Rosa's DopplerBench

Percentage Accurate: 72.3% → 98.1%

Time: 6.9s

Alternatives: 13

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-frac-neg2 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 99.1

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.1

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

4. Applied rewrites 99.1%

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

Alternative 2: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{+65} \lor \neg \left(t1 \leq 2.7 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3.8e+65) (not (<= t1 2.7e+55)))
   (/ v (fma -2.0 u (- t1)))
   (/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -3.8e+65) || !(t1 <= 2.7e+55)) {
		tmp = v / fma(-2.0, u, -t1);
	} else {
		tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -3.8e+65) || !(t1 <= 2.7e+55))
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -3.8e+65], N[Not[LessEqual[t1, 2.7e+55]], $MachinePrecision]], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.80000000000000011e65 or 2.69999999999999977e55 < t1

   1. Initial program 53.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 96.0

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

      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 96.0

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

      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 96.0

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

   4. Applied rewrites 96.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 96.0

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 91.0

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

   9. Applied rewrites 91.0%

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

   if -3.80000000000000011e65 < t1 < 2.69999999999999977e55

   1. Initial program 85.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.8 \cdot 10^{+65} \lor \neg \left(t1 \leq 2.7 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+56} \lor \neg \left(t1 \leq 6000000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7e+56) (not (<= t1 6000000000.0)))
   (/ v (fma -2.0 u (- t1)))
   (/ (* (/ t1 u) v) (- u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7e+56) || !(t1 <= 6000000000.0)) {
		tmp = v / fma(-2.0, u, -t1);
	} else {
		tmp = ((t1 / u) * v) / -u;
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7e+56) || !(t1 <= 6000000000.0))
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	else
		tmp = Float64(Float64(Float64(t1 / u) * v) / Float64(-u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7e+56], N[Not[LessEqual[t1, 6000000000.0]], $MachinePrecision]], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t1 / u), $MachinePrecision] * v), $MachinePrecision] / (-u)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.99999999999999999e56 or 6e9 < t1

   1. Initial program 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 94.9

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

      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 94.9

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

      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 94.9

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

   4. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 94.9

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

   6. Applied rewrites 94.9%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 88.6

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

   9. Applied rewrites 88.6%

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

   if -6.99999999999999999e56 < t1 < 6e9

   1. Initial program 85.7%

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

   3. Taylor expanded in 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%

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

   7. Applied rewrites 78.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+56} \lor \neg \left(t1 \leq 6000000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u} \cdot v}{-u}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+56} \lor \neg \left(t1 \leq 6000000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -7e+56) (not (<= t1 6000000000.0)))
   (/ v (fma -2.0 u (- t1)))
   (* (/ t1 u) (/ (- v) u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -7e+56) || !(t1 <= 6000000000.0)) {
		tmp = v / fma(-2.0, u, -t1);
	} else {
		tmp = (t1 / u) * (-v / u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -7e+56) || !(t1 <= 6000000000.0))
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	else
		tmp = Float64(Float64(t1 / u) * Float64(Float64(-v) / u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -7e+56], N[Not[LessEqual[t1, 6000000000.0]], $MachinePrecision]], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], N[(N[(t1 / u), $MachinePrecision] * N[((-v) / u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -6.99999999999999999e56 or 6e9 < t1

