Rosa's DopplerBench

Percentage Accurate: 73.2% → 98.0%

Time: 7.7s

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: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 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 72.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 99.0

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

   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 99.0

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{v}{t1 + u} \cdot t1}{-u}\\ \mathbf{if}\;u \leq -1.02 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+199}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (/ v (+ t1 u)) t1) (- u))))
   (if (<= u -1.02e+141)
     t_1
     (if (<= u 1.05e+199) (/ (- v) (fma (+ 2.0 (/ u t1)) u t1)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = ((v / (t1 + u)) * t1) / -u;
	double tmp;
	if (u <= -1.02e+141) {
		tmp = t_1;
	} else if (u <= 1.05e+199) {
		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(v / Float64(t1 + u)) * t1) / Float64(-u))
	tmp = 0.0
	if (u <= -1.02e+141)
		tmp = t_1;
	elseif (u <= 1.05e+199)
		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 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision] / (-u)), $MachinePrecision]}, If[LessEqual[u, -1.02e+141], t$95$1, If[LessEqual[u, 1.05e+199], 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 < -1.02e141 or 1.05e199 < u

   1. Initial program 77.7%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.0

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

   7. Applied rewrites 97.0%

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

   if -1.02e141 < u < 1.05e199

   1. Initial program 70.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 98.7

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

      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 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. lift-neg.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

      14. *-lft-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 96.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.5

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 96.5%

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

   7. Taylor expanded in u around 0

      \[\leadsto \frac{-v}{\color{blue}{t1 + u \cdot \left(2 + \frac{u}{t1}\right)}} \]
   8. 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 96.5

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

   9. Applied rewrites 96.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{v}{t1 + u} \cdot t1}{-u}\\ \mathbf{elif}\;u \leq 1.05 \cdot 10^{+199}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{t1 + u} \cdot t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -4.8 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 4.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -4.8e+140)
     t_1
     (if (<= t1 4.4e+68) (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -4.8e+140) {
		tmp = t_1;
	} else if (t1 <= 4.4e+68) {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -4.8e+140)
		tmp = t_1;
	elseif (t1 <= 4.4e+68)
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -4.8e+140], t$95$1, If[LessEqual[t1, 4.4e+68], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -4.7999999999999999e140 or 4.39999999999999974e68 < t1

   1. Initial program 52.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. lift-neg.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

      14. *-lft-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 97.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 95.5

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

   9. Applied rewrites 95.5%

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

   if -4.7999999999999999e140 < t1 < 4.39999999999999974e68

   1. Initial program 81.9%

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

Alternative 4: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-t1}{\frac{u}{v} \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -3.4e-85)
     t_1
     (if (<= t1 1.6e+68) (/ (- t1) (* (/ u v) u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -3.4e-85) {
		tmp = t_1;
	} else if (t1 <= 1.6e+68) {
		tmp = -t1 / ((u / v) * u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -3.4e-85)
		tmp = t_1;
	elseif (t1 <= 1.6e+68)
		tmp = Float64(Float64(-t1) / Float64(Float64(u / v) * u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e-85], t$95$1, If[LessEqual[t1, 1.6e+68], N[((-t1) / N[(N[(u / v), $MachinePrecision] * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.4e-85 or 1.59999999999999997e68 < t1

   1. Initial program 63.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. lift-neg.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

      14. *-lft-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 96.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.3

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 87.2

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

   9. Applied rewrites 87.2%

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

   if -3.4e-85 < t1 < 1.59999999999999997e68

   1. Initial program 80.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 81.4%

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

Alternative 5: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -3.4e-85)
     t_1
     (if (<= t1 1.6e+68) (* (/ (- v) u) (/ t1 u)) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -3.4e-85) {
		tmp = t_1;
	} else if (t1 <= 1.6e+68) {
		tmp = (-v / u) * (t1 / u);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -3.4e-85)
		tmp = t_1;
	elseif (t1 <= 1.6e+68)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e-85], t$95$1, If[LessEqual[t1, 1.6e+68], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.4e-85 or 1.59999999999999997e68 < t1

   1. Initial program 63.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. lift-neg.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

      14. *-lft-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 96.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.3

