Rosa's DopplerBench

Percentage Accurate: 71.6% → 97.9%

Time: 8.6s

Alternatives: 13

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: 71.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.3%

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

   1. lift-/.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 98.6

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

   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 98.6

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

   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 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ t_2 := \frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{if}\;t1 \leq -1.45 \cdot 10^{+102}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t1 \leq -4.5 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.66 \cdot 10^{-164}:\\ \;\;\;\;\frac{v}{\frac{-u}{t1} \cdot u}\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ u t1) (+ u t1))))
        (t_2 (/ (- v) (fma 2.0 u t1))))
   (if (<= t1 -1.45e+102)
     t_2
     (if (<= t1 -4.5e-140)
       t_1
       (if (<= t1 1.66e-164)
         (/ v (* (/ (- u) t1) u))
         (if (<= t1 3.8e+46) t_1 t_2))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((u + t1) * (u + t1));
	double t_2 = -v / fma(2.0, u, t1);
	double tmp;
	if (t1 <= -1.45e+102) {
		tmp = t_2;
	} else if (t1 <= -4.5e-140) {
		tmp = t_1;
	} else if (t1 <= 1.66e-164) {
		tmp = v / ((-u / t1) * u);
	} else if (t1 <= 3.8e+46) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(u + t1) * Float64(u + t1)))
	t_2 = Float64(Float64(-v) / fma(2.0, u, t1))
	tmp = 0.0
	if (t1 <= -1.45e+102)
		tmp = t_2;
	elseif (t1 <= -4.5e-140)
		tmp = t_1;
	elseif (t1 <= 1.66e-164)
		tmp = Float64(v / Float64(Float64(Float64(-u) / t1) * u));
	elseif (t1 <= 3.8e+46)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-t1) * v), $MachinePrecision] / N[(N[(u + t1), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -1.45e+102], t$95$2, If[LessEqual[t1, -4.5e-140], t$95$1, If[LessEqual[t1, 1.66e-164], N[(v / N[(N[((-u) / t1), $MachinePrecision] * u), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 3.8e+46], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -1.4500000000000001e102 or 3.7999999999999999e46 < t1

   1. Initial program 46.7%

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

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 95.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 95.0

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

   6. Applied rewrites 95.0%

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

   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. lower-fma.f64 88.4

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

   9. Applied rewrites 88.4%

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

   if -1.4500000000000001e102 < t1 < -4.50000000000000004e-140 or 1.6599999999999999e-164 < t1 < 3.7999999999999999e46

   1. Initial program 89.1%

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

   if -4.50000000000000004e-140 < t1 < 1.6599999999999999e-164

   1. Initial program 72.1%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 87.2

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

   5. Applied rewrites 87.2%

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

   7. Applied rewrites 91.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.45 \cdot 10^{+102}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq -4.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{elif}\;t1 \leq 1.66 \cdot 10^{-164}:\\ \;\;\;\;\frac{v}{\frac{-u}{t1} \cdot u}\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9e-8)
   (* (/ (- v) u) (/ t1 u))
   (if (<= u 3.4e+39) (/ (- v) t1) (/ (* (/ v u) t1) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9e-8) {
		tmp = (-v / u) * (t1 / u);
	} else if (u <= 3.4e+39) {
		tmp = -v / t1;
	} else {
		tmp = ((v / u) * t1) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-9d-8)) then
        tmp = (-v / u) * (t1 / u)
    else if (u <= 3.4d+39) then
        tmp = -v / t1
    else
        tmp = ((v / u) * t1) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9e-8) {
		tmp = (-v / u) * (t1 / u);
	} else if (u <= 3.4e+39) {
		tmp = -v / t1;
	} else {
		tmp = ((v / u) * t1) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9e-8:
		tmp = (-v / u) * (t1 / u)
	elif u <= 3.4e+39:
		tmp = -v / t1
	else:
		tmp = ((v / u) * t1) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9e-8)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	elseif (u <= 3.4e+39)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(v / u) * t1) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -9e-8)
		tmp = (-v / u) * (t1 / u);
	elseif (u <= 3.4e+39)
		tmp = -v / t1;
	else
		tmp = ((v / u) * t1) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9e-8], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.4e+39], N[((-v) / t1), $MachinePrecision], N[(N[(N[(v / u), $MachinePrecision] * t1), $MachinePrecision] / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.99999999999999986e-8

