Rosa's DopplerBench

Percentage Accurate: 72.3% → 98.2%

Time: 7.1s

Alternatives: 11

Speedup: 0.8×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-frac-neg2 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 99.1

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{v} \cdot \left(t1 + u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.6e+179)
   (/ (* (/ (- v) u) t1) u)
   (if (<= u 5e+147)
     (/ (- v) (fma (+ 2.0 (/ u t1)) u t1))
     (/ (- t1) (* (/ (+ t1 u) v) (+ t1 u))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.6e+179) {
		tmp = ((-v / u) * t1) / u;
	} else if (u <= 5e+147) {
		tmp = -v / fma((2.0 + (u / t1)), u, t1);
	} else {
		tmp = -t1 / (((t1 + u) / v) * (t1 + u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.6e+179)
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	elseif (u <= 5e+147)
		tmp = Float64(Float64(-v) / fma(Float64(2.0 + Float64(u / t1)), u, t1));
	else
		tmp = Float64(Float64(-t1) / Float64(Float64(Float64(t1 + u) / v) * Float64(t1 + u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.6e+179], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision], If[LessEqual[u, 5e+147], N[((-v) / N[(N[(2.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] * u + t1), $MachinePrecision]), $MachinePrecision], N[((-t1) / N[(N[(N[(t1 + u), $MachinePrecision] / v), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -1.6000000000000001e179

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 78.0

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

   7. Applied rewrites 78.0%

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

   9. Applied rewrites 98.2%

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

   if -1.6000000000000001e179 < u < 5.0000000000000002e147

   1. Initial program 71.4%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 35.5

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

   5. Applied rewrites 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 35.8

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

   7. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 35.6

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

   9. Applied rewrites 35.6%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 98.4

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

   12. Applied rewrites 98.4%

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

   if 5.0000000000000002e147 < u

   1. Initial program 61.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. frac-2neg N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-neg.f64 91.9

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 91.9

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

   4. Applied rewrites 91.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{t1 + u}{v} \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{if}\;u \leq -1.6 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (/ (- v) u) t1) u)))
   (if (<= u -1.6e+179)
     t_1
     (if (<= u 5e+147) (/ (- v) (fma (+ 2.0 (/ u t1)) u t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = ((-v / u) * t1) / u;
	double tmp;
	if (u <= -1.6e+179) {
		tmp = t_1;
	} else if (u <= 5e+147) {
		tmp = -v / fma((2.0 + (u / t1)), u, t1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(Float64(-v) / u) * t1) / u)
	tmp = 0.0
	if (u <= -1.6e+179)
		tmp = t_1;
	elseif (u <= 5e+147)
		tmp = Float64(Float64(-v) / fma(Float64(2.0 + Float64(u / t1)), u, t1));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision]}, If[LessEqual[u, -1.6e+179], t$95$1, If[LessEqual[u, 5e+147], N[((-v) / N[(N[(2.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] * u + t1), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -1.6000000000000001e179 or 5.0000000000000002e147 < u

   1. Initial program 61.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 99.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 81.5

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 92.7%

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

   if -1.6000000000000001e179 < u < 5.0000000000000002e147

   1. Initial program 71.4%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 35.5

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

   5. Applied rewrites 35.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 35.8

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

   7. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 35.6

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

   9. Applied rewrites 35.6%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 98.4

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

   12. Applied rewrites 98.4%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.6 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq 1.55 \cdot 10^{+50}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -7.2e+70)
   (/ (- v) (fma 2.0 u t1))
   (if (<= t1 1.55e+50) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) (/ (- v) t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -7.2e+70) {
		tmp = -v / fma(2.0, u, t1);
	} else if (t1 <= 1.55e+50) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -7.2e+70)
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	elseif (t1 <= 1.55e+50)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -7.2e+70], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 1.55e+50], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -7.1999999999999999e70

   1. Initial program 46.7%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 14.4

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

   5. Applied rewrites 14.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 14.9

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

   7. Applied rewrites 14.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 14.9

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

   9. Applied rewrites 14.9%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 94.3

          \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

   12. Applied rewrites 94.3%

       \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

   if -7.1999999999999999e70 < t1 < 1.55000000000000001e50

   1. Initial program 82.2%

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

   if 1.55000000000000001e50 < t1

   1. Initial program 49.0%

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

   3. Taylor expanded in u around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

Alternative 5: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{if}\;t1 \leq -2.65 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{-v}{u} \cdot t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma 2.0 u t1))))
   (if (<= t1 -2.65e+46)
     t_1
     (if (<= t1 1.26e-114) (/ (* (/ (- v) u) t1) u) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(2.0, u, t1);
	double tmp;
	if (t1 <= -2.65e+46) {
		tmp = t_1;
	} else if (t1 <= 1.26e-114) {
		tmp = ((-v / u) * t1) / u;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(2.0, u, t1))
	tmp = 0.0
	if (t1 <= -2.65e+46)
		tmp = t_1;
	elseif (t1 <= 1.26e-114)
		tmp = Float64(Float64(Float64(Float64(-v) / u) * t1) / u);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.65e+46], t$95$1, If[LessEqual[t1, 1.26e-114], N[(N[(N[((-v) / u), $MachinePrecision] * t1), $MachinePrecision] / u), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.64999999999999989e46 or 1.25999999999999992e-114 < t1

