Rosa's DopplerBench

Percentage Accurate: 72.3% → 98.2%

Time: 9.5s

Alternatives: 13

Speedup: 0.9×

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: 72.3% 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: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 98.1

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{+158}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -1.22e+154)
   (/ v (- (* u -2.0) t1))
   (if (<= t1 2.4e+158) (* v (/ t1 (* (+ t1 u) (- (- t1) u)))) (/ v (- t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.22e+154) {
		tmp = v / ((u * -2.0) - t1);
	} else if (t1 <= 2.4e+158) {
		tmp = v * (t1 / ((t1 + u) * (-t1 - u)));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-1.22d+154)) then
        tmp = v / ((u * (-2.0d0)) - t1)
    else if (t1 <= 2.4d+158) then
        tmp = v * (t1 / ((t1 + u) * (-t1 - u)))
    else
        tmp = v / -t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -1.22e+154) {
		tmp = v / ((u * -2.0) - t1);
	} else if (t1 <= 2.4e+158) {
		tmp = v * (t1 / ((t1 + u) * (-t1 - u)));
	} else {
		tmp = v / -t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -1.22e+154:
		tmp = v / ((u * -2.0) - t1)
	elif t1 <= 2.4e+158:
		tmp = v * (t1 / ((t1 + u) * (-t1 - u)))
	else:
		tmp = v / -t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -1.22e+154)
		tmp = Float64(v / Float64(Float64(u * -2.0) - t1));
	elseif (t1 <= 2.4e+158)
		tmp = Float64(v * Float64(t1 / Float64(Float64(t1 + u) * Float64(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 (t1 <= -1.22e+154)
		tmp = v / ((u * -2.0) - t1);
	elseif (t1 <= 2.4e+158)
		tmp = v * (t1 / ((t1 + u) * (-t1 - u)));
	else
		tmp = v / -t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -1.22e+154], N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.4e+158], N[(v * N[(t1 / N[(N[(t1 + u), $MachinePrecision] * N[((-t1) - u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / (-t1)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.22e154

   1. Initial program 46.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-neg.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied egg-rr 93.7%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}} \]

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.1

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

   9. Simplified 87.1%

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

   if -1.22e154 < t1 < 2.40000000000000008e158

   1. Initial program 85.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 91.5

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

   4. Applied egg-rr 91.5%

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

   if 2.40000000000000008e158 < t1

   1. Initial program 51.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

      4. lower-neg.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.22 \cdot 10^{+154}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 2.4 \cdot 10^{+158}:\\ \;\;\;\;v \cdot \frac{t1}{\left(t1 + u\right) \cdot \left(\left(-t1\right) - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{if}\;u \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- v) u) (/ t1 u))))
   (if (<= u -2.9e-6) t_1 (if (<= u 4.7e+45) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-v / u) * (t1 / u);
	double tmp;
	if (u <= -2.9e-6) {
		tmp = t_1;
	} else if (u <= 4.7e+45) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (-v / u) * (t1 / u)
    if (u <= (-2.9d-6)) then
        tmp = t_1
    else if (u <= 4.7d+45) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-v / u) * (t1 / u);
	double tmp;
	if (u <= -2.9e-6) {
		tmp = t_1;
	} else if (u <= 4.7e+45) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-v / u) * (t1 / u)
	tmp = 0
	if u <= -2.9e-6:
		tmp = t_1
	elif u <= 4.7e+45:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-v) / u) * Float64(t1 / u))
	tmp = 0.0
	if (u <= -2.9e-6)
		tmp = t_1;
	elseif (u <= 4.7e+45)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-v / u) * (t1 / u);
	tmp = 0.0;
	if (u <= -2.9e-6)
		tmp = t_1;
	elseif (u <= 4.7e+45)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.9e-6], t$95$1, If[LessEqual[u, 4.7e+45], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.9000000000000002e-6 or 4.70000000000000002e45 < u

   1. Initial program 86.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{t1 \cdot v}{{u}^{2}}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{t1 \cdot \frac{v}{{u}^{2}}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t1 \cdot \left(\mathsf{neg}\left(\frac{v}{{u}^{2}}\right)\right)} \]

