Rosa's DopplerBench

Percentage Accurate: 73.1% → 98.2%

Time: 8.1s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

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

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)) * Float64(-v)) / 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 69.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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 96.7

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

   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 96.7

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

   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 96.7

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

4. Applied rewrites 96.7%

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

5. Final simplification 96.7%

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

Alternative 2: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{if}\;u \leq -4.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq -5.2 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u + t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ u t1) (+ u t1)))))
   (if (<= u -4.1e+154)
     (* (/ (- v) u) (/ t1 u))
     (if (<= u -5.2e-62)
       t_1
       (if (<= u 5.2e-117)
         (/ v (fma -2.0 u (- t1)))
         (if (<= u 5e+141) t_1 (/ (* (/ t1 u) (- v)) (+ u t1))))))))

C

double code(double u, double v, double t1) {
	double t_1 = (-t1 * v) / ((u + t1) * (u + t1));
	double tmp;
	if (u <= -4.1e+154) {
		tmp = (-v / u) * (t1 / u);
	} else if (u <= -5.2e-62) {
		tmp = t_1;
	} else if (u <= 5.2e-117) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (u <= 5e+141) {
		tmp = t_1;
	} else {
		tmp = ((t1 / u) * -v) / (u + t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(Float64(Float64(-t1) * v) / Float64(Float64(u + t1) * Float64(u + t1)))
	tmp = 0.0
	if (u <= -4.1e+154)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	elseif (u <= -5.2e-62)
		tmp = t_1;
	elseif (u <= 5.2e-117)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (u <= 5e+141)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(t1 / u) * Float64(-v)) / Float64(u + t1));
	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]}, If[LessEqual[u, -4.1e+154], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, -5.2e-62], t$95$1, If[LessEqual[u, 5.2e-117], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 5e+141], t$95$1, N[(N[(N[(t1 / u), $MachinePrecision] * (-v)), $MachinePrecision] / N[(u + t1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if u < -4.1e154

   1. Initial program 58.2%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 93.2

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

   5. Applied rewrites 93.2%

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

   if -4.1e154 < u < -5.1999999999999999e-62 or 5.19999999999999966e-117 < u < 5.00000000000000025e141

   1. Initial program 86.2%

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

   if -5.1999999999999999e-62 < u < 5.19999999999999966e-117

   1. Initial program 55.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 83.7

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

   7. Applied rewrites 83.7%

      \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 83.7

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

   9. Applied rewrites 83.7%

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

   if 5.00000000000000025e141 < u

   1. Initial program 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-/l* N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 98.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in u around inf

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

      1. lower-/.f64 84.9

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

   7. Applied rewrites 84.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq -5.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot \left(-v\right)}{u + t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ t_2 := \frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{if}\;u \leq -4.1 \cdot 10^{+154}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;u \leq -5.2 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;u \leq 5.5 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* (- t1) v) (* (+ u t1) (+ u t1))))
        (t_2 (* (/ (- v) u) (/ t1 u))))
   (if (<= u -4.1e+154)
     t_2
     (if (<= u -5.2e-62)
       t_1
       (if (<= u 5.2e-117)
         (/ v (fma -2.0 u (- t1)))
         (if (<= u 5.5e+141) 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 / u) * (t1 / u);
	double tmp;
	if (u <= -4.1e+154) {
		tmp = t_2;
	} else if (u <= -5.2e-62) {
		tmp = t_1;
	} else if (u <= 5.2e-117) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (u <= 5.5e+141) {
		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(Float64(-v) / u) * Float64(t1 / u))
	tmp = 0.0
	if (u <= -4.1e+154)
		tmp = t_2;
	elseif (u <= -5.2e-62)
		tmp = t_1;
	elseif (u <= 5.2e-117)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (u <= 5.5e+141)
		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[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -4.1e+154], t$95$2, If[LessEqual[u, -5.2e-62], t$95$1, If[LessEqual[u, 5.2e-117], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 5.5e+141], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if u < -4.1e154 or 5.49999999999999967e141 < u

   1. Initial program 61.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 89.0

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

   5. Applied rewrites 89.0%

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

   if -4.1e154 < u < -5.1999999999999999e-62 or 5.19999999999999966e-117 < u < 5.49999999999999967e141