   1. Initial program 59.9%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 21.4

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

   7. Applied rewrites 21.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 21.4

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

   9. Applied rewrites 21.4%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 82.4

          \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

   12. Applied rewrites 82.4%

       \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

   if -2.64999999999999989e46 < t1 < 1.25999999999999992e-114

   1. Initial program 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-frac-neg2 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 98.1

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in u around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 69.5

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

   7. Applied rewrites 69.5%

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

   9. Applied rewrites 76.9%

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

Alternative 6: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{if}\;t1 \leq -2.65 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-114}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma 2.0 u t1))))
   (if (<= t1 -2.65e+46)
     t_1
     (if (<= t1 1.26e-114) (* (/ t1 u) (/ (- v) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(2.0, u, t1);
	double tmp;
	if (t1 <= -2.65e+46) {
		tmp = t_1;
	} else if (t1 <= 1.26e-114) {
		tmp = (t1 / u) * (-v / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(2.0, u, t1))
	tmp = 0.0
	if (t1 <= -2.65e+46)
		tmp = t_1;
	elseif (t1 <= 1.26e-114)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -2.65e+46], t$95$1, If[LessEqual[t1, 1.26e-114], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.64999999999999989e46 or 1.25999999999999992e-114 < t1

   1. Initial program 59.9%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 21.0

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

   5. Applied rewrites 21.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 21.4

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

   7. Applied rewrites 21.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 21.4

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

   9. Applied rewrites 21.4%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 82.4

          \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

   12. Applied rewrites 82.4%

       \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

   if -2.64999999999999989e46 < t1 < 1.25999999999999992e-114

   1. Initial program 80.5%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.65 \cdot 10^{+46}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq 1.26 \cdot 10^{-114}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.22 \cdot 10^{-114}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma 2.0 u t1))))
   (if (<= t1 -5.2e-94)
     t_1
     (if (<= t1 1.22e-114) (* (/ (- t1) (* u u)) v) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(2.0, u, t1);
	double tmp;
	if (t1 <= -5.2e-94) {
		tmp = t_1;
	} else if (t1 <= 1.22e-114) {
		tmp = (-t1 / (u * u)) * v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(2.0, u, t1))
	tmp = 0.0
	if (t1 <= -5.2e-94)
		tmp = t_1;
	elseif (t1 <= 1.22e-114)
		tmp = Float64(Float64(Float64(-t1) / Float64(u * u)) * v);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -5.2e-94], t$95$1, If[LessEqual[t1, 1.22e-114], N[(N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.19999999999999988e-94 or 1.22e-114 < t1

   1. Initial program 65.4%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 26.7

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

   5. Applied rewrites 26.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 26.6

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

   7. Applied rewrites 26.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 26.6

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

   9. Applied rewrites 26.6%

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

   10. Taylor expanded in u around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 77.8

          \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

   12. Applied rewrites 77.8%

       \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

   if -5.19999999999999988e-94 < t1 < 1.22e-114

   1. Initial program 77.5%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 73.9

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

   7. Applied rewrites 73.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq 1.22 \cdot 10^{-114}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 0.8× speedup

Math

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

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)) * 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[(N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision] / (-N[(t1 + u), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 69.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-neg-in N/A

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 9: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{-v}{\mathsf{fma}\left(2, u, t1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

3. Taylor expanded in u around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 41.3

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

5. Applied rewrites 41.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 41.4

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

7. Applied rewrites 41.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-frac-neg2 N/A

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

   9. remove-double-neg N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 41.2

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

9. Applied rewrites 41.2%

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

10. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 61.9

       \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]

12. Applied rewrites 61.9%

    \[\leadsto \frac{-v}{\color{blue}{\mathsf{fma}\left(2, u, t1\right)}} \]
13. Add Preprocessing

Alternative 10: 61.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-neg-in N/A

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 98.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u around 0

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

7. Applied rewrites 61.7%

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

8. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 61.7

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

10. Applied rewrites 61.7%

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

11. Final simplification 61.7%

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

Alternative 11: 54.5% accurate, 2.1× 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(Float64(-v) / t1)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{v}{t1}} \]
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 56.1

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

5. Applied rewrites 56.1%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))