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 87.2

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

   9. Applied rewrites 87.2%

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

   if -3.4e-85 < t1 < 1.59999999999999997e68

   1. Initial program 80.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -6.5 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -6.5e-86)
     t_1
     (if (<= t1 1.6e+68) (* (/ (- v) (* u u)) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -6.5e-86) {
		tmp = t_1;
	} else if (t1 <= 1.6e+68) {
		tmp = (-v / (u * u)) * t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -6.5e-86)
		tmp = t_1;
	elseif (t1 <= 1.6e+68)
		tmp = Float64(Float64(Float64(-v) / Float64(u * u)) * t1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[((-v) / N[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -6.5e-86], t$95$1, If[LessEqual[t1, 1.6e+68], N[(N[((-v) / N[(u * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.50000000000000028e-86 or 1.59999999999999997e68 < t1

   1. Initial program 63.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. lift-neg.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

      14. *-lft-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 96.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.3

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 87.2

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

   9. Applied rewrites 87.2%

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

   if -6.50000000000000028e-86 < t1 < 1.59999999999999997e68

   1. Initial program 80.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 71.7

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

   5. Applied rewrites 71.7%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.6

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

   7. Applied rewrites 75.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-v}{u \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{if}\;t1 \leq -3.4 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma u 2.0 t1))))
   (if (<= t1 -3.4e-85)
     t_1
     (if (<= t1 1.6e+68) (* (/ (- t1) (* u u)) v) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(u, 2.0, t1);
	double tmp;
	if (t1 <= -3.4e-85) {
		tmp = t_1;
	} else if (t1 <= 1.6e+68) {
		tmp = (-t1 / (u * u)) * v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(-v) / fma(u, 2.0, t1))
	tmp = 0.0
	if (t1 <= -3.4e-85)
		tmp = t_1;
	elseif (t1 <= 1.6e+68)
		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[(u * 2.0 + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -3.4e-85], t$95$1, If[LessEqual[t1, 1.6e+68], N[(N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.4e-85 or 1.59999999999999997e68 < t1

   1. Initial program 63.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. lift-neg.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

      14. *-lft-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 96.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.3

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 87.2

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

   9. Applied rewrites 87.2%

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

   if -3.4e-85 < t1 < 1.59999999999999997e68

   1. Initial program 80.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 71.7

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

   5. Applied rewrites 71.7%

      \[\leadsto \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]
   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 71.8

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

   7. Applied rewrites 71.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \mathbf{elif}\;t1 \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(u, 2, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. times-frac N/A

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

   9. lift-neg.f64 N/A

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

   10. frac-2neg N/A

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

   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

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-/.f64 98.1

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 9: 56.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+195}:\\ \;\;\;\;\frac{-v}{2 \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -1.4e+195) (/ (- v) (* 2.0 u)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+195) {
		tmp = -v / (2.0 * u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-1.4d+195)) then
        tmp = -v / (2.0d0 * u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -1.4e+195) {
		tmp = -v / (2.0 * u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -1.4e+195:
		tmp = -v / (2.0 * u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -1.4e+195)
		tmp = Float64(Float64(-v) / Float64(2.0 * u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -1.4e+195)
		tmp = -v / (2.0 * u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -1.4e+195], N[((-v) / N[(2.0 * u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -1.3999999999999999e195

   1. Initial program 72.6%

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

      1. lift-/.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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-times N/A

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

      10. clear-num N/A

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

      11. lift-neg.f64 N/A

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

      12. frac-2neg N/A

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

      13. frac-times N/A

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

      14. *-lft-identity N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 81.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 81.7

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 81.7%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 47.8

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

   9. Applied rewrites 47.8%

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

   10. Taylor expanded in u around inf

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

   12. Applied rewrites 43.7%

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

   if -1.3999999999999999e195 < u

   1. Initial program 72.0%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 57.7

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

   5. Applied rewrites 57.7%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -1.4 \cdot 10^{+195}:\\ \;\;\;\;\frac{-v}{2 \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 99.0

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

   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 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. *-commutative N/A

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

   7. remove-double-neg N/A

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

   8. lift-neg.f64 N/A

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

   9. frac-times N/A

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

   10. clear-num N/A

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

   11. lift-neg.f64 N/A

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

   12. frac-2neg N/A

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

   13. frac-times N/A

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

   14. *-lft-identity N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 93.0

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 93.0

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

6. Applied rewrites 93.0%

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

7. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 59.3

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

9. Applied rewrites 59.3%

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

Alternative 11: 53.9% 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 72.0%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 54.4

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

5. Applied rewrites 54.4%

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

Reproduce

herbie shell --seed 2024273 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))