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto t1 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{v}{{u}^{2}}\right)\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto t1 \cdot \color{blue}{\frac{v}{\mathsf{neg}\left({u}^{2}\right)}} \]

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto t1 \cdot \frac{v}{\color{blue}{\mathsf{neg}\left({u}^{2}\right)}} \]

      11. unpow2 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 84.4

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

   5. Simplified 84.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 87.4

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

   7. Applied egg-rr 87.4%

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

   if -2.9000000000000002e-6 < u < 4.70000000000000002e45

   1. Initial program 68.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

      4. lower-neg.f64 80.5

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

   5. Simplified 80.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.25 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-111}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (- (* u -2.0) t1))))
   (if (<= t1 -1.25e-57)
     t_1
     (if (<= t1 5.8e-111) (* v (/ t1 (* u (- u)))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.25e-57) {
		tmp = t_1;
	} else if (t1 <= 5.8e-111) {
		tmp = v * (t1 / (u * -u));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = v / ((u * (-2.0d0)) - t1)
    if (t1 <= (-1.25d-57)) then
        tmp = t_1
    else if (t1 <= 5.8d-111) then
        tmp = v * (t1 / (u * -u))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = v / ((u * -2.0) - t1);
	double tmp;
	if (t1 <= -1.25e-57) {
		tmp = t_1;
	} else if (t1 <= 5.8e-111) {
		tmp = v * (t1 / (u * -u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = v / ((u * -2.0) - t1)
	tmp = 0
	if t1 <= -1.25e-57:
		tmp = t_1
	elif t1 <= 5.8e-111:
		tmp = v * (t1 / (u * -u))
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(v / Float64(Float64(u * -2.0) - t1))
	tmp = 0.0
	if (t1 <= -1.25e-57)
		tmp = t_1;
	elseif (t1 <= 5.8e-111)
		tmp = Float64(v * Float64(t1 / Float64(u * Float64(-u))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = v / ((u * -2.0) - t1);
	tmp = 0.0;
	if (t1 <= -1.25e-57)
		tmp = t_1;
	elseif (t1 <= 5.8e-111)
		tmp = v * (t1 / (u * -u));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.25e-57], t$95$1, If[LessEqual[t1, 5.8e-111], N[(v * N[(t1 / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.25e-57 or 5.80000000000000003e-111 < t1

   1. Initial program 72.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. times-frac N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-frac-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-neg.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-neg.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}} \]

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.2

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

   9. Simplified 83.2%

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

   if -1.25e-57 < t1 < 5.80000000000000003e-111

   1. Initial program 82.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.1

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in t1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.2

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

   7. Simplified 82.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{-111}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot \left(-u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{if}\;u \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ v (* u (- u))))))
   (if (<= u -2.9e-6) t_1 (if (<= u 4.7e+45) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (v / (u * -u));
	double tmp;
	if (u <= -2.9e-6) {
		tmp = t_1;
	} else if (u <= 4.7e+45) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t1 * (v / (u * -u))
    if (u <= (-2.9d-6)) then
        tmp = t_1
    else if (u <= 4.7d+45) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * (v / (u * -u));
	double tmp;
	if (u <= -2.9e-6) {
		tmp = t_1;
	} else if (u <= 4.7e+45) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * (v / (u * -u))
	tmp = 0
	if u <= -2.9e-6:
		tmp = t_1
	elif u <= 4.7e+45:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(v / Float64(u * Float64(-u))))
	tmp = 0.0
	if (u <= -2.9e-6)
		tmp = t_1;
	elseif (u <= 4.7e+45)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * (v / (u * -u));
	tmp = 0.0;
	if (u <= -2.9e-6)
		tmp = t_1;
	elseif (u <= 4.7e+45)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(v / N[(u * (-u)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -2.9e-6], t$95$1, If[LessEqual[u, 4.7e+45], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -2.9000000000000002e-6 or 4.70000000000000002e45 < u

   1. Initial program 86.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{t1 \cdot v}{{u}^{2}}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{t1 \cdot \frac{v}{{u}^{2}}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t1 \cdot \left(\mathsf{neg}\left(\frac{v}{{u}^{2}}\right)\right)} \]

