%

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

      12. sqrt-unprod 23.1%

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

      13. add-sqr-sqrt 35.0%

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

      14. sub-neg 35.0%

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

      15. +-commutative 35.0%

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

      16. add-sqr-sqrt 11.9%

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

      17. sqrt-unprod 47.9%

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

      18. sqr-neg 47.9%

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

      19. sqrt-unprod 45.3%

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

      20. add-sqr-sqrt 18.7%

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

      21. sqrt-unprod 44.5%

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

      22. sqr-neg 44.5%

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

      23. sqrt-unprod 31.0%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t1 around inf 89.6%

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

      1. mul-1-neg 89.6%

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

   9. Simplified 89.6%

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

   if -5.20000000000000042e43 < t1 < 6.0000000000000002e38

   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 96.3%

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

      2. distribute-frac-neg 96.3%

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

      3. distribute-neg-frac2 96.3%

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

      4. +-commutative 96.3%

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

      5. distribute-neg-in 96.3%

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

      6. unsub-neg 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in t1 around 0 75.5%

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

   6. Taylor expanded in t1 around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. mul-1-neg 78.0%

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

   8. Simplified 78.0%

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

      1. distribute-frac-neg 78.0%

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

      2. distribute-frac-neg2 78.0%

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

      3. associate-*l/ 78.9%

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

      4. add-sqr-sqrt 39.6%

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

      5. sqrt-unprod 56.1%

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

      6. sqr-neg 56.1%

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

      7. sqrt-unprod 19.0%

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

      8. add-sqr-sqrt 40.1%

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

   10. Applied egg-rr 40.1%

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

       1. add-sqr-sqrt 17.8%

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

       2. sqrt-unprod 48.1%

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

       3. sqr-neg 48.1%

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

       4. sqrt-unprod 42.3%

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

       5. add-sqr-sqrt 78.9%

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

       6. distribute-lft-neg-in 78.9%

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

   12. Applied egg-rr 78.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{+43} \lor \neg \left(t1 \leq 6 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+64} \lor \neg \left(u \leq 10^{+138}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -4.2e+64) (not (<= u 1e+138)))
   (/ v (* u (/ u t1)))
   (/ (- v) (+ t1 u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.2e+64) || !(u <= 1e+138)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-4.2d+64)) .or. (.not. (u <= 1d+138))) then
        tmp = v / (u * (u / t1))
    else
        tmp = -v / (t1 + u)
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -4.2e+64) || !(u <= 1e+138)) {
		tmp = v / (u * (u / t1));
	} else {
		tmp = -v / (t1 + u);
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -4.2e+64) or not (u <= 1e+138):
		tmp = v / (u * (u / t1))
	else:
		tmp = -v / (t1 + u)
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -4.2e+64) || !(u <= 1e+138))
		tmp = Float64(v / Float64(u * Float64(u / t1)));
	else
		tmp = Float64(Float64(-v) / Float64(t1 + u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -4.2e+64) || ~((u <= 1e+138)))
		tmp = v / (u * (u / t1));
	else
		tmp = -v / (t1 + u);
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -4.2e+64], N[Not[LessEqual[u, 1e+138]], $MachinePrecision]], N[(v / N[(u * N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -4.2000000000000001e64 or 1e138 < u

   1. Initial program 78.6%

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

      1. times-frac 97.4%

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

      2. distribute-frac-neg 97.4%

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

      3. distribute-neg-frac2 97.4%

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

      4. +-commutative 97.4%

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

      5. distribute-neg-in 97.4%

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

      6. unsub-neg 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in t1 around 0 84.6%

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

   6. Taylor expanded in t1 around 0 84.2%

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

      1. associate-*r/ 84.2%

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

      2. mul-1-neg 84.2%

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

   8. Simplified 84.2%

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

      1. clear-num 83.7%

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

      2. frac-times 74.1%

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

      3. *-un-lft-identity 74.1%

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

      4. add-sqr-sqrt 38.2%

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

      5. sqrt-unprod 61.9%

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

      6. sqr-neg 61.9%

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

      7. sqrt-unprod 30.9%

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

      8. add-sqr-sqrt 63.6%

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

   10. Applied egg-rr 63.6%

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

   if -4.2000000000000001e64 < u < 1e138

   1. Initial program 71.2%

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

      1. associate-*l/ 76.3%

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

      2. *-commutative 76.3%

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

   3. Simplified 76.3%

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

      1. associate-*r/ 71.2%

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

      2. times-frac 98.1%

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

      3. *-commutative 98.1%

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

      4. frac-2neg 98.1%

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

      5. +-commutative 98.1%

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

      6. distribute-neg-in 98.1%

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

      7. sub-neg 98.1%

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

      8. associate-*r/ 99.2%

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

      9. add-sqr-sqrt 50.8%

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

      10. sqrt-unprod 30.1%

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

      11. sqr-neg 30.1%

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

      12. sqrt-unprod 12.7%

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

      13. add-sqr-sqrt 22.1%

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

      14. sub-neg 22.1%

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

      15. +-commutative 22.1%

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

      16. add-sqr-sqrt 9.4%

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

      17. sqrt-unprod 40.1%

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

      18. sqr-neg 40.1%

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

      19. sqrt-unprod 36.7%

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

      20. add-sqr-sqrt 18.1%

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

      21. sqrt-unprod 43.3%

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

      22. sqr-neg 43.3%

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

      23. sqrt-unprod 25.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in t1 around inf 69.7%