   1. Initial program 86.2%

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

   if -5.1999999999999999e-62 < u < 5.19999999999999966e-117

   1. Initial program 55.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 83.7

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

   7. Applied rewrites 83.7%

      \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 83.7

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

   9. Applied rewrites 83.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq -5.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;u \leq 5.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(u + t1\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+228}:\\ \;\;\;\;\frac{-t1}{\frac{u}{v} \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u} \cdot v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= u -3.4e+24)
   (* (/ (- v) u) (/ t1 u))
   (if (<= u 1.35e+17)
     (/ v (fma -2.0 u (- t1)))
     (if (<= u 4.8e+228) (/ (- t1) (* (/ u v) u)) (/ (* (/ (- t1) u) v) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (u <= -3.4e+24) {
		tmp = (-v / u) * (t1 / u);
	} else if (u <= 1.35e+17) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (u <= 4.8e+228) {
		tmp = -t1 / ((u / v) * u);
	} else {
		tmp = ((-t1 / u) * v) / u;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (u <= -3.4e+24)
		tmp = Float64(Float64(Float64(-v) / u) * Float64(t1 / u));
	elseif (u <= 1.35e+17)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (u <= 4.8e+228)
		tmp = Float64(Float64(-t1) / Float64(Float64(u / v) * u));
	else
		tmp = Float64(Float64(Float64(Float64(-t1) / u) * v) / u);
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[u, -3.4e+24], N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 1.35e+17], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.8e+228], N[((-t1) / N[(N[(u / v), $MachinePrecision] * u), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-t1) / u), $MachinePrecision] * v), $MachinePrecision] / u), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if u < -3.4000000000000001e24

   1. Initial program 69.9%

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

   3. Taylor expanded in u around inf

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

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

   if -3.4000000000000001e24 < u < 1.35e17

   1. Initial program 67.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 71.4

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

   7. Applied rewrites 71.4%

      \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 71.4

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

   9. Applied rewrites 71.4%

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

   if 1.35e17 < u < 4.79999999999999977e228

   1. Initial program 79.8%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 82.4%

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

   if 4.79999999999999977e228 < u

   1. Initial program 52.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 95.8

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

   5. Applied rewrites 95.8%

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

   7. Applied rewrites 95.8%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+228}:\\ \;\;\;\;\frac{-t1}{\frac{u}{v} \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{u} \cdot v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{if}\;u \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+228}:\\ \;\;\;\;\frac{-t1}{\frac{u}{v} \cdot u}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- v) u) (/ t1 u))))
   (if (<= u -3.4e+24)
     t_1
     (if (<= u 1.35e+17)
       (/ v (fma -2.0 u (- t1)))
       (if (<= u 4.8e+228) (/ (- t1) (* (/ u v) u)) t_1)))))

C

double code(double u, double v, double t1) {
	double t_1 = (-v / u) * (t1 / u);
	double tmp;
	if (u <= -3.4e+24) {
		tmp = t_1;
	} else if (u <= 1.35e+17) {
		tmp = v / fma(-2.0, u, -t1);
	} else if (u <= 4.8e+228) {
		tmp = -t1 / ((u / v) * u);
	} 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 <= -3.4e+24)
		tmp = t_1;
	elseif (u <= 1.35e+17)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	elseif (u <= 4.8e+228)
		tmp = Float64(Float64(-t1) / Float64(Float64(u / v) * u));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.4e+24], t$95$1, If[LessEqual[u, 1.35e+17], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[u, 4.8e+228], N[((-t1) / N[(N[(u / v), $MachinePrecision] * u), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if u < -3.4000000000000001e24 or 4.79999999999999977e228 < u

   1. Initial program 66.9%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

   if -3.4000000000000001e24 < u < 1.35e17

   1. Initial program 67.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 71.4

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

   7. Applied rewrites 71.4%

      \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 71.4

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

   9. Applied rewrites 71.4%

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

   if 1.35e17 < u < 4.79999999999999977e228

   1. Initial program 79.8%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 82.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+17}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;u \leq 4.8 \cdot 10^{+228}:\\ \;\;\;\;\frac{-t1}{\frac{u}{v} \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{+169}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 1.48 \cdot 10^{+85}:\\ \;\;\;\;\frac{-t1}{\frac{u + t1}{v} \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -2.4e+169)
   (/ (- v) t1)
   (if (<= t1 1.48e+85)
     (/ (- t1) (* (/ (+ u t1) v) (+ u t1)))
     (/ v (fma -2.0 u (- t1))))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (t1 <= -2.4e+169) {
		tmp = -v / t1;
	} else if (t1 <= 1.48e+85) {
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	} else {
		tmp = v / fma(-2.0, u, -t1);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if (t1 <= -2.4e+169)
		tmp = Float64(Float64(-v) / t1);
	elseif (t1 <= 1.48e+85)
		tmp = Float64(Float64(-t1) / Float64(Float64(Float64(u + t1) / v) * Float64(u + t1)));
	else
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[t1, -2.4e+169], N[((-v) / t1), $MachinePrecision], If[LessEqual[t1, 1.48e+85], N[((-t1) / N[(N[(N[(u + t1), $MachinePrecision] / v), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.3999999999999998e169

   1. Initial program 43.3%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if -2.3999999999999998e169 < t1 < 1.48e85

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. frac-2neg N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

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

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

      13. neg-mul-1 N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-neg.f64 88.6