   1. Initial program 67.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   if -8.99999999999999986e-8 < u < 3.3999999999999999e39

   1. Initial program 67.4%

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

   3. Taylor expanded in u around 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 81.2

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

   5. Applied rewrites 81.2%

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

   if 3.3999999999999999e39 < u

   1. Initial program 76.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in u around inf

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

      1. lower-/.f64 85.0

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

   10. Applied rewrites 85.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9e-8)
   (* (/ (- v) u) (/ t1 u))
   (if (<= u 3.4e+39) (/ (- v) t1) (/ (/ (* t1 v) u) (- u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9e-8) {
		tmp = (-v / u) * (t1 / u);
	} else if (u <= 3.4e+39) {
		tmp = -v / t1;
	} else {
		tmp = ((t1 * v) / u) / -u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-9d-8)) then
        tmp = (-v / u) * (t1 / u)
    else if (u <= 3.4d+39) then
        tmp = -v / t1
    else
        tmp = ((t1 * v) / u) / -u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9e-8) {
		tmp = (-v / u) * (t1 / u);
	} else if (u <= 3.4e+39) {
		tmp = -v / t1;
	} else {
		tmp = ((t1 * v) / u) / -u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9e-8:
		tmp = (-v / u) * (t1 / u)
	elif u <= 3.4e+39:
		tmp = -v / t1
	else:
		tmp = ((t1 * v) / u) / -u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9e-8)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	elseif (u <= 3.4e+39)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(t1 * v) / u) / Float64(-u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -9e-8)
		tmp = (-v / u) * (t1 / u);
	elseif (u <= 3.4e+39)
		tmp = -v / t1;
	else
		tmp = ((t1 * v) / u) / -u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9e-8], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.4e+39], N[((-v) / t1), $MachinePrecision], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / (-u)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.99999999999999986e-8

   1. Initial program 67.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   if -8.99999999999999986e-8 < u < 3.3999999999999999e39

   1. Initial program 67.4%

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

   3. Taylor expanded in u around 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 81.2

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

   5. Applied rewrites 81.2%

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

   if 3.3999999999999999e39 < u

   1. Initial program 76.4%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in u around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 83.7

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

   10. Applied rewrites 83.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u} \cdot v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -9e-8)
   (* (/ (- v) u) (/ t1 u))
   (if (<= u 3.4e+39) (/ (- v) t1) (/ (* (/ (- t1) u) v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9e-8) {
		tmp = (-v / u) * (t1 / u);
	} else if (u <= 3.4e+39) {
		tmp = -v / t1;
	} else {
		tmp = ((-t1 / u) * v) / u;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (u <= (-9d-8)) then
        tmp = (-v / u) * (t1 / u)
    else if (u <= 3.4d+39) then
        tmp = -v / t1
    else
        tmp = ((-t1 / u) * v) / u
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (u <= -9e-8) {
		tmp = (-v / u) * (t1 / u);
	} else if (u <= 3.4e+39) {
		tmp = -v / t1;
	} else {
		tmp = ((-t1 / u) * v) / u;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if u <= -9e-8:
		tmp = (-v / u) * (t1 / u)
	elif u <= 3.4e+39:
		tmp = -v / t1
	else:
		tmp = ((-t1 / u) * v) / u
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -9e-8)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	elseif (u <= 3.4e+39)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = Float64(Float64(Float64(Float64(-t1) / u) * v) / u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (u <= -9e-8)
		tmp = (-v / u) * (t1 / u);
	elseif (u <= 3.4e+39)
		tmp = -v / t1;
	else
		tmp = ((-t1 / u) * v) / u;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -9e-8], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 3.4e+39], N[((-v) / t1), $MachinePrecision], N[(N[(N[((-t1) / u), $MachinePrecision] * v), $MachinePrecision] / u), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if u < -8.99999999999999986e-8