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

      1. mul-1-neg 69.7%

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

   9. Simplified 69.7%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+64} \lor \neg \left(u \leq 10^{+138}\right):\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double u, double v, double t1) {
	return (v / (-1.0 - (u / t1))) / (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 / ((-1.0d0) - (u / t1))) / (t1 + u)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. associate-*l/ 75.1%

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

   2. *-commutative 75.1%

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

3. Simplified 75.1%

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

   1. associate-*r/ 74.0%

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

   2. times-frac 97.9%

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

   3. *-commutative 97.9%

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

   4. frac-2neg 97.9%

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

   5. +-commutative 97.9%

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

   6. distribute-neg-in 97.9%

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

   7. sub-neg 97.9%

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

   8. associate-*r/ 98.5%

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

   9. add-sqr-sqrt 50.3%

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

   10. sqrt-unprod 42.9%

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

   11. sqr-neg 42.9%

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

   12. sqrt-unprod 20.1%

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

   13. add-sqr-sqrt 38.4%

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

   14. sub-neg 38.4%

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

   15. +-commutative 38.4%

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

   16. add-sqr-sqrt 18.3%

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

   17. sqrt-unprod 49.9%

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

   18. sqr-neg 49.9%

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

   19. sqrt-unprod 36.4%

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

   20. add-sqr-sqrt 19.9%

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

   21. sqrt-unprod 39.9%

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

   22. sqr-neg 39.9%

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

   23. sqrt-unprod 22.9%

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

6. Applied egg-rr 98.5%

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

   1. clear-num 98.3%

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

   2. inv-pow 98.3%

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

8. Applied egg-rr 98.3%

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

   1. unpow-1 98.3%

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

10. Simplified 98.3%

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

11. Taylor expanded in v around 0 84.0%

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

    1. mul-1-neg 84.0%

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

    2. associate-*r/ 98.4%

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

    3. *-commutative 98.4%

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

    4. associate-/r/ 98.4%

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

    5. distribute-neg-frac2 98.4%

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

    6. neg-sub0 98.4%

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

    7. *-lft-identity 98.4%

       \[\leadsto \frac{\frac{v}{0 - \frac{\color{blue}{1 \cdot \left(t1 + u\right)}}{t1}}}{t1 + u} \]

    8. associate-*l/ 98.3%

       \[\leadsto \frac{\frac{v}{0 - \color{blue}{\frac{1}{t1} \cdot \left(t1 + u\right)}}}{t1 + u} \]

    9. distribute-lft-in 98.3%

       \[\leadsto \frac{\frac{v}{0 - \color{blue}{\left(\frac{1}{t1} \cdot t1 + \frac{1}{t1} \cdot u\right)}}}{t1 + u} \]

    10. lft-mult-inverse 98.4%

        \[\leadsto \frac{\frac{v}{0 - \left(\color{blue}{1} + \frac{1}{t1} \cdot u\right)}}{t1 + u} \]

    11. associate--r+ 98.4%

        \[\leadsto \frac{\frac{v}{\color{blue}{\left(0 - 1\right) - \frac{1}{t1} \cdot u}}}{t1 + u} \]

    12. metadata-eval 98.4%

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

    13. associate-*l/ 98.4%

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

    14. *-lft-identity 98.4%

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

13. Simplified 98.4%

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

14. Final simplification 98.4%

    \[\leadsto \frac{\frac{v}{-1 - \frac{u}{t1}}}{t1 + u} \]
15. Add Preprocessing

Alternative 9: 40.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.45 \cdot 10^{-197}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 1.45e-197) (/ v (+ t1 u)) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1.45e-197) {
		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 (v <= 1.45d-197) 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 (v <= 1.45e-197) {
		tmp = v / (t1 + u);
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 1.45e-197:
		tmp = v / (t1 + u)
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 1.45e-197)
		tmp = Float64(v / Float64(t1 + u));
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 1.45e-197)
		tmp = v / (t1 + u);
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 1.45e-197], N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.45000000000000011e-197

   1. Initial program 78.2%

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

      1. times-frac 98.7%

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

      2. distribute-frac-neg 98.7%

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

      3. distribute-neg-frac2 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. unsub-neg 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in t1 around inf 67.7%