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto t1 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{v}{{u}^{2}}\right)\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto t1 \cdot \color{blue}{\frac{v}{\mathsf{neg}\left({u}^{2}\right)}} \]

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto t1 \cdot \frac{v}{\color{blue}{\mathsf{neg}\left({u}^{2}\right)}} \]

      11. unpow2 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 84.4

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

   5. Simplified 84.4%

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

   if -2.9000000000000002e-6 < u < 4.70000000000000002e45

   1. Initial program 68.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

      4. lower-neg.f64 80.5

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

   5. Simplified 80.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot \left(-u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 97.3

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

4. Applied egg-rr 97.3%

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

5. Final simplification 97.3%

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

Alternative 7: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -4 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* t1 (/ v (* u u)))))
   (if (<= u -4e+75) t_1 (if (<= u 2.2e+56) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = t1 * (v / (u * u));
	double tmp;
	if (u <= -4e+75) {
		tmp = t_1;
	} else if (u <= 2.2e+56) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t1 * (v / (u * u))
    if (u <= (-4d+75)) then
        tmp = t_1
    else if (u <= 2.2d+56) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = t1 * (v / (u * u));
	double tmp;
	if (u <= -4e+75) {
		tmp = t_1;
	} else if (u <= 2.2e+56) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = t1 * (v / (u * u))
	tmp = 0
	if u <= -4e+75:
		tmp = t_1
	elif u <= 2.2e+56:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(t1 * Float64(v / Float64(u * u)))
	tmp = 0.0
	if (u <= -4e+75)
		tmp = t_1;
	elseif (u <= 2.2e+56)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = t1 * (v / (u * u));
	tmp = 0.0;
	if (u <= -4e+75)
		tmp = t_1;
	elseif (u <= 2.2e+56)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(t1 * N[(v / N[(u * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4e+75], t$95$1, If[LessEqual[u, 2.2e+56], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.99999999999999971e75 or 2.20000000000000016e56 < u

   1. Initial program 85.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{t1 \cdot v}{{u}^{2}}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{t1 \cdot \frac{v}{{u}^{2}}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{t1 \cdot \left(\mathsf{neg}\left(\frac{v}{{u}^{2}}\right)\right)} \]

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

         \[\leadsto t1 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{v}{{u}^{2}}\right)\right)} \]

      7. distribute-neg-frac2 N/A

         \[\leadsto t1 \cdot \color{blue}{\frac{v}{\mathsf{neg}\left({u}^{2}\right)}} \]

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto t1 \cdot \frac{v}{\color{blue}{\mathsf{neg}\left({u}^{2}\right)}} \]

      11. unpow2 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 85.3

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

   5. Simplified 85.3%

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

   7. Applied egg-rr 75.2%

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

   if -3.99999999999999971e75 < u < 2.20000000000000016e56

   1. Initial program 70.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

      4. lower-neg.f64 74.2

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

   5. Simplified 74.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+75}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 2.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 98.1

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

4. Applied egg-rr 98.1%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-neg.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. distribute-lft-neg-out N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. lift-neg.f64 N/A

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

   15. associate-/r* N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

6. Applied egg-rr 95.9%

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

Alternative 9: 56.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u}\\ \mathbf{if}\;u \leq -3.8 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+105}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) u)))
   (if (<= u -3.8e+79) t_1 (if (<= u 5e+105) (/ v (- t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / u;
	double tmp;
	if (u <= -3.8e+79) {
		tmp = t_1;
	} else if (u <= 5e+105) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = -v / u
    if (u <= (-3.8d+79)) then
        tmp = t_1
    else if (u <= 5d+105) then
        tmp = v / -t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = -v / u;
	double tmp;
	if (u <= -3.8e+79) {
		tmp = t_1;
	} else if (u <= 5e+105) {
		tmp = v / -t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = -v / u
	tmp = 0
	if u <= -3.8e+79:
		tmp = t_1
	elif u <= 5e+105:
		tmp = v / -t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / u)
	tmp = 0.0
	if (u <= -3.8e+79)
		tmp = t_1;
	elseif (u <= 5e+105)
		tmp = Float64(v / Float64(-t1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = -v / u;
	tmp = 0.0;
	if (u <= -3.8e+79)
		tmp = t_1;
	elseif (u <= 5e+105)
		tmp = v / -t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / u), $MachinePrecision]}, If[LessEqual[u, -3.8e+79], t$95$1, If[LessEqual[u, 5e+105], N[(v / (-t1)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.8000000000000002e79 or 5.00000000000000046e105 < u