   1. Initial program 67.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   if -8.99999999999999986e-8 < u < 3.3999999999999999e39

   1. Initial program 67.4%

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

   3. Taylor expanded in u around 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 81.2

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

   5. Applied rewrites 81.2%

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

   if 3.3999999999999999e39 < u

   1. Initial program 76.4%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 83.6%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u} \cdot v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{if}\;u \leq -9 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- v) u) (/ t1 u))))
   (if (<= u -9e-8) t_1 (if (<= u 3.4e+39) (/ (- v) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-v / u) * (t1 / u);
	double tmp;
	if (u <= -9e-8) {
		tmp = t_1;
	} else if (u <= 3.4e+39) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-v / u) * (t1 / u)
    if (u <= (-9d-8)) then
        tmp = t_1
    else if (u <= 3.4d+39) then
        tmp = -v / t1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double t_1 = (-v / u) * (t1 / u);
	double tmp;
	if (u <= -9e-8) {
		tmp = t_1;
	} else if (u <= 3.4e+39) {
		tmp = -v / t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	t_1 = (-v / u) * (t1 / u)
	tmp = 0
	if u <= -9e-8:
		tmp = t_1
	elif u <= 3.4e+39:
		tmp = -v / t1
	else:
		tmp = t_1
	return tmp

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-v) / u) * Float64(t1 / u))
	tmp = 0.0
	if (u <= -9e-8)
		tmp = t_1;
	elseif (u <= 3.4e+39)
		tmp = Float64(Float64(-v) / t1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	t_1 = (-v / u) * (t1 / u);
	tmp = 0.0;
	if (u <= -9e-8)
		tmp = t_1;
	elseif (u <= 3.4e+39)
		tmp = -v / t1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -9e-8], t$95$1, If[LessEqual[u, 3.4e+39], N[((-v) / t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -8.99999999999999986e-8 or 3.3999999999999999e39 < u

   1. Initial program 71.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 82.1

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

   5. Applied rewrites 82.1%

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

   if -8.99999999999999986e-8 < u < 3.3999999999999999e39

   1. Initial program 67.4%

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

   3. Taylor expanded in u around 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 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+191}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -2e+191)
   (* (/ t1 u) (/ (- v) (+ u t1)))
   (/ (- v) (fma (+ 2.0 (/ u t1)) u t1))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -2e+191) {
		tmp = (t1 / u) * (-v / (u + t1));
	} else {
		tmp = -v / fma((2.0 + (u / t1)), u, t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -2e+191)
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / Float64(u + t1)));
	else
		tmp = Float64(Float64(-v) / fma(Float64(2.0 + Float64(u / t1)), u, t1));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -2e+191], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-v) / N[(N[(2.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision] * u + t1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.00000000000000015e191

   1. Initial program 59.9%

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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in u around inf

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

      1. lower-/.f64 94.0

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

   7. Applied rewrites 94.0%

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

   if -2.00000000000000015e191 < u

   1. Initial program 70.5%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. 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 98.3

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

      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 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 97.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 97.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.6

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

   6. Applied rewrites 97.6%

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

   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 97.6

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

   9. Applied rewrites 97.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+191}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2 + \frac{u}{t1}, u, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma 2.0 u t1))))
   (if (<= t1 -1.08e+86)
     t_1
     (if (<= t1 1.1e-142) (* (/ (- 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 <= -1.08e+86) {
		tmp = t_1;
	} else if (t1 <= 1.1e-142) {
		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 <= -1.08e+86)
		tmp = t_1;
	elseif (t1 <= 1.1e-142)
		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, -1.08e+86], t$95$1, If[LessEqual[t1, 1.1e-142], N[(N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.07999999999999993e86 or 1.10000000000000008e-142 < t1