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 88.6

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

   4. Applied rewrites 88.6%

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

   if 1.48e85 < t1

   1. Initial program 47.8%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 90.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 90.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 76.6

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

   7. Applied rewrites 76.6%

      \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 76.6

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

   9. Applied rewrites 76.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{+169}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 1.48 \cdot 10^{+85}:\\ \;\;\;\;\frac{-t1}{\frac{u + t1}{v} \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{if}\;u \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (* (/ (- v) u) (/ t1 u))))
   (if (<= u -3.4e+24) t_1 (if (<= u 2.8e-55) (/ v (fma -2.0 u (- t1))) t_1))))

C

double code(double u, double v, double t1) {
	double t_1 = (-v / u) * (t1 / u);
	double tmp;
	if (u <= -3.4e+24) {
		tmp = t_1;
	} else if (u <= 2.8e-55) {
		tmp = v / fma(-2.0, u, -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 <= -3.4e+24)
		tmp = t_1;
	elseif (u <= 2.8e-55)
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := Block[{t$95$1 = N[(N[((-v) / u), $MachinePrecision] * N[(t1 / u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[u, -3.4e+24], t$95$1, If[LessEqual[u, 2.8e-55], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if u < -3.4000000000000001e24 or 2.79999999999999984e-55 < u

   1. Initial program 73.3%

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

   3. Taylor expanded in u around 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 77.0

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

   5. Applied rewrites 77.0%

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

   if -3.4000000000000001e24 < u < 2.79999999999999984e-55

   1. Initial program 64.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 75.7

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

   7. Applied rewrites 75.7%

      \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 75.7

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

   9. Applied rewrites 75.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;u \leq 2.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -80000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -80000.0)
     t_1
     (if (<= t1 6.2e-36) (* (/ (- 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 <= -80000.0) {
		tmp = t_1;
	} else if (t1 <= 6.2e-36) {
		tmp = (-t1 / (u * u)) * v;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -80000.0)
		tmp = t_1;
	elseif (t1 <= 6.2e-36)
		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, -80000.0], t$95$1, If[LessEqual[t1, 6.2e-36], N[(N[((-t1) / N[(u * u), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -8e4 or 6.1999999999999997e-36 < t1

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 90.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 90.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 75.9

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

   7. Applied rewrites 75.9%

      \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 75.9

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

   9. Applied rewrites 75.9%

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

   if -8e4 < t1 < 6.1999999999999997e-36

   1. Initial program 79.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 79.5

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 79.5

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

   4. Applied rewrites 79.5%

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

   5. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.5

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

   7. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 68.1

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

   9. Applied rewrites 68.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -80000:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-t1}{u \cdot u} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{if}\;t1 \leq -6.3 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (fma -2.0 u (- t1)))))
   (if (<= t1 -6.3e-74)
     t_1
     (if (<= t1 6.2e-36) (* (/ 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 <= -6.3e-74) {
		tmp = t_1;
	} else if (t1 <= 6.2e-36) {
		tmp = (v / (-u * u)) * t1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	t_1 = Float64(v / fma(-2.0, u, Float64(-t1)))
	tmp = 0.0
	if (t1 <= -6.3e-74)
		tmp = t_1;
	elseif (t1 <= 6.2e-36)
		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, -6.3e-74], t$95$1, If[LessEqual[t1, 6.2e-36], N[(N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.30000000000000003e-74 or 6.1999999999999997e-36 < t1

   1. Initial program 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 92.1

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 92.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 73.3

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

   7. Applied rewrites 73.3%

      \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-lft-identity 73.3

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

   9. Applied rewrites 73.3%

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

   if -6.30000000000000003e-74 < t1 < 6.1999999999999997e-36

   1. Initial program 75.8%

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

   3. Taylor expanded in u around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 68.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -6.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{elif}\;t1 \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{v}{\left(-u\right) \cdot u} \cdot t1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. frac-2neg N/A

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

   11. lift-neg.f64 N/A

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

   12. remove-double-neg N/A

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

   13. lower-/.f64 N/A

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

   14. lower-neg.f64 91.0

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 91.0

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 91.0

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

4. Applied rewrites 91.0%

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

5. Taylor expanded in u around 0

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

   1. lower-fma.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 55.4

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

7. Applied rewrites 55.4%

   \[\leadsto \frac{1 \cdot v}{\color{blue}{\mathsf{fma}\left(-2, u, -t1\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-lft-identity 55.4

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

9. Applied rewrites 55.4%

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

Alternative 11: 61.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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 96.7

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

   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 96.7

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

   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 96.7

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

4. Applied rewrites 96.7%

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

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 54.6

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

7. Applied rewrites 54.6%

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

Alternative 12: 54.3% 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 69.9%

   \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
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 47.2

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

5. Applied rewrites 47.2%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))