   1. Initial program 84.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 97.8

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 61.1

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

   7. Simplified 61.1%

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

   8. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{v}{u}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(u\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{v}{\color{blue}{\mathsf{neg}\left(u\right)}} \]

      6. lower-neg.f64 45.5

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

   10. Simplified 45.5%

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

   if -3.8000000000000002e79 < u < 5.00000000000000046e105

   1. Initial program 72.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

      4. lower-neg.f64 70.5

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

   5. Simplified 70.5%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+105}:\\ \;\;\;\;\frac{v}{-t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 22.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -9.2e+120) (/ v t1) (if (<= t1 3e+106) (/ (- v) u) (/ v t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.2e+120) {
		tmp = v / t1;
	} else if (t1 <= 3e+106) {
		tmp = -v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (t1 <= (-9.2d+120)) then
        tmp = v / t1
    else if (t1 <= 3d+106) then
        tmp = -v / u
    else
        tmp = v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -9.2e+120) {
		tmp = v / t1;
	} else if (t1 <= 3e+106) {
		tmp = -v / u;
	} else {
		tmp = v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if t1 <= -9.2e+120:
		tmp = v / t1
	elif t1 <= 3e+106:
		tmp = -v / u
	else:
		tmp = v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -9.2e+120)
		tmp = Float64(v / t1);
	elseif (t1 <= 3e+106)
		tmp = Float64(Float64(-v) / u);
	else
		tmp = Float64(v / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (t1 <= -9.2e+120)
		tmp = v / t1;
	elseif (t1 <= 3e+106)
		tmp = -v / u;
	else
		tmp = v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -9.2e+120], N[(v / t1), $MachinePrecision], If[LessEqual[t1, 3e+106], N[((-v) / u), $MachinePrecision], N[(v / t1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.1999999999999997e120 or 3.0000000000000001e106 < t1

   1. Initial program 56.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t1 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

      4. lower-neg.f64 89.5

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

   5. Simplified 89.5%

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

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{v}{t1}\right)} \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(t1\right)}} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{v}{\color{blue}{\mathsf{neg}\left(t1\right)}} \]

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 89.4

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

   7. Applied egg-rr 89.4%

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

      1. associate-*l/ N/A

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

      2. neg-mul-1 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

      3. +-lft-identity N/A

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

      4. flip3-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\color{blue}{\frac{{0}^{3} + {t1}^{3}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}}} \]

      5. sqr-pow N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + \color{blue}{{t1}^{\left(\frac{3}{2}\right)} \cdot {t1}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

      6. pow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + \color{blue}{{\left(t1 \cdot t1\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

      7. sqr-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\color{blue}{\left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + \color{blue}{{\left(\mathsf{neg}\left(t1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(t1\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

      11. sqr-pow N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + \color{blue}{{\left(\mathsf{neg}\left(t1\right)\right)}^{3}}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

      12. sqr-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right)} - 0 \cdot t1\right)}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot t1\right)}} \]

      14. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} - 0 \cdot t1\right)}} \]

      15. remove-double-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t1\right)\right)\right)\right)}\right)}} \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}\right)\right)\right)}} \]

      17. neg-mul-1 N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \color{blue}{\left(-1 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}\right)}} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - \color{blue}{\left(0 \cdot -1\right) \cdot \left(\mathsf{neg}\left(t1\right)\right)}\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - \color{blue}{0} \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}} \]

      20. flip3-+ N/A

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

   9. Applied egg-rr 45.4%

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

   if -9.1999999999999997e120 < t1 < 3.0000000000000001e106

   1. Initial program 85.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 96.3

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in t1 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 57.3

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

   7. Simplified 57.3%

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

   8. Taylor expanded in t1 around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{v}{u}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(u\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{v}{\color{blue}{\mathsf{neg}\left(u\right)}} \]

      6. lower-neg.f64 21.4

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

   10. Simplified 21.4%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{v}{u \cdot -2 - t1} \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ v (- (* u -2.0) t1)))