   1. Initial program 63.0%

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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 94.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 94.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 94.9

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

   6. Applied rewrites 94.9%

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

   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. lower-fma.f64 80.8

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

   9. Applied rewrites 80.8%

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

   if -1.07999999999999993e86 < t1 < 1.10000000000000008e-142

   1. Initial program 77.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 67.0

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

   5. Applied rewrites 67.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg N/A

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

      8. neg-mul-1 N/A

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

      9. times-frac N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. neg-mul-1 N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.08 \cdot 10^{+86}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{if}\;t1 \leq -27000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (- v) (fma 2.0 u t1))))
   (if (<= t1 -27000000000000.0)
     t_1
     (if (<= t1 1.1e-142) (* (/ v (* (- u) u)) t1) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = -v / fma(2.0, u, t1);
	double tmp;
	if (t1 <= -27000000000000.0) {
		tmp = t_1;
	} else if (t1 <= 1.1e-142) {
		tmp = (v / (-u * u)) * t1;
	} 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 <= -27000000000000.0)
		tmp = t_1;
	elseif (t1 <= 1.1e-142)
		tmp = Float64(Float64(v / Float64(Float64(-u) * u)) * t1);
	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, -27000000000000.0], t$95$1, If[LessEqual[t1, 1.1e-142], N[(N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.7e13 or 1.10000000000000008e-142 < t1

   1. Initial program 63.9%

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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 94.5

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 94.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 94.5

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

   6. Applied rewrites 94.5%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 78.4

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

   9. Applied rewrites 78.4%

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

   if -2.7e13 < t1 < 1.10000000000000008e-142

   1. Initial program 77.7%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 82.5

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 71.9%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -27000000000000:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{elif}\;t1 \leq 1.1 \cdot 10^{-142}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.3%

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

   1. lift-/.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 98.5

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

   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 98.5

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

4. Applied rewrites 98.5%

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

Alternative 11: 61.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 4.9 \cdot 10^{+234}:\\ \;\;\;\;\frac{-v}{\mathsf{fma}\left(2, u, t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 4.9e+234) (/ (- v) (fma 2.0 u t1)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 4.9e+234) {
		tmp = -v / fma(2.0, u, t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 4.9e+234)
		tmp = Float64(Float64(-v) / fma(2.0, u, t1));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 4.9e+234], N[((-v) / N[(2.0 * u + t1), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 4.89999999999999989e234

   1. Initial program 70.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 98.8

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

      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 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 95.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 95.1

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

   6. Applied rewrites 95.1%

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

   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. lower-fma.f64 61.9

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

   9. Applied rewrites 61.9%

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

   if 4.89999999999999989e234 < v

   1. Initial program 50.4%

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

   3. Taylor expanded in u around 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 45.6

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

   5. Applied rewrites 45.6%

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

Alternative 12: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 5.8 \cdot 10^{+225}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 5.8e+225) (/ (- v) (+ u t1)) (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 5.8e+225) {
		tmp = -v / (u + t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (v <= 5.8d+225) then
        tmp = -v / (u + t1)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 5.8e+225) {
		tmp = -v / (u + t1);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 5.8e+225:
		tmp = -v / (u + t1)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 5.8e+225)
		tmp = Float64(Float64(-v) / Float64(u + t1));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 5.8e+225)
		tmp = -v / (u + t1);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 5.8e+225], N[((-v) / N[(u + t1), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 5.8000000000000003e225

   1. Initial program 70.7%

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

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

      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 98.6

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

      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 98.6

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

      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 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot v}}{u + t1} \]
   6. 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 61.4

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

   7. Applied rewrites 61.4%

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

   if 5.8000000000000003e225 < v

   1. Initial program 52.9%

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

   3. Taylor expanded in u around 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 43.7

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

   5. Applied rewrites 43.7%

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

Alternative 13: 53.8% 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 69.3%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 55.0

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

5. Applied rewrites 55.0%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))