   1. Initial program 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 94.9

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

      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 94.9

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

      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 94.9

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

   4. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 94.9

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

   6. Applied rewrites 94.9%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 88.6

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

   9. Applied rewrites 88.6%

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

   if -6.99999999999999999e56 < t1 < 6e9

   1. Initial program 85.7%

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

   3. Taylor expanded in 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%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{+56} \lor \neg \left(t1 \leq 6000000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 10^{+250}:\\ \;\;\;\;\frac{v}{\left(-1 - \frac{u}{t1}\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u + t1}{v} \cdot \left(u + t1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (<= v 1e+250)
   (/ v (* (- -1.0 (/ u t1)) (+ u t1)))
   (/ (- t1) (* (/ (+ u t1) v) (+ u t1)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1e+250) {
		tmp = v / ((-1.0 - (u / t1)) * (u + t1));
	} else {
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if (v <= 1d+250) then
        tmp = v / (((-1.0d0) - (u / t1)) * (u + t1))
    else
        tmp = -t1 / (((u + t1) / v) * (u + t1))
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if (v <= 1e+250) {
		tmp = v / ((-1.0 - (u / t1)) * (u + t1));
	} else {
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if v <= 1e+250:
		tmp = v / ((-1.0 - (u / t1)) * (u + t1))
	else:
		tmp = -t1 / (((u + t1) / v) * (u + t1))
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if (v <= 1e+250)
		tmp = Float64(v / Float64(Float64(-1.0 - Float64(u / t1)) * Float64(u + t1)));
	else
		tmp = Float64(Float64(-t1) / Float64(Float64(Float64(u + t1) / v) * Float64(u + t1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if (v <= 1e+250)
		tmp = v / ((-1.0 - (u / t1)) * (u + t1));
	else
		tmp = -t1 / (((u + t1) / v) * (u + t1));
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[LessEqual[v, 1e+250], N[(v / N[(N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-t1) / N[(N[(N[(u + t1), $MachinePrecision] / v), $MachinePrecision] * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 9.9999999999999992e249

   1. Initial program 75.2%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 94.8

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

      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 94.8

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

      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 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 94.8

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

   6. Applied rewrites 94.8%

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

   7. Taylor expanded in u around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 94.8

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

   9. Applied rewrites 94.8%

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

   if 9.9999999999999992e249 < v

   1. Initial program 48.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. 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 90.7

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

      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 90.7

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

   4. Applied rewrites 90.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 10^{+250}:\\ \;\;\;\;\frac{v}{\left(-1 - \frac{u}{t1}\right) \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u + t1}{v} \cdot \left(u + t1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-67} \lor \neg \left(t1 \leq 5600000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.15e-67) (not (<= t1 5600000000.0)))
   (/ v (fma -2.0 u (- t1)))
   (/ (* (- t1) v) (* u u))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.15e-67) || !(t1 <= 5600000000.0)) {
		tmp = v / fma(-2.0, u, -t1);
	} else {
		tmp = (-t1 * v) / (u * u);
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.15e-67) || !(t1 <= 5600000000.0))
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	else
		tmp = Float64(Float64(Float64(-t1) * v) / Float64(u * u));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.15e-67], N[Not[LessEqual[t1, 5600000000.0]], $MachinePrecision]], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], N[(N[((-t1) * v), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.15e-67 or 5.6e9 < t1

   1. Initial program 60.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 93.6

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

      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 93.6

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

      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 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 93.6

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 82.4

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

   9. Applied rewrites 82.4%

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

   if -1.15e-67 < t1 < 5.6e9

   1. Initial program 86.9%

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

   3. Taylor expanded in u around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-67} \lor \neg \left(t1 \leq 5600000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{u \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-67} \lor \neg \left(t1 \leq 5600000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -1.15e-67) (not (<= t1 5600000000.0)))
   (/ v (fma -2.0 u (- t1)))
   (* t1 (/ v (* (- u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -1.15e-67) || !(t1 <= 5600000000.0)) {
		tmp = v / fma(-2.0, u, -t1);
	} else {
		tmp = t1 * (v / (-u * u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -1.15e-67) || !(t1 <= 5600000000.0))
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	else
		tmp = Float64(t1 * Float64(v / Float64(Float64(-u) * u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -1.15e-67], N[Not[LessEqual[t1, 5600000000.0]], $MachinePrecision]], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], N[(t1 * N[(v / N[((-u) * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -1.15e-67 or 5.6e9 < t1

   1. Initial program 60.6%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 93.6

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

      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 93.6

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

      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 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 93.6

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 82.4

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

   9. Applied rewrites 82.4%

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

   if -1.15e-67 < t1 < 5.6e9

   1. Initial program 86.9%

      \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 78.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -1.15 \cdot 10^{-67} \lor \neg \left(t1 \leq 5600000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{\left(-u\right) \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{-79} \lor \neg \left(t1 \leq 5600000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{\left(-u\right) \cdot u}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.7e-79) (not (<= t1 5600000000.0)))
   (/ v (fma -2.0 u (- t1)))
   (* v (/ t1 (* (- u) u)))))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((t1 <= -2.7e-79) || !(t1 <= 5600000000.0)) {
		tmp = v / fma(-2.0, u, -t1);
	} else {
		tmp = v * (t1 / (-u * u));
	}
	return tmp;
}