C

double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = v / ((u * (-2.0d0)) - t1)
end function

Java

public static double code(double u, double v, double t1) {
	return v / ((u * -2.0) - t1);
}

Python

def code(u, v, t1):
	return v / ((u * -2.0) - t1)

Julia

function code(u, v, t1)
	return Float64(v / Float64(Float64(u * -2.0) - t1))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = v / ((u * -2.0) - t1);
end

Wolfram

code[u_, v_, t1_] := N[(v / N[(N[(u * -2.0), $MachinePrecision] - t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. times-frac N/A

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

   5. frac-2neg N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-frac-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-neg.f64 98.1

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

4. Applied egg-rr 98.1%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-neg.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. distribute-lft-neg-out N/A

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

   11. lift-neg.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. lift-neg.f64 N/A

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

   15. associate-/r* N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

6. Applied egg-rr 95.9%

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

7. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

      \[\leadsto \frac{v}{-2 \cdot u + \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}} \]

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 63.1

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

9. Simplified 63.1%

   \[\leadsto \frac{v}{\color{blue}{u \cdot -2 - t1}} \]
10. Add Preprocessing

Alternative 12: 60.7% accurate, 1.8× speedup

Math

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

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(v / Float64(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 76.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. neg-mul-1 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 97.3

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

4. Applied egg-rr 97.3%

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

5. Taylor expanded in t1 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 66.9

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

7. Simplified 66.9%

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

8. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-+.f64 62.8

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

10. Simplified 62.8%

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

11. Final simplification 62.8%

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

Alternative 13: 13.2% accurate, 2.5× 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 76.7%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
2. Add Preprocessing

3. Taylor expanded in t1 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

   4. lower-neg.f64 53.9

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

5. Simplified 53.9%

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

   1. distribute-frac-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{v}{t1}\right)} \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \color{blue}{\frac{v}{\mathsf{neg}\left(t1\right)}} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{v}{\color{blue}{\mathsf{neg}\left(t1\right)}} \]

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lift-neg.f64 N/A

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

   10. remove-double-neg N/A

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

   11. lower-/.f64 53.8

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

7. Applied egg-rr 53.8%

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

   1. associate-*l/ N/A

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

   2. neg-mul-1 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(v\right)}}{t1} \]

   3. +-lft-identity N/A

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

   4. flip3-+ N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\color{blue}{\frac{{0}^{3} + {t1}^{3}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}}} \]

   5. sqr-pow N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + \color{blue}{{t1}^{\left(\frac{3}{2}\right)} \cdot {t1}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

   6. pow-prod-down N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + \color{blue}{{\left(t1 \cdot t1\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

   7. sqr-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\color{blue}{\left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

   9. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

   10. pow-prod-down N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + \color{blue}{{\left(\mathsf{neg}\left(t1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(t1\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

   11. sqr-pow N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + \color{blue}{{\left(\mathsf{neg}\left(t1\right)\right)}^{3}}}{0 \cdot 0 + \left(t1 \cdot t1 - 0 \cdot t1\right)}} \]

   12. sqr-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right)} - 0 \cdot t1\right)}} \]

   13. lift-neg.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot t1\right)}} \]

   14. lift-neg.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(t1\right)\right)} - 0 \cdot t1\right)}} \]

   15. remove-double-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t1\right)\right)\right)\right)}\right)}} \]

   16. lift-neg.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(t1\right)\right)}\right)\right)\right)}} \]

   17. neg-mul-1 N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - 0 \cdot \color{blue}{\left(-1 \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}\right)}} \]

   18. associate-*r* N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - \color{blue}{\left(0 \cdot -1\right) \cdot \left(\mathsf{neg}\left(t1\right)\right)}\right)}} \]

   19. metadata-eval N/A

       \[\leadsto \frac{\mathsf{neg}\left(v\right)}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(t1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(t1\right)\right) \cdot \left(\mathsf{neg}\left(t1\right)\right) - \color{blue}{0} \cdot \left(\mathsf{neg}\left(t1\right)\right)\right)}} \]

   20. flip3-+ N/A

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

9. Applied egg-rr 15.7%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))