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

      1. clear-num 68.3%

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

      2. clear-num 67.1%

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

      3. frac-times 67.7%

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

      4. metadata-eval 67.7%

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

      5. sub-neg 67.7%

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

      6. distribute-neg-in 67.7%

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

      7. +-commutative 67.7%

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

      8. add-sqr-sqrt 34.5%

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

      9. sqrt-unprod 32.1%

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

      10. sqr-neg 32.1%

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

      11. sqrt-unprod 15.6%

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

      12. add-sqr-sqrt 34.2%

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

      13. frac-2neg 34.2%

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

   7. Applied egg-rr 34.2%

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

      1. *-commutative 34.2%

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

      2. associate-/r* 33.6%

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

      3. associate-/r/ 33.5%

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

      4. associate-*r/ 37.9%

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

      5. associate-/r/ 36.0%

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

      6. *-commutative 36.0%

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

      7. associate-*r* 28.1%

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

      8. lft-mult-inverse 28.1%

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

      9. *-lft-identity 28.1%

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

   9. Simplified 28.1%

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

   if 1.45000000000000011e-197 < v

   1. Initial program 67.2%

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

      1. associate-*l/ 65.8%

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

      2. *-commutative 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in t1 around inf 44.6%

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

      1. associate-*r/ 44.6%

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

      2. neg-mul-1 44.6%

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

   7. Simplified 44.6%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.45 \cdot 10^{-197}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+215}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.15e+215) (/ (- v) u) (/ v (- t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+215) {
		tmp = -v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.15d+215)) then
        tmp = -v / u
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.15e+215) {
		tmp = -v / u;
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.15e+215:
		tmp = -v / u
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.15e+215)
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(v / Float64(-t1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.15e+215)
		tmp = -v / u;
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.15e+215], N[((-v) / u), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.1500000000000001e215

   1. Initial program 92.8%

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

      1. times-frac 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t1 around 0 100.0%

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

   6. Taylor expanded in t1 around inf 41.8%

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

      1. associate-*r/ 41.8%

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

      2. mul-1-neg 41.8%

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

   8. Simplified 41.8%

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

   if -1.1500000000000001e215 < u

   1. Initial program 73.0%

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

      1. associate-*l/ 74.2%

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

      2. *-commutative 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in t1 around inf 54.5%

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

      1. associate-*r/ 54.5%

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

      2. neg-mul-1 54.5%

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

   7. Simplified 54.5%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+215}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% 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 74.0%

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

   1. associate-*l/ 75.1%

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

   2. *-commutative 75.1%

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

3. Simplified 75.1%

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

   1. associate-*r/ 74.0%

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

   2. times-frac 97.9%

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

   3. *-commutative 97.9%

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

   4. frac-2neg 97.9%

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

   5. +-commutative 97.9%

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

   6. distribute-neg-in 97.9%

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

   7. sub-neg 97.9%

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

   8. associate-*r/ 98.5%

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

   9. add-sqr-sqrt 50.3%

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

   10. sqrt-unprod 42.9%

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

   11. sqr-neg 42.9%

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

   12. sqrt-unprod 20.1%

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

   13. add-sqr-sqrt 38.4%

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

   14. sub-neg 38.4%

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

   15. +-commutative 38.4%

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

   16. add-sqr-sqrt 18.3%

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

   17. sqrt-unprod 49.9%

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

   18. sqr-neg 49.9%

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

   19. sqrt-unprod 36.4%

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

   20. add-sqr-sqrt 19.9%

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

   21. sqrt-unprod 39.9%

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

   22. sqr-neg 39.9%

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

   23. sqrt-unprod 22.9%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in t1 around inf 57.7%

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

   1. mul-1-neg 57.7%

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

9. Simplified 57.7%

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

10. Final simplification 57.7%

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

Alternative 12: 55.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(u, v, t1)
	return Float64(v / Float64(-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 74.0%

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

   1. associate-*l/ 75.1%

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

   2. *-commutative 75.1%

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

3. Simplified 75.1%

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

5. Taylor expanded in t1 around inf 52.2%

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

   1. associate-*r/ 52.2%

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

   2. neg-mul-1 52.2%

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

7. Simplified 52.2%

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

8. Final simplification 52.2%

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

Alternative 13: 14.1% 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 74.0%

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

   1. times-frac 97.9%

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

   2. distribute-frac-neg 97.9%

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

   3. distribute-neg-frac2 97.9%

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

   4. +-commutative 97.9%

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

   5. distribute-neg-in 97.9%

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

   6. unsub-neg 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in t1 around inf 52.6%

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

6. Taylor expanded in u around inf 15.3%

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

7. Final simplification 15.3%

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

Reproduce

herbie shell --seed 2024130 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))