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((t1 <= -2.7e-79) || !(t1 <= 5600000000.0))
		tmp = Float64(v / fma(-2.0, u, Float64(-t1)));
	else
		tmp = Float64(v * Float64(t1 / Float64(Float64(-u) * u)));
	end
	return tmp
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[t1, -2.7e-79], N[Not[LessEqual[t1, 5600000000.0]], $MachinePrecision]], N[(v / N[(-2.0 * u + (-t1)), $MachinePrecision]), $MachinePrecision], N[(v * N[(t1 / N[((-u) * u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.7000000000000002e-79 or 5.6e9 < t1

   1. Initial program 61.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 93.1

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

      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 93.1

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

      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 93.1

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

   4. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 93.1

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

   6. Applied rewrites 93.1%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 81.5

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

   9. Applied rewrites 81.5%

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

   if -2.7000000000000002e-79 < t1 < 5.6e9

   1. Initial program 86.5%

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

   3. Taylor expanded in u around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 81.8

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

   5. Applied rewrites 81.8%

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 77.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{-79} \lor \neg \left(t1 \leq 5600000000\right):\\ \;\;\;\;\frac{v}{\mathsf{fma}\left(-2, u, -t1\right)}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{\left(-u\right) \cdot u}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. associate-/l* N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 99.1

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

   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 99.1

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

   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 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 10: 94.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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 93.3

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

   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 93.3

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

   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 93.3

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

4. Applied rewrites 93.3%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 93.3

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

6. Applied rewrites 93.3%

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

7. Taylor expanded in u around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 93.3

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

9. Applied rewrites 93.3%

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

Alternative 11: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+127} \lor \neg \left(u \leq 2.5 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{v}{-2 \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \end{array} \]

FPCore

(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -2.7e+127) (not (<= u 2.5e+223)))
   (/ v (* -2.0 u))
   (/ (- v) t1)))

C

double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e+127) || !(u <= 2.5e+223)) {
		tmp = v / (-2.0 * u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Fortran

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    real(8) :: tmp
    if ((u <= (-2.7d+127)) .or. (.not. (u <= 2.5d+223))) then
        tmp = v / ((-2.0d0) * u)
    else
        tmp = -v / t1
    end if
    code = tmp
end function

Java

public static double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -2.7e+127) || !(u <= 2.5e+223)) {
		tmp = v / (-2.0 * u);
	} else {
		tmp = -v / t1;
	}
	return tmp;
}

Python

def code(u, v, t1):
	tmp = 0
	if (u <= -2.7e+127) or not (u <= 2.5e+223):
		tmp = v / (-2.0 * u)
	else:
		tmp = -v / t1
	return tmp

Julia

function code(u, v, t1)
	tmp = 0.0
	if ((u <= -2.7e+127) || !(u <= 2.5e+223))
		tmp = Float64(v / Float64(-2.0 * u));
	else
		tmp = Float64(Float64(-v) / t1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, v, t1)
	tmp = 0.0;
	if ((u <= -2.7e+127) || ~((u <= 2.5e+223)))
		tmp = v / (-2.0 * u);
	else
		tmp = -v / t1;
	end
	tmp_2 = tmp;
end

Wolfram

code[u_, v_, t1_] := If[Or[LessEqual[u, -2.7e+127], N[Not[LessEqual[u, 2.5e+223]], $MachinePrecision]], N[(v / N[(-2.0 * u), $MachinePrecision]), $MachinePrecision], N[((-v) / t1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if u < -2.7000000000000002e127 or 2.49999999999999992e223 < u

   1. Initial program 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 85.8

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

      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 85.8

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

      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 85.8

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

   4. Applied rewrites 85.8%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 85.8

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

   6. Applied rewrites 85.8%

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

   7. Taylor expanded in u around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 39.8

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

   9. Applied rewrites 39.8%

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

   10. Taylor expanded in u around inf

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

   12. Applied rewrites 36.0%

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

   if -2.7000000000000002e127 < u < 2.49999999999999992e223

   1. Initial program 71.0%

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

   3. Taylor expanded in u around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.2

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

   5. Applied rewrites 64.2%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+127} \lor \neg \left(u \leq 2.5 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{v}{-2 \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.5% 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 73.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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 93.3

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

   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 93.3

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

   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 93.3

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

4. Applied rewrites 93.3%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 93.3

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

6. Applied rewrites 93.3%

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

7. Taylor expanded in u around 0

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

   1. lower-fma.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 60.4

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

9. Applied rewrites 60.4%

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

Alternative 13: 53.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 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 53.5

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

5. Applied rewrites 53.5%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))